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Robust D-stability via positivity
Automatica 35 (1999) 1477}1484
Technical Communique
Robust D-stability via positivity夽 D.D. S[ iljak*, D.M. StipanovicH Department of Electrical Engineering, Santa Clara University, Santa Clara, CA 95053-0569, USA Received 19 November 1998; received in "nal form 8 February 1999
Abstract The main objective of this paper is to convert the general problem of robust D-stability of a complex polynomial to positivity in the real domain of the corresponding magnitude function. In particular, the obtained criterion for Hurwitz stability is applied to polynomials with interval parameters and polynomic uncertainty structures. The robust stability is veri"ed by testing positivity of a real polynomial using the Bernstein subdivision algorithm. A new feature in this context is the stopping criterion, which is applied whenever the algorithm is inconclusive after a large number of iterations, but we can show that at least one zero of the polynomial is closer to the imaginary axis than a prescribed limit. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Robust stability; Interval uncertainty; Polynomic uncertainty structure; Bernstein subdivision algorithm
1. Introduction Converting D-stability of an uncertain polynomial to positivity of a multivariable polynomial opens up a possibility to apply the powerful Bernstein expansion algorithms to robust stability problems. This idea has been introduced by Vicino and Milanese (1990) and has achieved recently a high level of sophistication and e$ciency in the work of Garlo! and his co}workers (Garlo!, 1993; Zettler and Garlo!, 1996; Garlo! et al., 1997). The approach to robust stability is based upon a large number of results on Bernstein polynomials; in particular, the subdivision algorithm, which was introduced by Lane and Riesenfeld (1981) and considerably extended for stability applications in Garlo! (1986), Malan et al. (1992, 1997), Fiorio et al. (1993) and Garlo! et al. (1997). Among many comparisons that one can make with respect to numerous results available in the context of robust stability (e.g., Milanese et al., 1989; Zeheb, 1990; Vicino and Milanese, 1990; Barmish and Tempo, 1990;
***** * Corresponding author. Tel. #1-408-554-4488; e-mail: dsiljak@ scu.edu. 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato.
Tesi and Vicino, 1991; Ackermann, 1993; Barmish, 1994; Bhattacharyya et al., 1995; Polyak and Kogan, 1995), the ability of the Bernstein expansion algorithms to handle with relative ease and accuracy the polynomials with polynomic uncertainty structures has a particular signi"cance. A straightforward way to determine stability of a real polynomial is to establish positivity of the next to the last Hurwitz determinant, which is a multivariable polynomial in uncertain parameters (Vicino and Milanese, 1990; Garlo!, 1993; Zettler and Garlo!, 1996; Garlo! et al., 1997). If the uncertain polynomial is complex, stability can be shown by simultaneous analysis of the zeros of its real and imaginary parts (Zettler and Garlo!, 1996). Stability of an uncertain matrix can be analyzed in the same way by considering its characteristic polynomial (Vicino and Milanese, 1990), because we are not concerned with the fact that the resulting uncertainty structure is polynomic. Our principal contribution to the Bernstein algorithms for robust stability analysis is the use of the magnitude function of a given uncertain polynomial to convert the general problem of robust D-stability to positivity. This function, which was used successfully in the root}"nding algorithms (Stolan, 1995), o!ers new possibilities in the context of robust D-stability. First, D-stability of complex uncertain polynomials with respect to general stability regions and having coe$cients as arbitrary continuous functions of uncertain parameters, can be considered
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 4 2 - 4
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using only real arithmetic. In case of Hurwitz stability and polynomic uncertainty structure, the magnitude function results in only the doubling of the powers of uncertain parameters regardless of the degree of the polynomial. This is not the case with Hurwitz determinants, which are, in addition, more di$cult to compute than the magnitude function. Furthermore, the positivity test of the function is essentially the same whether the polynomial is real or complex. Second, the magnitude function provides a suitable stopping criterion in cases when the Bernstein algorithm is inconclusive after a large number of subdivisions, but the computations show that at least one zero of the polynomial is closer to the imaginary axis than the acceptable limit. Finally, positivity analysis of the magnitude function can be facilitated by a number of results involving minorizing polynomials, the modi"ed Routh array, and convex value sets, which are available in the context of positivity of uncertain polynomials (S[ iljak, 1989; S[ iljak and S[ iljak, 1997). The organization of the paper is as follows. In the next section, we shall reformulate robust D-stability of a complex polynomial as positivity of a real polynomial using the magnitude function. In Section 3, we specialize the stability region to the left half plane and analyze robust Hurwitz stability by verifying positivity of the real magnitude polynomial on the real line. By a simple minorization of the magnitude polynomial we can determine the intervals for positivity testing via Bernstein subdivision algorithm. The testing is discussed in Section 4, where the stopping criterion for the algorithm is provided. In Section 5, we extend the approach to robust Schur stability, and consider general D regions with boundaries made up of straight lines and circles.
2. D-stability via positivity Let us consider a complex polynomial L f (s, p)" a (p)sG, G G
(2.1)
where the coe$cients a (p) are complex continuous funcG tions of uncertain parameter vector p31l, and s3". We assume that p belongs to a bounded and pathwise connected set PL1l, and that f (s, p) has invariant degree, that is, deg f (s, p)"n for all p3P. We de"ne next the family of polynomials F"+ f ( ), p): p3P,
(2.2)
and specify an open simply connected region DL". De5nition 1. A family of polynomials F is D-stable if all zeros of f (s, p) lie in D for all p3P.
To establish conditions of robust D-stability via polynomial positivity we de"ne the function L L fK (s, p)"f (s, p) f (s, p)" a (p)aN (p)sGsN H, (2.3) G H G H where overbar denotes conjugation. We note immediately that fK (s, p)"" f (s, p)", which means that fK (s, p) is nonnegative for all s3". This fact is essential to our development. Functions fK (s, p) form a family F< "+ f K ( ), p): p3P,
(2.4)
which we consider on a Jordan arc J. De5nition 2. A family of functions F< is J-positive if fK (s, p) is positive on J for all p3P. We denote by *D the boundary of D and prove the following: Theorem 1. Suppose that there exists a p3P such that f (s, p) has all zeros in D. ¹hen, a family of polynomials F is D-stable if and only if the corresponding family F< is *D-positive. Proof. To prove the &&only if '' part of the theorem, we assume that F is D-stable, but F< is not *D-positive. Then, f K (s*, p*)"0 for some p*3P and s*3*D, which implies that f (s*, p*)"0. Since D is open, the last statement implies that F is not stable, which is a contradiction. To prove the &&if '' part of the theorem, we again proceed with contradiction. We assume that for some p3P, f (s, p) does not have all zeros within D, but F< is *Dpositive. Since P is pathwise connected, there exists a continuous function h : [0, 1]PP such that h(0)"p and h(1)"p. We recall that continuity of coe$cients a (p) implies that s (p), s (p), 2, s (p), which represent the G L zeros of f (s, p), are continuous functions, as well. Now, since f (s, p) has zeros outside D, there is a zero s * (p) for G some i*3+1, 2, 2, n, which is not in D. On the other hand, s * (p) is within D by assumption. Observe then G that for j3[0, 1], s * (h(j)) describes a continuously G varying zero which starts at s * (h(0)) within D and G ends up at s * (h(1)) outside D. Due to continuity, G there is a j*3[0, 1] such that s * (h(j*))3*D. Taking G p*"h(j*) we have that f (s * (p*), p*)"0 and, hence, G fK (s * (p*), p*)"0 for s * (p*)3*D, which is the contradicG G tion we want. ) A similar theorem was established in S[ iljak and S[ iljak (1997), which linked robust Hurwitz stability to positivity. In that theorem, the polynomial g' (s, p)"f (s, p) fM (!s, p)
(2.5)
D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
was used instead of function fK (s, p). The polynomial g' (s, p) is nonnegative on the imaginary axis in ", but not necessarily o! the axis. For this reason, it could not be used to establish robust D-stability. We also note that the proofs of both theorems are conceptually similar to the standard proof of the Zero Exclusion Theorem (Zadeh and Desoer, 1963).
3. Robust Hurwitz stability When D is the open left half of ", which we denote by " "+s3": Re s(0,, \ then *D is the imaginary axis
(3.1)
I"+s3": Re s"0,.
(3.2)
On I, the function f K ( ju, p)"" f ( ju, p)" is a real polynomial
L g(u, p)" a (p)( ju)G . (3.3) G G We now assume that a (p) are complex multivariable G polynomials in p, and rewrite g(u, p) as L g(u, p)" b (p)uG, (3.4) G G where u31, and b (p) are real multivariable polynomials G in p. We recall that a polynomial is said to be Hurwitz if all of its zeros are in " . From De"nition 1, we have \ an analogous de"nition for the Hurwitz property of a family F. We de"ne the polynomial family G"+g( ), p): p3P,
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The problem is to "nd the two bounding numerical polynomials de"ned as L g>(u)" b>uG4g(u, p), ∀u50, ∀p3P G G and
(3.7)
L (3.8) g\(u)" b\uG4g(u, p), ∀u40, ∀p3P G G and compute the interval [u, u ] outside of which they are both positive. One way to do it is to use a transformation (Sideris and Pen a, 1989) which expands the parameter space so that the new polynomial has multia$ne uncertainty structure. The minorizing polynomial is one of the vertex polynomials of the expanded uncertainty box in the new parameter space (S[ iljak and S[ iljak, 1997). We propose a less conservative approach to produce g>(u) and g\(u), which applies the Bernstein expansion. The idea is to minimize or maximize all b over the box G P by using the Bernstein subdivision algorithm (Malan et al., 1992). For u50, we minimize each b independentG ly and form g>(u) using the obtained minima b>. For G u40, we get g\(u) by maximizing b when i is odd and G minimizing b when i is even, which produces b\. G G If it so happens that both minorizing polynomials g>(u) and g\(u) are positive for all u, then the family G is positive and we are done. Positivity of the polynomials can be tested by the modi"ed Routh array (S[ iljak, 1989; S[ iljak and S[ iljak, 1997) or by a root "nding algorithm (e.g., Stolan, 1995). If the two polynomials g>(u) and g\(u) are not both positive for all u, they can be used to compute a positivity interval [ u, u ]. Simply, u is the largest nonnegative zero of g>(u), while u is the smallest nonpositive zero of g\(u).
(3.5)
and state the obvious. Corollary 1. Suppose that there exists a p3P such that f (s, p) is Hurwitz. ¹hen, a family F is Hurwitz if and only if the corresponding family G is 1-positive. Our test for robust stability is based on the numerical methods for verifying positivity of multivariable polynomials via Bernstein polynomials (Malan et al., 1992). This requires that all variables in the polynomial g(u, p) be restricted to intervals. While it is standard to restrict the parameter vector p to a box (3.6) P"+p31l: p 3[ p , pN ],, G G G the variable u must be allowed to vary over the entire real line 1. A way to restrict u to a "nite interval [u, u ] for all p3P, is to use the minorizing polynomials (S[ iljak and S[ iljak, 1997)
Example 1. To illustrate the role of minorizing polynomials, let us consider the polynomial from Barmish (1994): f (s, p)"s#(3p#pp #p p #3p #10)s #(4p#p#15)s#16p p #17 and choose the uncertainty box
(3.9)
P"+p31: p 3[!0.8,!0.6], p 3[!0.4, 1],. (3.10) Since the magnitude function g(u, p)"" f ( ju, p)" is a real even polynomial, by replacing u by u we get the polynomial to be tested for positivity as g(u, p)"u#(9p#6pp #pp#6pp #18p #2pp#6pp #60p#pp#26pp #p#20p p #60p !2p#70)u #(!36pp #16p!12pp!102p
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!4pp#30p!70pp #120p!154p p !102p #30p!115)u#36pp #204p p #289 (3.11) which has the same degree as the original polynomial f (s, p); in the real case, no doubling of degree takes place when robust stability of f (s, p) is replaced by positivity of g(u, p). Furthermore, since f (s, p) is real, we have g>(u)"g\(u) O g(u), and g(u) needs to be tested for positivity only for u50. Using the expanding transformation (Sideris and Pen a, 1989) we get the minorizing polynomial
was obtained by applying the expanding transformation (Sideris and Pen a, 1989) to the individual coe$cients of g(u, p). Since the polynomial f (s, p) is real, we need to be concerned only with u31 . The zeros of gJ (u) are 3.866, > !4.794 and 6.292, and the interval bounds are u"3.866 and u "6.292. Once the interval for u is speci"ed, we can treat u as just another parameter. Then, minimization of the polynomial g(u, p) over a box can proceed by using directly the multivariable version of the Bernstein algorithm (Malan et al., 1992).
gJ (u)"u!5.3635u!24.3690u#116.5840.
4. Bernstein algorithm: the stopping criterion
(3.12)
The modi"ed Routh array (S[ iljak, 1989) for gJ (u) is computed as !1 !5.3635 24.3690 116.5840 !3 !10.7270 24.3690 !1.7878 16.2460 116.5840 !37.9884 171.2636 8.1860 116.5840 721.2897 116.5840
(3.13)
Y"[ u, u ];P.
Since the array has one sign variation in the "rst column, the polynomial gJ (u) has two positive zeros and, therefore, gJ (u) is not 1 -positive. Stability of the original > family F, which involves the polynomial f (s, p) of Eq. (3.9) and the box P of Eq. (3.10), remains inconclusive. Applying the Bernstein subdivision algorithm to minimize each individual coe$cient of g(u, p) in Eq. (3.11), we obtain g(u)"u!4.2027u!6.4982u#148.84.
(3.14)
The array is now !1 !4.2022 6.4982 148.8400 !3 !8.4044 6.4982 !1.4 4.3321 148.8400 !17.6875 !312.4446 20.3985 148.8400 !183.3857 148.8400
Let us "rst outline the algorithm (Malan et al., 1992) for bounding a real multivariable polynomial using the Bernstein polynomials. We shall recall only the elements of the algorithm that are necessary in our application. By assuming that u3[ u, u ], where the bounds u and u are computed using the minorizing polynomials, we can consider g(u, p) as a real multivariable polynomial z(y) in y"(u, p)31l> restricted to a box
(3.15)
which has three variations in the "rst column. The polynomial g(u) is 1 -positive and the family G is 1 > > positive as well. Since f (s, p) is real, 1 -positivity is all > that is required. Finally, for p"(!0.8, 1) the numerical polynomial f (s, p) is Hurwitz and the family F is Hurwitz as well. To illustrate the computation of the interval [ u, u ] for u, when the minorizing polynomials are not positive on the entire real line, let us assume that the polynomial gJ (u)
(4.1)
Since any box Y of dimension l#1 can be mapped a$nely onto the unit box U"[0, 1]l>, using the same mapping z(y) is mapped to the multivariable polynomial h(x) of the same degree N, given by l , > c 2 l xGH , h(x)" (4.2) > GG G H G G 2 Gl> H where x3U. The objective of the algorithm is to compute
(4.3) m"min h(x). VZ U For simplicity, let us use r to denote the l#1-tuple of nonnegative integers (i , i , 2, il ) with the range > K"+(i , i , 2, il ): i , i , 2, il "0, 1, 2, k,. Then, > > Bernstein polynomials of a degree k5N are de"ned as
l > k BI(x)" xGH (1!x )I\GH, r3K. (4.4) P H i H H H We expand the polynomial h(x) using Bernstein polynomials to get
h(x)" bIBI(x), x3U, P P PZ K where the Bernstein coe$cients are de"ned as
(4.5)
G G Gl> l> i 2 bI" H o 2l c 2l P q O O O > OO O > l > O O O H H (4.6)
D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
with
l > k \ o 2l " (4.7) OO O > q H H and c 2 l "0 if some q is such that q 'N. OO O > H H In search of the minimum m of h(x), the algorithm provides iteratively improved bounds on m at each step. This is done by either increasing the degree k of Bernstein polynomials, or by subdivision of the box U. Experience suggests (Malan et al., 1992) that the algorithm is more e$cient if k is held constant (usually at k"N) and U is subdivided into smaller and smaller boxes. If mI denotes the minimum of the Bernstein coe$cients of h(x) on all subboxes at a "xed subdivision level k, with k"0 at the level of U, then the following result (Malan et al., 1992) applies.
Theorem 2. For each k we have (i) mI4m4mI#a2\I,
(4.8)
where a is a constant independent of k; and (ii) mI"m if and only if h(x) assumes its minimum at a vertex of a subbox. From Eq. (4.8) we see that the sequence +mI, converges quadratically to m with respect to the level of subdivision k. The algorithm is terminated when mI'0. Then, we know that family G is 1-positive. If, in addition, we "nd a p3P such that f (s, p) is Hurwitz, we conclude that the original family F is Hurwitz as well. In applying the algorithm di$culties arise when the minimum of h(x) in U is either zero or is very close to zero, and convergence of the subdivision algorithm is too slow. Abandoning the search for the minimum in this case can be justi"ed on practical grounds: the minimum of h(x) close to zero, even if it is positive, implies that the original uncertain polynomial f (s, p) has a zero close to the imaginary axis, implying further that f (s, p) is poorly stable. We suggest that the algorithm be terminated whenever, for a prescribed constant e'0, we arrive at min " ju!s (p)"4e, (4.9) G 2 K GZ SZ 1 NZ P which implies that at least one zero of f (s, p) has the absolute value of its real part less than e. For simplicity, we assume that a "1 and prove the L following: Theorem 3. ¸et us de,ne cI as cI"min+MI, mI#a2\I,,
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Proof. It is obvious that for any k, MI is greater or equal to the minimum m of h(x), that is, g(u, p). Using this fact and Eq. (4.8), we compute eL5min+MI, mI#a2\I, 5 min " f ( ju, p)" SZ S P S
NZ L 5 min " ju!s (p)" G SZ SN P S G NZ L 5 min " ju!s (p)" . G 2 L+ GZ 1 P SZ NZ
(4.11)
The last inequality implies Eq. (4.9). 䊐 Theorem 3 provides a stopping criterion when, after a large number of subdivision levels, mI remains negative. The quality of the stopping criterion depends on the size of a which, in turn, depends only on the degree and coe$cients of the original polynomial. Although the computation of a can be involved, it is completed o!-line before the subdivision algorithm is initiated. If the quality of the stopping criterion deteriorates due to a large value of a, a better estimate of the upper bound of the minimum is provided by MI. To show how the criterion works we present the following example: Example 2. Let us consider a polynomial family F with f (s, p)"s#(p #p#1!j)s #[p p#2p #p#j(p!2p )]s #2p p!2p#4#j(2p!4p #4) and P"+p31: p 3[1, 1.15], p 3[0.4, 0.5],. Then, the magnitude polynomial is
(4.12)
(4.13)
g(u, p)"u!2u#(p#p!2p #2)u #(!4p#8p #2p!8)u #(8p#pp!16p !2p p#6p)u #(!16p#32p #8p)u#16p!32p #4pp!8p p#8p#32. (4.14) To determine the u interval, we apply the Bernstein subdivision algorithm to each coe$cient of g(u, p) and compute the two minorizing polynomials g>(u)"u!2u#1.0256u!4.0388u!7.8720u
(4.10)
where MI denotes the minimum of the maximums of the Bernstein coe.cients over all subboxes at sublevel k. If mI(0 and cI'0, then (cI) L4e implies Eq. (4.9).
#15.8448u#16.1024, g\(u)"u!2u#1.0256u!3.8750u!7.8720u #16.5000u#16.1024.
(4.15)
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From g>(u) and g\(u) we get u3[!1.0917, !0.9111], so that Y"[!1.0917, !0.9111][1, 1.15][0.4, 0.5] .
(4.16)
Let us choose the threshold e"0.05. For p"1.15 and p"0.4, the zeros of f (s, p) are stable with the real parts smaller than !0.05. Before we apply the Bernstein algorithm, let us note that the minima of the magnitude polynomial g(u, p) over Y must be nonnegative. This feature of g(u, p) provides for a great savings in memory requirements. During the subdivision process we can continually discard all the subboxes which have positive minima. By exploiting this fact in case of g(u, p) in Eq. (4.14), at the 12th level we have retained only 4096 subboxes having negative minima, and m"!3.2747;10\,
a2\"4.7336;10\.
Fig. 1. Value sets (10 subintervals).
(4.17) At this point we required only 11.2 MB of memory. Since both M and m#a2\ are larger than eL"(0.05)"1.5625;10\, we cannot terminate the algorithm at this point. Continuation of the subdivision process, however, places rapidly increasing memory demands at each subsequent level. To avoid the memory problem we determined the maximum M"1.1058; 10\ in the box where we reached the minimum at 12th level. It turned out that this maximum was equal to M. For this reason, we applied the subdivision algorithm at 13th and 14th level only to the subbox where the minimum occurred at the 12th level. This produced a box in which the minimum m and maximum M were such that m4m"!1.3889;10\(0, (4.18)
Fig. 2. Value sets (10,000 subintervals).
M"9.2186;10\(1.5625;10\. This meant that M4M(eL"1.5625;10\
(4.19)
and we terminated the process because Eqs. (4.18) and (4.19) by Theorem 3 implied that there was at least one zero of f (s, p) with the absolute value of its real part being smaller than the prescribed threshold e"0.05. Finally, we should provide the interpretation of the positivity analysis in terms of value sets and the Zero Exclusion Theorem. By converting stability to positivity of the magnitude polynomial, we get the value sets to be intervals (S[ iljak and S[ iljak, 1997) with bounds produced directly by the Bernstein algorithm. To illustrate the degree of simpli"cation of the original stability problem, we consider again the polynomial f (s, p) in Eq. (4.12) and plot the value sets in Fig. 1 of the magnitude polynomial g(u, p) of Eq. (4.14). The nominal u interval [!1.0917, !0.9111] was split into 10 subintervals pro-
ducing 11 points. By further splitting the interval into 10,000 subintervals, we obtain the value sets between the 6th and 7th point of Fig. 1 and show only 11 of them in Fig. 2. From this diagram we reach the same conclusion as before that the zero does not belong to the value sets (yet!), but at least one zero of f (s, p) is closer to the imaginary axis than the allowed limit e"0.05.
5. D regions: lines and circles Before we consider more general shapes of the region D, which are bounded by straight lines and circular arcs, let us apply the Bernstein algorithm to the magnitude function arising in stability with respect to the unit circle, that is, Schur stability. We use the standard bilinear transformation to convert Schur stability to Hurwitz stability, which is accomplished without changing the
D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
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nature of the polynomic uncertainty structure of the original polynomial. Let us denote the interior of the unit circle as C"+s3": "s"(1,
(5.1)
with the boundary *C"+e F: h3[0, 2n],.
(5.2)
Using the bilinear transformation
e F"
j!u , j#u
sin h 3(!R,#R), 1#cos h . h3[0, 2n]!+n,
!1,
h"n
u"
(5.3)
we can write
L j!u G a (p) , G j#u G " f (e F, p)""
u31, h3[0, 2n]!+n,,
" f (!1, p)",
h"n.
Fig. 3. Region D.
(5.4)
Let us now de"ne the polynomial L d(u, p)" d (p)uG , G G such that
bounded by straight lines and circular arcs. This is illustrated by the following: (5.5) Example 3. Consider the polynomial
L d(u, p)" a (p)( j!u)G( j#u)L\G , G G and d "" f (!1, p)". Then, L " f (eHF, p)"'0 , ∀h3[0, 2n], ∀p3P
f (s, p)"s#[2p #p #j(p !p)]s (5.6)
#j(p#p p !p p!p)]s #p p#pp !j(p p!pp )
(5.7)
(5.10)
with the uncertainty box
if and only if d(u, p)'0,
#[p#2p p #p p
∀u31, ∀p3P.
(5.8)
P"+p31: p 3[2.5, 3], p 3[1.5, 2],
(5.11)
It is important to note that coe$cients d (p), as de"ned by G Eq. (5.6), are polynomial functions in p if a (p) are, and G positivity of " f (s, p)" on *C is equivalent to positivity of polynomial d(u, p) on 1 having the same uncertainty structure. We de"ne a polynomial family
and the region D shown in Fig. 3. Various segments of *D are as follows:
D"+d( ), p): p3P,
BC"
(5.9)
AB"+!1#ju: u3[!(3, (3],,
1 (3 ! #j u: u3[2, 6] , 2 2
and recall that a polynomial family is Schur stable if each member of the family has all zeros inside the unit circle C, that is, it is Schur stable. From Theorem 1 we get
2n 4n CD" 6e F: h3 , 3 3
Corollary 2. Suppose that there exists a p3P such that f (s, p) is Schur stable. ¹hen, a family F is Schur stable if and only if the corresponding family D is 1-positive.
DA"
By combining this corollary with Corollary 1, we can consider fairly general shapes of D-stability regions
(5.12)
,
1 (3 ! !j u: u3[2, 6] . 2 2
When we use the Bernstein algorithm on each segment, we conclude that " f (s, p)" is positive on *D. Since f (s, p) has all zeros within D when p"(3, 2), the family F is D}stable with respect to the region D shown in Fig. 3.
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6. Conclusions By reformulating D-stability of a complex polynomial as positivity of its magnitude function, we made stability testing suitable for the Bernstein subdivision algorithm. In particular, the proposed approach requires only real arithmetic, allows for a simple and meaningful stopping criterion, and leads to value sets which are intervals and, thus, convex. Future research should further exploit these aspects of the approach in the context of robust stability of nonlinear control systems (S[ iljak, 1989; S[ iljak and S[ iljak, 1997; Wada et al., 1995, 1997).
Acknowledgements This work has been supported by the National Science Foundation under the grant ECS-9526142.
References Ackermann, J. (1993). Robust control: Systems with uncertain physical parameters. London, UK: Springer. Barmish, B. R. (1994). New tools for robustness of linear systems. New York: Macmillan. Barmish, B. R., & Tempo, R. (1990). The robust root locus. Automatica, 26, 283}292. Bhattacharyya, S. P., Chapellat, H., & Keel, L. H. (1995). Robust control: ¹he parametric approach. Upper Saddle River, NJ: Prentice}Hall. Fiorio, G., Malan, S., Milanese, M., & Taragna, M., (1993). Robust performance design of "xed structure controllers for systems with uncertain parameters. Proceedings of the 33rd IEEE CDC, pp. 3029}3031. Garlo!, J. (1986). Convergent bounds for the range of multivariate polynomials. ¸ecture Notes in Computer Science (Vol. 212, pp. 37}56). Berlin: Springer. Garlo!, J. (1993). The Bernstein algorithm, Interval Computations, 2, 154}168. Garlo!, J., Graf, B., & Zettler, M. (1997). Speeding up an algorithm for checking robust stability of polynomials. Proceedings of the 2nd Symposium on Robust Control Design, Budapest, Hungary, (25}27).
Lane, J. M., & Riesenfeld, R. F. (1981). Bounds on a polynomial. BI¹, 21, 112}117. Malan, S., Milanese, M., & Taragna, M. (1997). Robust analysis and design of control systems using interval arithmetics, Automatica, 33, 1363}1372. Malan, S., Milanese, M., Taragna, M., & Garlo!, J. (1992). B algorithm for robust performance analysis in presence of mixed parametric and dynamic perturbations. Proceedings of the 31st IEEE CDC (pp. 128}133). Milanese, M., Tempo, R., & Vicino, A. (1989). Robustness in identi,cation and control. New York: Plenum. Polyak, B. T., & Kogan, J. (1995). Necessary and su$cient conditions for robust stability of linear systems with multia$ne uncertainty structure. IEEE ¹ransactions on Automatic Control, 40, 1255}1260. Sideris, A., & Pen a, R. S. (1989). Fast computation of the multivariable stability margin for real interrelated uncertain parameters. IEEE ¹ransactions on Automatic Control, 34, 1272}1276. S[ iljak, D. D. (1989). Polytopes of nonnegative polynomials. Proceedings of the 1988 ACC (pp. 193}199). S[ iljak, D. D., & S[ iljak, M. D. (1997). Nonnegativity of uncertain polynomials. Mathematical Problems in Engineering, 3, 1}29. Stolan, J. A. (1995). An improved S[ iljak's algorithm for solving polynomial equations converges quadratically to multiple zeros. Journal of Computational and Applied Mathematics, 64, 247}268. Tesi, A., & Vicino, A. (1991). Design of optimally robust controllers with few degrees of freedom. Proceedings of the IFAC Symposium on Design Methods of Control Systems, Zurich, Switzerland (pp. 264}269). Vicino, V., & Milanese, M. (1990). Robust stability of linear state space models via Bernstein polynomials. In Control of ;ncertain Systems (pp. 329}343). D. Hinrichsen, & B. Martensson (Eds.), Boston: Birkhauser. Wada, T., Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1998). Parametric absolute stability of Lur'e systems. IEEE ¹ransactions on Automatic Control, 43, 1649}1653. Wada, T., Ikeda, M., Ohta, Y., & S[ iljak, D. D. (1997). Parametric absolute stability of multivariable Lur'e systems. Automatica, submitted. Zadeh, L., & Desoer, C. A. (1963). ¸inear system theory. New York: McGraw-Hill. Zeheb, E. (1990). Necessary and su$cient conditions for robust stability of a continuous system * The continuous dependency case illustrated via multilinear dependency. IEEE ¹ransactions on Circuits and Systems, 37, 47}53. Zettler, M., & Garlo!, J. (1996). Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansions. IEEE ¹ransactions on Automatic Control, 43, 425}431.
Technical Communique
Robust D-stability via positivity夽 D.D. S[ iljak*, D.M. StipanovicH Department of Electrical Engineering, Santa Clara University, Santa Clara, CA 95053-0569, USA Received 19 November 1998; received in "nal form 8 February 1999
Abstract The main objective of this paper is to convert the general problem of robust D-stability of a complex polynomial to positivity in the real domain of the corresponding magnitude function. In particular, the obtained criterion for Hurwitz stability is applied to polynomials with interval parameters and polynomic uncertainty structures. The robust stability is veri"ed by testing positivity of a real polynomial using the Bernstein subdivision algorithm. A new feature in this context is the stopping criterion, which is applied whenever the algorithm is inconclusive after a large number of iterations, but we can show that at least one zero of the polynomial is closer to the imaginary axis than a prescribed limit. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Robust stability; Interval uncertainty; Polynomic uncertainty structure; Bernstein subdivision algorithm
1. Introduction Converting D-stability of an uncertain polynomial to positivity of a multivariable polynomial opens up a possibility to apply the powerful Bernstein expansion algorithms to robust stability problems. This idea has been introduced by Vicino and Milanese (1990) and has achieved recently a high level of sophistication and e$ciency in the work of Garlo! and his co}workers (Garlo!, 1993; Zettler and Garlo!, 1996; Garlo! et al., 1997). The approach to robust stability is based upon a large number of results on Bernstein polynomials; in particular, the subdivision algorithm, which was introduced by Lane and Riesenfeld (1981) and considerably extended for stability applications in Garlo! (1986), Malan et al. (1992, 1997), Fiorio et al. (1993) and Garlo! et al. (1997). Among many comparisons that one can make with respect to numerous results available in the context of robust stability (e.g., Milanese et al., 1989; Zeheb, 1990; Vicino and Milanese, 1990; Barmish and Tempo, 1990;
***** * Corresponding author. Tel. #1-408-554-4488; e-mail: dsiljak@ scu.edu. 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato.
Tesi and Vicino, 1991; Ackermann, 1993; Barmish, 1994; Bhattacharyya et al., 1995; Polyak and Kogan, 1995), the ability of the Bernstein expansion algorithms to handle with relative ease and accuracy the polynomials with polynomic uncertainty structures has a particular signi"cance. A straightforward way to determine stability of a real polynomial is to establish positivity of the next to the last Hurwitz determinant, which is a multivariable polynomial in uncertain parameters (Vicino and Milanese, 1990; Garlo!, 1993; Zettler and Garlo!, 1996; Garlo! et al., 1997). If the uncertain polynomial is complex, stability can be shown by simultaneous analysis of the zeros of its real and imaginary parts (Zettler and Garlo!, 1996). Stability of an uncertain matrix can be analyzed in the same way by considering its characteristic polynomial (Vicino and Milanese, 1990), because we are not concerned with the fact that the resulting uncertainty structure is polynomic. Our principal contribution to the Bernstein algorithms for robust stability analysis is the use of the magnitude function of a given uncertain polynomial to convert the general problem of robust D-stability to positivity. This function, which was used successfully in the root}"nding algorithms (Stolan, 1995), o!ers new possibilities in the context of robust D-stability. First, D-stability of complex uncertain polynomials with respect to general stability regions and having coe$cients as arbitrary continuous functions of uncertain parameters, can be considered
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 4 2 - 4
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using only real arithmetic. In case of Hurwitz stability and polynomic uncertainty structure, the magnitude function results in only the doubling of the powers of uncertain parameters regardless of the degree of the polynomial. This is not the case with Hurwitz determinants, which are, in addition, more di$cult to compute than the magnitude function. Furthermore, the positivity test of the function is essentially the same whether the polynomial is real or complex. Second, the magnitude function provides a suitable stopping criterion in cases when the Bernstein algorithm is inconclusive after a large number of subdivisions, but the computations show that at least one zero of the polynomial is closer to the imaginary axis than the acceptable limit. Finally, positivity analysis of the magnitude function can be facilitated by a number of results involving minorizing polynomials, the modi"ed Routh array, and convex value sets, which are available in the context of positivity of uncertain polynomials (S[ iljak, 1989; S[ iljak and S[ iljak, 1997). The organization of the paper is as follows. In the next section, we shall reformulate robust D-stability of a complex polynomial as positivity of a real polynomial using the magnitude function. In Section 3, we specialize the stability region to the left half plane and analyze robust Hurwitz stability by verifying positivity of the real magnitude polynomial on the real line. By a simple minorization of the magnitude polynomial we can determine the intervals for positivity testing via Bernstein subdivision algorithm. The testing is discussed in Section 4, where the stopping criterion for the algorithm is provided. In Section 5, we extend the approach to robust Schur stability, and consider general D regions with boundaries made up of straight lines and circles.
2. D-stability via positivity Let us consider a complex polynomial L f (s, p)" a (p)sG, G G
(2.1)
where the coe$cients a (p) are complex continuous funcG tions of uncertain parameter vector p31l, and s3". We assume that p belongs to a bounded and pathwise connected set PL1l, and that f (s, p) has invariant degree, that is, deg f (s, p)"n for all p3P. We de"ne next the family of polynomials F"+ f ( ), p): p3P,
(2.2)
and specify an open simply connected region DL". De5nition 1. A family of polynomials F is D-stable if all zeros of f (s, p) lie in D for all p3P.
To establish conditions of robust D-stability via polynomial positivity we de"ne the function L L fK (s, p)"f (s, p) f (s, p)" a (p)aN (p)sGsN H, (2.3) G H G H where overbar denotes conjugation. We note immediately that fK (s, p)"" f (s, p)", which means that fK (s, p) is nonnegative for all s3". This fact is essential to our development. Functions fK (s, p) form a family F< "+ f K ( ), p): p3P,
(2.4)
which we consider on a Jordan arc J. De5nition 2. A family of functions F< is J-positive if fK (s, p) is positive on J for all p3P. We denote by *D the boundary of D and prove the following: Theorem 1. Suppose that there exists a p3P such that f (s, p) has all zeros in D. ¹hen, a family of polynomials F is D-stable if and only if the corresponding family F< is *D-positive. Proof. To prove the &&only if '' part of the theorem, we assume that F is D-stable, but F< is not *D-positive. Then, f K (s*, p*)"0 for some p*3P and s*3*D, which implies that f (s*, p*)"0. Since D is open, the last statement implies that F is not stable, which is a contradiction. To prove the &&if '' part of the theorem, we again proceed with contradiction. We assume that for some p3P, f (s, p) does not have all zeros within D, but F< is *Dpositive. Since P is pathwise connected, there exists a continuous function h : [0, 1]PP such that h(0)"p and h(1)"p. We recall that continuity of coe$cients a (p) implies that s (p), s (p), 2, s (p), which represent the G L zeros of f (s, p), are continuous functions, as well. Now, since f (s, p) has zeros outside D, there is a zero s * (p) for G some i*3+1, 2, 2, n, which is not in D. On the other hand, s * (p) is within D by assumption. Observe then G that for j3[0, 1], s * (h(j)) describes a continuously G varying zero which starts at s * (h(0)) within D and G ends up at s * (h(1)) outside D. Due to continuity, G there is a j*3[0, 1] such that s * (h(j*))3*D. Taking G p*"h(j*) we have that f (s * (p*), p*)"0 and, hence, G fK (s * (p*), p*)"0 for s * (p*)3*D, which is the contradicG G tion we want. ) A similar theorem was established in S[ iljak and S[ iljak (1997), which linked robust Hurwitz stability to positivity. In that theorem, the polynomial g' (s, p)"f (s, p) fM (!s, p)
(2.5)
D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
was used instead of function fK (s, p). The polynomial g' (s, p) is nonnegative on the imaginary axis in ", but not necessarily o! the axis. For this reason, it could not be used to establish robust D-stability. We also note that the proofs of both theorems are conceptually similar to the standard proof of the Zero Exclusion Theorem (Zadeh and Desoer, 1963).
3. Robust Hurwitz stability When D is the open left half of ", which we denote by " "+s3": Re s(0,, \ then *D is the imaginary axis
(3.1)
I"+s3": Re s"0,.
(3.2)
On I, the function f K ( ju, p)"" f ( ju, p)" is a real polynomial
L g(u, p)" a (p)( ju)G . (3.3) G G We now assume that a (p) are complex multivariable G polynomials in p, and rewrite g(u, p) as L g(u, p)" b (p)uG, (3.4) G G where u31, and b (p) are real multivariable polynomials G in p. We recall that a polynomial is said to be Hurwitz if all of its zeros are in " . From De"nition 1, we have \ an analogous de"nition for the Hurwitz property of a family F. We de"ne the polynomial family G"+g( ), p): p3P,
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The problem is to "nd the two bounding numerical polynomials de"ned as L g>(u)" b>uG4g(u, p), ∀u50, ∀p3P G G and
(3.7)
L (3.8) g\(u)" b\uG4g(u, p), ∀u40, ∀p3P G G and compute the interval [u, u ] outside of which they are both positive. One way to do it is to use a transformation (Sideris and Pen a, 1989) which expands the parameter space so that the new polynomial has multia$ne uncertainty structure. The minorizing polynomial is one of the vertex polynomials of the expanded uncertainty box in the new parameter space (S[ iljak and S[ iljak, 1997). We propose a less conservative approach to produce g>(u) and g\(u), which applies the Bernstein expansion. The idea is to minimize or maximize all b over the box G P by using the Bernstein subdivision algorithm (Malan et al., 1992). For u50, we minimize each b independentG ly and form g>(u) using the obtained minima b>. For G u40, we get g\(u) by maximizing b when i is odd and G minimizing b when i is even, which produces b\. G G If it so happens that both minorizing polynomials g>(u) and g\(u) are positive for all u, then the family G is positive and we are done. Positivity of the polynomials can be tested by the modi"ed Routh array (S[ iljak, 1989; S[ iljak and S[ iljak, 1997) or by a root "nding algorithm (e.g., Stolan, 1995). If the two polynomials g>(u) and g\(u) are not both positive for all u, they can be used to compute a positivity interval [ u, u ]. Simply, u is the largest nonnegative zero of g>(u), while u is the smallest nonpositive zero of g\(u).
(3.5)
and state the obvious. Corollary 1. Suppose that there exists a p3P such that f (s, p) is Hurwitz. ¹hen, a family F is Hurwitz if and only if the corresponding family G is 1-positive. Our test for robust stability is based on the numerical methods for verifying positivity of multivariable polynomials via Bernstein polynomials (Malan et al., 1992). This requires that all variables in the polynomial g(u, p) be restricted to intervals. While it is standard to restrict the parameter vector p to a box (3.6) P"+p31l: p 3[ p , pN ],, G G G the variable u must be allowed to vary over the entire real line 1. A way to restrict u to a "nite interval [u, u ] for all p3P, is to use the minorizing polynomials (S[ iljak and S[ iljak, 1997)
Example 1. To illustrate the role of minorizing polynomials, let us consider the polynomial from Barmish (1994): f (s, p)"s#(3p#pp #p p #3p #10)s #(4p#p#15)s#16p p #17 and choose the uncertainty box
(3.9)
P"+p31: p 3[!0.8,!0.6], p 3[!0.4, 1],. (3.10) Since the magnitude function g(u, p)"" f ( ju, p)" is a real even polynomial, by replacing u by u we get the polynomial to be tested for positivity as g(u, p)"u#(9p#6pp #pp#6pp #18p #2pp#6pp #60p#pp#26pp #p#20p p #60p !2p#70)u #(!36pp #16p!12pp!102p
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!4pp#30p!70pp #120p!154p p !102p #30p!115)u#36pp #204p p #289 (3.11) which has the same degree as the original polynomial f (s, p); in the real case, no doubling of degree takes place when robust stability of f (s, p) is replaced by positivity of g(u, p). Furthermore, since f (s, p) is real, we have g>(u)"g\(u) O g(u), and g(u) needs to be tested for positivity only for u50. Using the expanding transformation (Sideris and Pen a, 1989) we get the minorizing polynomial
was obtained by applying the expanding transformation (Sideris and Pen a, 1989) to the individual coe$cients of g(u, p). Since the polynomial f (s, p) is real, we need to be concerned only with u31 . The zeros of gJ (u) are 3.866, > !4.794 and 6.292, and the interval bounds are u"3.866 and u "6.292. Once the interval for u is speci"ed, we can treat u as just another parameter. Then, minimization of the polynomial g(u, p) over a box can proceed by using directly the multivariable version of the Bernstein algorithm (Malan et al., 1992).
gJ (u)"u!5.3635u!24.3690u#116.5840.
4. Bernstein algorithm: the stopping criterion
(3.12)
The modi"ed Routh array (S[ iljak, 1989) for gJ (u) is computed as !1 !5.3635 24.3690 116.5840 !3 !10.7270 24.3690 !1.7878 16.2460 116.5840 !37.9884 171.2636 8.1860 116.5840 721.2897 116.5840
(3.13)
Y"[ u, u ];P.
Since the array has one sign variation in the "rst column, the polynomial gJ (u) has two positive zeros and, therefore, gJ (u) is not 1 -positive. Stability of the original > family F, which involves the polynomial f (s, p) of Eq. (3.9) and the box P of Eq. (3.10), remains inconclusive. Applying the Bernstein subdivision algorithm to minimize each individual coe$cient of g(u, p) in Eq. (3.11), we obtain g(u)"u!4.2027u!6.4982u#148.84.
(3.14)
The array is now !1 !4.2022 6.4982 148.8400 !3 !8.4044 6.4982 !1.4 4.3321 148.8400 !17.6875 !312.4446 20.3985 148.8400 !183.3857 148.8400
Let us "rst outline the algorithm (Malan et al., 1992) for bounding a real multivariable polynomial using the Bernstein polynomials. We shall recall only the elements of the algorithm that are necessary in our application. By assuming that u3[ u, u ], where the bounds u and u are computed using the minorizing polynomials, we can consider g(u, p) as a real multivariable polynomial z(y) in y"(u, p)31l> restricted to a box
(3.15)
which has three variations in the "rst column. The polynomial g(u) is 1 -positive and the family G is 1 > > positive as well. Since f (s, p) is real, 1 -positivity is all > that is required. Finally, for p"(!0.8, 1) the numerical polynomial f (s, p) is Hurwitz and the family F is Hurwitz as well. To illustrate the computation of the interval [ u, u ] for u, when the minorizing polynomials are not positive on the entire real line, let us assume that the polynomial gJ (u)
(4.1)
Since any box Y of dimension l#1 can be mapped a$nely onto the unit box U"[0, 1]l>, using the same mapping z(y) is mapped to the multivariable polynomial h(x) of the same degree N, given by l , > c 2 l xGH , h(x)" (4.2) > GG G H G G 2 Gl> H where x3U. The objective of the algorithm is to compute
(4.3) m"min h(x). VZ U For simplicity, let us use r to denote the l#1-tuple of nonnegative integers (i , i , 2, il ) with the range > K"+(i , i , 2, il ): i , i , 2, il "0, 1, 2, k,. Then, > > Bernstein polynomials of a degree k5N are de"ned as
l > k BI(x)" xGH (1!x )I\GH, r3K. (4.4) P H i H H H We expand the polynomial h(x) using Bernstein polynomials to get
h(x)" bIBI(x), x3U, P P PZ K where the Bernstein coe$cients are de"ned as
(4.5)
G G Gl> l> i 2 bI" H o 2l c 2l P q O O O > OO O > l > O O O H H (4.6)
D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
with
l > k \ o 2l " (4.7) OO O > q H H and c 2 l "0 if some q is such that q 'N. OO O > H H In search of the minimum m of h(x), the algorithm provides iteratively improved bounds on m at each step. This is done by either increasing the degree k of Bernstein polynomials, or by subdivision of the box U. Experience suggests (Malan et al., 1992) that the algorithm is more e$cient if k is held constant (usually at k"N) and U is subdivided into smaller and smaller boxes. If mI denotes the minimum of the Bernstein coe$cients of h(x) on all subboxes at a "xed subdivision level k, with k"0 at the level of U, then the following result (Malan et al., 1992) applies.
Theorem 2. For each k we have (i) mI4m4mI#a2\I,
(4.8)
where a is a constant independent of k; and (ii) mI"m if and only if h(x) assumes its minimum at a vertex of a subbox. From Eq. (4.8) we see that the sequence +mI, converges quadratically to m with respect to the level of subdivision k. The algorithm is terminated when mI'0. Then, we know that family G is 1-positive. If, in addition, we "nd a p3P such that f (s, p) is Hurwitz, we conclude that the original family F is Hurwitz as well. In applying the algorithm di$culties arise when the minimum of h(x) in U is either zero or is very close to zero, and convergence of the subdivision algorithm is too slow. Abandoning the search for the minimum in this case can be justi"ed on practical grounds: the minimum of h(x) close to zero, even if it is positive, implies that the original uncertain polynomial f (s, p) has a zero close to the imaginary axis, implying further that f (s, p) is poorly stable. We suggest that the algorithm be terminated whenever, for a prescribed constant e'0, we arrive at min " ju!s (p)"4e, (4.9) G 2 K GZ SZ 1 NZ P which implies that at least one zero of f (s, p) has the absolute value of its real part less than e. For simplicity, we assume that a "1 and prove the L following: Theorem 3. ¸et us de,ne cI as cI"min+MI, mI#a2\I,,
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Proof. It is obvious that for any k, MI is greater or equal to the minimum m of h(x), that is, g(u, p). Using this fact and Eq. (4.8), we compute eL5min+MI, mI#a2\I, 5 min " f ( ju, p)" SZ S P S
NZ L 5 min " ju!s (p)" G SZ SN P S G NZ L 5 min " ju!s (p)" . G 2 L+ GZ 1 P SZ NZ
(4.11)
The last inequality implies Eq. (4.9). 䊐 Theorem 3 provides a stopping criterion when, after a large number of subdivision levels, mI remains negative. The quality of the stopping criterion depends on the size of a which, in turn, depends only on the degree and coe$cients of the original polynomial. Although the computation of a can be involved, it is completed o!-line before the subdivision algorithm is initiated. If the quality of the stopping criterion deteriorates due to a large value of a, a better estimate of the upper bound of the minimum is provided by MI. To show how the criterion works we present the following example: Example 2. Let us consider a polynomial family F with f (s, p)"s#(p #p#1!j)s #[p p#2p #p#j(p!2p )]s #2p p!2p#4#j(2p!4p #4) and P"+p31: p 3[1, 1.15], p 3[0.4, 0.5],. Then, the magnitude polynomial is
(4.12)
(4.13)
g(u, p)"u!2u#(p#p!2p #2)u #(!4p#8p #2p!8)u #(8p#pp!16p !2p p#6p)u #(!16p#32p #8p)u#16p!32p #4pp!8p p#8p#32. (4.14) To determine the u interval, we apply the Bernstein subdivision algorithm to each coe$cient of g(u, p) and compute the two minorizing polynomials g>(u)"u!2u#1.0256u!4.0388u!7.8720u
(4.10)
where MI denotes the minimum of the maximums of the Bernstein coe.cients over all subboxes at sublevel k. If mI(0 and cI'0, then (cI) L4e implies Eq. (4.9).
#15.8448u#16.1024, g\(u)"u!2u#1.0256u!3.8750u!7.8720u #16.5000u#16.1024.
(4.15)
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D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
From g>(u) and g\(u) we get u3[!1.0917, !0.9111], so that Y"[!1.0917, !0.9111][1, 1.15][0.4, 0.5] .
(4.16)
Let us choose the threshold e"0.05. For p"1.15 and p"0.4, the zeros of f (s, p) are stable with the real parts smaller than !0.05. Before we apply the Bernstein algorithm, let us note that the minima of the magnitude polynomial g(u, p) over Y must be nonnegative. This feature of g(u, p) provides for a great savings in memory requirements. During the subdivision process we can continually discard all the subboxes which have positive minima. By exploiting this fact in case of g(u, p) in Eq. (4.14), at the 12th level we have retained only 4096 subboxes having negative minima, and m"!3.2747;10\,
a2\"4.7336;10\.
Fig. 1. Value sets (10 subintervals).
(4.17) At this point we required only 11.2 MB of memory. Since both M and m#a2\ are larger than eL"(0.05)"1.5625;10\, we cannot terminate the algorithm at this point. Continuation of the subdivision process, however, places rapidly increasing memory demands at each subsequent level. To avoid the memory problem we determined the maximum M"1.1058; 10\ in the box where we reached the minimum at 12th level. It turned out that this maximum was equal to M. For this reason, we applied the subdivision algorithm at 13th and 14th level only to the subbox where the minimum occurred at the 12th level. This produced a box in which the minimum m and maximum M were such that m4m"!1.3889;10\(0, (4.18)
Fig. 2. Value sets (10,000 subintervals).
M"9.2186;10\(1.5625;10\. This meant that M4M(eL"1.5625;10\
(4.19)
and we terminated the process because Eqs. (4.18) and (4.19) by Theorem 3 implied that there was at least one zero of f (s, p) with the absolute value of its real part being smaller than the prescribed threshold e"0.05. Finally, we should provide the interpretation of the positivity analysis in terms of value sets and the Zero Exclusion Theorem. By converting stability to positivity of the magnitude polynomial, we get the value sets to be intervals (S[ iljak and S[ iljak, 1997) with bounds produced directly by the Bernstein algorithm. To illustrate the degree of simpli"cation of the original stability problem, we consider again the polynomial f (s, p) in Eq. (4.12) and plot the value sets in Fig. 1 of the magnitude polynomial g(u, p) of Eq. (4.14). The nominal u interval [!1.0917, !0.9111] was split into 10 subintervals pro-
ducing 11 points. By further splitting the interval into 10,000 subintervals, we obtain the value sets between the 6th and 7th point of Fig. 1 and show only 11 of them in Fig. 2. From this diagram we reach the same conclusion as before that the zero does not belong to the value sets (yet!), but at least one zero of f (s, p) is closer to the imaginary axis than the allowed limit e"0.05.
5. D regions: lines and circles Before we consider more general shapes of the region D, which are bounded by straight lines and circular arcs, let us apply the Bernstein algorithm to the magnitude function arising in stability with respect to the unit circle, that is, Schur stability. We use the standard bilinear transformation to convert Schur stability to Hurwitz stability, which is accomplished without changing the
D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
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nature of the polynomic uncertainty structure of the original polynomial. Let us denote the interior of the unit circle as C"+s3": "s"(1,
(5.1)
with the boundary *C"+e F: h3[0, 2n],.
(5.2)
Using the bilinear transformation
e F"
j!u , j#u
sin h 3(!R,#R), 1#cos h . h3[0, 2n]!+n,
!1,
h"n
u"
(5.3)
we can write
L j!u G a (p) , G j#u G " f (e F, p)""
u31, h3[0, 2n]!+n,,
" f (!1, p)",
h"n.
Fig. 3. Region D.
(5.4)
Let us now de"ne the polynomial L d(u, p)" d (p)uG , G G such that
bounded by straight lines and circular arcs. This is illustrated by the following: (5.5) Example 3. Consider the polynomial
L d(u, p)" a (p)( j!u)G( j#u)L\G , G G and d "" f (!1, p)". Then, L " f (eHF, p)"'0 , ∀h3[0, 2n], ∀p3P
f (s, p)"s#[2p #p #j(p !p)]s (5.6)
#j(p#p p !p p!p)]s #p p#pp !j(p p!pp )
(5.7)
(5.10)
with the uncertainty box
if and only if d(u, p)'0,
#[p#2p p #p p
∀u31, ∀p3P.
(5.8)
P"+p31: p 3[2.5, 3], p 3[1.5, 2],
(5.11)
It is important to note that coe$cients d (p), as de"ned by G Eq. (5.6), are polynomial functions in p if a (p) are, and G positivity of " f (s, p)" on *C is equivalent to positivity of polynomial d(u, p) on 1 having the same uncertainty structure. We de"ne a polynomial family
and the region D shown in Fig. 3. Various segments of *D are as follows:
D"+d( ), p): p3P,
BC"
(5.9)
AB"+!1#ju: u3[!(3, (3],,
1 (3 ! #j u: u3[2, 6] , 2 2
and recall that a polynomial family is Schur stable if each member of the family has all zeros inside the unit circle C, that is, it is Schur stable. From Theorem 1 we get
2n 4n CD" 6e F: h3 , 3 3
Corollary 2. Suppose that there exists a p3P such that f (s, p) is Schur stable. ¹hen, a family F is Schur stable if and only if the corresponding family D is 1-positive.
DA"
By combining this corollary with Corollary 1, we can consider fairly general shapes of D-stability regions
(5.12)
,
1 (3 ! !j u: u3[2, 6] . 2 2
When we use the Bernstein algorithm on each segment, we conclude that " f (s, p)" is positive on *D. Since f (s, p) has all zeros within D when p"(3, 2), the family F is D}stable with respect to the region D shown in Fig. 3.
1484
D.D. S[ iljak, D.M. StipanovicH /Automatica 35 (1999) 1477}1484
6. Conclusions By reformulating D-stability of a complex polynomial as positivity of its magnitude function, we made stability testing suitable for the Bernstein subdivision algorithm. In particular, the proposed approach requires only real arithmetic, allows for a simple and meaningful stopping criterion, and leads to value sets which are intervals and, thus, convex. Future research should further exploit these aspects of the approach in the context of robust stability of nonlinear control systems (S[ iljak, 1989; S[ iljak and S[ iljak, 1997; Wada et al., 1995, 1997).
Acknowledgements This work has been supported by the National Science Foundation under the grant ECS-9526142.
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