Multidimensional BIBO Stability and Jury’s Conjecture Martin Scheicher∗, Ulrich Oberst† February 9, 2009 Abstract Twenty years ago E. I. Jury conjectured by analogy to the case of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function has the form H = 1/P where P is a very strict Hurwitz polynomial (VSHP). In more detail he conjectured that the impulse response of the filter is an absolutely integrable function. However, he did not specify the exact equations of these filters and did not prove the existence of the impulse response. In the present paper we generalise Jury’s conjecture to arbitrary proper transfer functions H = Q/P where P is a bivariate VSHP and prove this generalisation. In particular, we show the existence of a suitable impulse response or fundamental solution for any multivariate proper rational function. However, this impulse response is a measure and not a function. We have not succeeded to prove an analogue of Jury’s conjecture in higher dimensions than two yet, but we propose a new conjecture in context with the robustly stable multivariate polynomials investigated by Kharitonov et. al. For the discrete case we prove that the structurally stable rational functions after Bose, Lin, et al. coincide with the stable rational functions discussed in context with the stabilisation of discrete input/output systems. These rational functions are BIBO stable, but the converse is not true as established by several authors. Keywords: multidimensional input/output system, impulse response, fundamental solution, BIBO stability, Jury’s conjecture. AMS classification: 93C20, 93D25, 35E05, 39A11.
1 Introduction We refer to the surveys [3], [4], and [19] for comprehensive lists of contributors to stability and stabilisation theory of multidimensional systems. The present bibliography contains only those references which specifically treat multidimensional BIBO stability and which are actually used in the present paper. We consider the linear partial differential resp. difference equation with constant coefficients P ◦ y = Q ◦ u, (1) ∗ Institut für Mathematik, Universität Innsbruck Technikerstraße 13, 6020 Innsbruck, Austria
[email protected] Financial support of M. Scheicher through the Austrian FWF-project P18974 is gratefully acknowledged. † Institut für Mathematik, Universität Innsbruck Technikerstraße 13, 6020 Innsbruck, Austria
[email protected] 1
where P and Q are polynomials over C in r variables s = (s1 , . . . , sr ) and the input y and the output u are elements of suitable signal spaces, which are, in this paper, the arbitrarily often differentiable functions C ∞ (Rr , C) or the space of distributions D′ (Rr , C) in the continuous case of differential equations and the space of multiser quences CN = {y : Nr −→ C} in the discrete case of difference equations. The polynomials act on these spaces via partial differentiation sρ ◦ y := ∂ρ y, ρ = 1, . . . , r, in the continuous case and by shifts (sρ ◦ y)α := y
ρ
α+(0,...,0,1,0,...,0)
, ρ = 1, . . . , r, α ∈ Nr ,
in the discrete case. The transfer function of the equation or system (1) is the rational function H :=
Q ∈ C(s). P
It is proper if it is a power series in −1 z = (z1 , . . . , zr ) := s−1 := (s−1 1 , . . . , sr ).
A rational function is strictly proper if it is contained in CJzKz1 · · · zr . For a disjoint decomposition Cr = Λ 1 ⊎ Λ 2 of Cr we call a polynomial stable with respect to this decomposition if it has no zeros in Λ2 . For the standard cases of linear partial differential resp. difference equations the usual choices for Λ2 are Λ2 := {w ∈ C; ℜ(w) > 0}r = (C+ )r for the continuous case and Λ2 := {w ∈ C; |w| > 1}r for the discrete case. The stable polynomials in the continuous case are also called strict Hurwitz polynomials ([3, p. 12], [14, Def. 5 on p. 140], [21, p. 79]). In the continous case the impulse response or fundamental solution h is a solution of the equation P ◦h = Q◦δ with the r-dimensional delta-distribution δ. Given the impulse response h of the system (1) and an input u, one can obtain one output y via convolution y = h ∗ u, if the convolution is defined for this specific input u. It is not the only possible output since one can always add a solution of the homogeneous problem P ◦ y = 0. The PDE (1) is called BIBO (bounded input, bounded output) stable if for a continuous input u bounded on Rr>0 the output y = h ∗ u is also bounded on Rr>0 . The precise assumptions will be discussed in Section 2. In 1986 E. I. Jury proposed the following conjecture in the last section 8 on “Stability of Multidimensional Continuous Systems” of the survey paper [14] by analogy with the theory of digital filters and an algebraic, non-analytic reduction to the discrete case of partial difference equations.
2
To formulate Jury’s conjecture we need the notion of a bivariate very strict Hurwitz polynomial (VSHP). To explain this let P ∈ C[s1 , s2 ] = C[s2 ][s1 ] = C[s1 ][s2 ] be a non-zero bivariate polynomial of degree ni in the indeterminates si , i = 1, 2. Hence P can be written as P =
n1 X
b1i (s2 )si1 =
with
b2j (s1 )sj2
j=0
i=0
b1n1
n2 X
∈ C[s2 ] \ {0} and b2n2 ∈ C[s1 ] \ {0}.
The polynomial is called strict Hurwitz if P (s1 , s2 ) 6= 0 for all (s1 , s2 ) ∈ C2 with ℜ(s1 ) > 0 and ℜ(s2 ) > 0. The polynomial is called a VSHP if in addition 1. deg(b1n1 ) = n2 which implies also deg(b2n2 ) = n1 and 2. b1n1 and b2n2 are Hurwitz in the one-dimensional sense, i.e., have zeros only in the open left half plane of C. The first condition signifies that the rational function H = P1 ∈ C(s1 , s2 ) is proper, −1 i.e., a power series in s−1 1 and s2 . 2 Let, for example, P = s1 + s2 . Then n1 = 2, n2 = 1, and P = 1s21 + s2 s01 = 1s2 + s21 s02 , thus b12 = 1 and b21 = 1. Since deg(b1n1 ) = 0 < n2 and deg(b2n2 ) = 0 < n1 the polynomial P is not a VSHP. Conjecture 1 (Jury, [14, Def. 6, Th. 23, and Remark]). Let P be a bivariate VSHP and H = P1 ∈ C(s1 , s2 ). Then the impulse response of H exists, is an absolutely integrable function on the positive quadrant R2>0 = {(t1 , t2 ) ∈ R2 ; t1 > 0, t2 > 0} and is, in particular, BIBO stable. However, the exact nature of the considered continuous systems is not specified by Jury and the (usually difficult) existence of the impulse response is not addressed in [14]. In [21, Th. 6.1] Jury’s conjecture is quoted as a theorem. In Section 2 of this paper we show that the impulse response h of a multivariate proper rational function H = Q P exists, but is a measure and not a function. We give an explicit formula for it and develop criteria for BIBO stability. Section 3 is devoted to the proof of Jury’s conjecture and its generalisation to arbitrary bivariate strictly proper rational transfer functions with very strict Hurwitz denominators. Then we derive BIBO stability of a class of multivariate (r ∈ N arbitrary) proper rational functions once it has been shown for strictly proper rational functions in the class (Section 4). The VSHPs form such a class and thus we generalise Jury’s conjecture to bivariate proper rational functions with a VSHP as denominator.
3
Our proof of Jury’s conjecture cannot be directly extended to r > 2. Hence the proofs of Theorems 11 and 17 in higher dimensions remain open. In the concluding section we propose a conjecture on continuous BIBO stability of a class of proper real multivariate rational functions in context with the robustly stable multivariate polynomials which were introduced and investigated by Kharitonov et. al. [15]. In Section 5 we treat BIBO stability for partial difference equations. It is a wellknown fact that transfer functions in the ring of structurally stable rational functions are BIBO stable ([10], [16]). We show that the proper stable rational functions in the sense of [18] and the present paper indeed coincide with the structurally stable ones and are therefore BIBO stable. The proof uses multivariate Laurent series. We thank the associate editor and the two referees for their valuable remarks and hints to new literature.
2 Impulse Response and BIBO Stability In the following we will show the existence of the impulse response, give an explicit formula for it and define BIBO stability of transfer functions. Furthermore we will lay the groundwork for the proof of Jury’s conjecture. Let A be an integral domain. We use the variables −1 s = (s1 , . . . , sr ), z = (z1 , . . . , zr ) := s−1 = (s−1 1 , . . . , sr ), and t = (t1 , . . . , tr )
which we consider both as indeterminates and as vectors in Ar . Every polynomial f ∈ A[s] can be seen as a polynomial in one variable sρ with coefficients in A[s1 , . . . , sbρ , . . . , sr ] for every ρ ∈ {1, . . . , r}. The univariate degree of f with respect to sρ is written as degsρ . The multivariate or component-wise degree of a polynomial is the function deg :
A[s] \ {0} −→ Nr f 7−→ degs1 (f ), . . . , degsr (f ) .
This P degree was also used in [15, p. 8]. It satisfies deg(f g) = deg(f ) + deg(g). If f = µ∈Nr aµ sµ the coefficient lc(f ) := adeg(f ) is the leading coefficient of f . The degree of a monomial sµ := s1µ1 · · · srµr is its exponent: deg(sµ ) = µ. On Nr we use the order relation µ 6cw ν :⇐⇒ ∃γ ∈ Nr , ν = µ + γ ⇐⇒ sµ |sν . In contrast to degrees induced by total orderings of Nr as needed,Pe.g., for Gröbner bases computations, the multivariate degree of a polynomial f = µ∈Nr αµ sµ does not necessarily satisfy deg(f ) ∈ supp(f ) := {µ, αµ 6= 0} ⊆ Nr , e.g., deg(s1 + s2 ) = (1, 1) ∈ / supp(s1 + s2 ) = {(1, 0), (0, 1)}. In other words, the leading coefficient may be zero. We call the polynomials satisfying deg(f ) ∈ supp(f ) component-wise unital, in short cw-unital, see [17, p. 115, Theorem and Definition 50]. According to one of the referees, cw-unital polynomials were already used by Pontryagin as “polynomials with principal term present”. With the exception of Section 4 we use A = C in this paper.
4
In [17, p. 117ff, Theorem and Definition 60] the ring of proper rational functions is characterised as f ; f, g ∈ C[s], g is cw-unital, deg(f ) 6cw deg(g) . (2) P := C(s) ∩ CJzK = g The spaces of power series X X BR := f = aµ z µ ∈ CJzK; kf kR := |aµ |Rµ < ∞ , R ∈ Rr>0 , r r µ∈N
(3)
µ∈N
are subalgebras of CJzK and indeed Banach algebras with respect to kf kR , [9, Satz 1 on p. 16]. According to [9, p. 27] let [ BR ⊂ CJzK Chzi := R∈Rr>0
denote the algebra of (locally) convergent power series. The set Z := {z µ ; µ ∈ Nr } is a multiplicatively closed subset of C[z] ⊂ Chzi ⊂ CJzK and gives rise to the corresponding quotient rings with elements of Z as denominators: C[z]Z = C[z, s] = C[z, z −1 ] = ⊕µ∈Zr Cz µ , CJzKZ = CJzK[s] = {sµ H(z); H ∈ CJzK, µ ∈ Nr }, and ChziZ = Chzi[s] = {sµ H(z); H ∈ Chzi, µ ∈ Nr }. The elements of these rings are called Laurent polynomials resp. power series. We need certain preparations concerning distributions from [22, pp. 170–180] and ′ ′ consider the subspace D+ := D+ (Rr ) ⊂ D′ := D′ (Rr ) of all distributions with left bounded support, i.e., whose support is contained in a subset t + Rr>0 for a t ∈ ′ is a commutative integral domain with respect to the convolution Rr . The space D+ product ∗ and the identity is 1D+′ = δ, [22, Th. XIV on p. 173]. For univariate functions ϕ1 , . . . , ϕr ∈ D(R) = C0∞ (R) and univariate distributions T1 , . . . , Tr ∈ D′ (R) their tensor products are defined [22, Ch. IV] as follows: ϕ := ⊗rρ=1 ϕρ ∈ D(Rr ),
ϕ(t1 , . . . , tr ) =
T := ⊗rρ=1 Tρ ∈ D′ (Rr ),
T (ϕ) =
r Y
ρ=1 r Y
ϕρ (tρ ), Tρ (ϕρ ).
ρ=1
′ If the Tρ have even left bounded support, i.e. Tρ ∈ D+ (R) for all ρ, then so has their tensor product. The Heaviside step function in one variable t1 ∈ R, 1 if t1 > 0 Y (t1 ) := , 0 otherwise ′ is invertible with respect to ∗, its inverse Y ∗(−1) = δ ′ in (D+ (R), ∗) is the derivative ′ of δ. The powers of Y in (D+ (R), ∗) are ( k−1 t1 if k > 0 ∗k (k−1)! Y . (4) Y := δ (−k) if k 6 0
5
For k > 0 these distribution are functions with Z ∞ k−1 t1 ϕ(t1 ) dt1 , ϕ ∈ D(R). Y ∗k (ϕ) = (k − 1)! 0 Schwartz ([22, (II, 2; 31)]) defines the Y ∗k for all complex numbers k, but we do not need this here. For the multivariate analogues of these distributions we need some notations. For a set X, x = (x1 , . . . , xr ) ∈ X r , S ⊆ {1, . . . , r} and S ′ := {1, . . . , r} \ S we define xS := (xρ )ρ∈S and identify ′
x = (xS , xS ′ ) ∈ X r = X S × X S , for instance µ = (µ1 , . . . , µr ), 1 := (1, . . . , 1) ∈ Nr µS = (µρ )ρ∈S , µ = (µS , µS ′ ), 1S ∈ NS , 1 = (1S , 1S ′ ). The ring homomorphisms ′ ′ (D+ (R, tρ ), ∗) −→ (D+ (Rr ), ∗) Tρ 7−→ δ(t1 ) ⊗ · · · ⊗ δ(tρ−1 ) ⊗ Tρ ⊗ δ(tρ+1 ) ⊗ · · · ⊗ δ(tr )
induce the ring homomorphism ′ ⊗rρ=1 D+ (R) T1 ⊗ · · · ⊗ Tr
−→ 7−→
′ D+ (Rr ) T1 ⊗ · · · ⊗ Tr .
In particular, the distributions Yρ := δ(t1 ) ⊗ · · · ⊗ δ(tρ−1 ) ⊗ Y (tρ ) ⊗ δ(tρ+1 ) ⊗ · · · ⊗ δ(tr ) ′ are invertible in (D+ (Rr ), ∗) and satisfy
Y ∗µ := Y1∗µ1 ∗ · · · ∗ Yr∗µr = Y (t1 )∗µ1 ⊗ · · · ⊗ Y (tr )∗µr for all µ ∈ Zr . The support of the distributions Y1 , . . . , Yr is contained in Rr>0 and so is that of the powers Y ∗µ . For µ ∈ Nr , ϕ ∈ D(Rr ) and S := S(µ) := supp(µ) := {ρ; µρ 6= 0} equation (4) implies tµSS −1S ⊗ρ∈S Y (tρ ) ⊗ ⊗ρ∈S ′ δ(tρ ), (µS − 1S )! Z ∞S tµSS −1S Y ∗µ (ϕ) = ϕ(tS , 0S ′ ) dtS (µS − 1S )! 0S Y ∗µ =
where tµSS −1S = Z
Y
tµρ ρ −1 ,
(µS − 1S )! =
ρ∈S ∞S
0S
(−) dtS :=
Z
∞
···
0
Z
0
6
Y
(µρ − 1)! and
ρ∈S ∞
(−)
Y
ρ∈S
dtρ .
(5)
′ The invertible elements Y1 , . . . , Yr ∈ D+ (Rr ) induce the C-algebra substitution homomorphism
f=
C[z]Z P
µ∈Zr
fµ z
µ
′ r −→ DP + (R ) 7−→ f (Y ) := µ∈Zr fµ Y ∗µ .
(6)
This map is injective as will be shown in Theorem 2 and is going to be extended to ChziZ in the next theorem. P A convergent power series H = µ∈Nr Hµ z µ can be considered as an analytic functional on Cr ([13, Ch. 4, §5], [17, pp. 64–70]), i.e., as the continuous linear map H: u= where
O(Cr ) P
µ µ∈Nr uµ z
O(Cr ) :=
\
−→ C P 7−→ H(u) := µ∈Nr Hµ uµ BR ⊂ Chzi =
[
BR
R∈Rr>0
R∈Rr>0
is the algebra of entire functions or everywhere convergent power series in z. Its Laplace transform [13, Def. 4.5.2] X tµ b H(t) := Hz ez•t = Hµ , where z • t := z1 t1 + · · · + zr tr , µ! r µ∈N
is contained in O(Cr , exp), i.e., is an entire holomorphic function of exponential type, and the map H=
Chzi µ µ∈Nr Hµ z
P
∼ = 7−→
O(Cr , exp) P µ b H(t) = Hz (ez•t ) = µ∈Nr Hµ tµ!
(7)
is an isomorphism [17, (4.28)]. The notation Hz (ez•t ) signifies that H acts as operator on functions of the variables z1 , . . . , zr , whereas t1 , . . . , tr are considered as parameters. Theorem 2 (Functional calculus). The substitution homomorphism (6) can be extended to algebra homomorphisms X X ′ Chzi → D+ , H= Hµ z µ 7→ H(Y ) := Hµ Y ∗µ , and µ∈Nr
µ∈Nr
′ ChziZ → D+ , a = sµ H 7→ a(Y ) := Y ∗(−µ) ∗ H(Y ) = δ (µ) ∗ H(Y ) = sµ ◦ H(Y ).
(8)
These homomorphisms are injective. The first of them is continuous in the sense that it transforms analytically convergent sequences in Chzi into convergent sequences of distributions, and is indeed the unique continuous linear extension of the substitution homomorphism. ′ The second homomorphism makes D+ (Rr ) a ChziZ -module via a◦T := a(Y )∗T . For a ∈ C[s] ⊂ ChziZ this action coincides with the action by partial differential operators. Each H(Y ), H ∈ Chzi, is indeed a measure , i.e., a continuous linear map on C00 (Rr ). Its support like those of the Y ∗µ is contained in Rr>0 . If ϕ is a continuous function with compact support in K := {t ∈ Rr ; ∀ρ = 1, . . . , r : −Rρ 6 tρ 6 Rρ }, R ∈ Rr>0 , 7
and kϕkK := supt∈K |ϕ(t)| is its maximum norm, then X X Rµ ∗µ 6 kϕkK H Y (ϕ) |H | µ µ µ! r r µ∈N
where
P
µ∈Nr
µ∈N
P µ µ b |Hµ | Rµ! < ∞ since H(t) = µ∈Nr Hµ tµ! is entire. If
H=
X
µ>cw 1
e ∗1 where Hµ z µ ∈ Chziz1 · · · zr then H(Y ) = HY
Y ∗1 = Y1 ∗ · · · ∗ Yr = Y (t1 ) ⊗ · · · ⊗ Y (tr ) and X tµ−1 e ∈ O(Cr ). H(t) := Hµ (µ − 1)!
(9)
µ>cw 1
Proof.
1. For H :=
P
µ∈Nr
Hµ z µ ∈ Chzi and ϕ ∈ C00 (Rr ) we define
H(Y )(ϕ) :=
X
Hµ Y ∗µ (ϕ)
µ∈Nr
and have to show that this series converges absolutely. But for µ ∈ Nr and S := supp(µ) := {ρ; µρ > 0} equation (5) furnishes Y ∗µ (ϕ) =
Z
∞S
0S
tSµS −1 ϕ(tS , 0S ′ )dtS . (µS − 1)!
For the support K and the norm kϕkK of ϕ this implies |Y
∗µ
(ϕ)| 6 kϕkK
Z
RS 0S
RµS tSµS −1 Rµ dtS = S kϕkK = kϕkK . (µS − 1)! µS ! µ!
b Since the Laplace transform H(t) = H is entire we obtain the estimate |H(Y )(ϕ)| 6
X
P
µ
µ∈Nr
|Hµ ||Y ∗µ (ϕ)| 6
µ∈Nr
Hµ tµ! of the analytical functional X
µ∈Nr
|Hµ |
Rµ kϕkK < ∞ µ!
(10)
which implies convergence. It is clear that the map ϕ 7→ H(Y )(ϕ) is linear. The preceding inequality also shows that it is continuous on C00 (Rr ) and thus a measure. The convolution product T1 ∗ T2 is continuous in both variables on the subspace ′ of distributions in D+ (Rr ) whose support is contained in Rr>0 [22, Th.XIII on p.28]. In particular, this applies to the convergent series X H(Y ) = Hµ Y ∗µ , H ∈ Chzi µ∈Nr
and implies that (H1 H2 )(Y ) = H1 (Y ) ∗ H2 (Y ) or that Y 7→ H(Y ) is a ring homomorphism. 8
2. Equation (9) follows at once from (5). P 3. The map H 7→ H(Y ) is injective on Chzi: From µ∈Nr Hµ Y ∗µ = 0 we infer X X ∗1 b 0 = Y ∗1 ∗ Hµ Y ∗µ = Hµ Y ∗(µ+1) = H(t)Y µ∈Nr
µ∈Nr
b where the last equality follows from (9). But then H(t) = 0 for all t ∈ Rr>0 . b is analytic it is identically zero and hence H = 0. Since H
4. Since the Y1 , . . . , Yr and all Y ∗µ are invertible the monomorphism Y 7→ H(Y ) on Chzi can be uniquely extended to a monomorphism on the Laurent power series algebra ChziZ by z −µ H 7→ Y ∗(−µ) ∗ H(Y ) = δ (µ) ∗ H(Y ) = H(Y )(µ) = sµ ◦ H(Y ), µ ∈ Nr . 5. The map Y 7→ H(Y ) is continuous with respect to analytic convergence on Chzi: A sequence H k ∈ Chzi, k = 0, 1, 2, . . ., converges analytically to zero if it is contained in some Banach algebra BT , T ∈ Rr>0 , [9, p. 31] and converges to zero in BT . The norm of H ∈ BT is X kHkT := |Hµ |T µ , hence |Hµ | 6 kHkT T −µ . µ∈Nr
Equation (10) then implies |H(Y )(ϕ)| 6
X
|Hµ ||Y ∗µ (ϕ)| 6
µ∈Nr
X
|Hµ |
µ∈Nr
6 kHkT kϕkK
Rµ kϕkK µ!
X Rµ T −µ µ! r
µ∈N
= kHkT kϕkK exp(R1 T1−1 + · · · + Rr Tr−1 ). This estimate implies the asserted analytic convergence. Since H is the analytic limit of its partial sums we infer that the map H 7→ H(Y ) on Chzi is the unique continuous linear extension of the substitution homomorphism on C[z]. Corollary and Definition 3 (Impulse response and BIBO stability). For a proper rational function H=
X Q(s) = Hµ z µ ∈ P = C(s) ∩ CJzK ⊂ Chzi P (s) r µ∈N
′ and an input u ∈ D+ (Rr ) the output
y := H ◦ u = H(Y ) ∗ u = (H ◦ δ) ∗ u ′ is the unique solution of the partial differential equation P ◦ y = Q ◦ u in D+ (Rr ). For obvious reasons the solution
h := H ◦ δ = H(Y ) =
X
µ∈Nr
µ
Hµ
S(µ) tS(µ)
−1S(µ)
(µS(µ) − 1S(µ) )! 9
⊗ρ∈S(µ) Y (tρ ) ⊗ ⊗ρ∈S(µ)′ δ(tρ )
of P ◦ h = Q ◦ δ is called the fundamental solution or the impulse response of the differential equation. The transfer function H is called BIBO stable if for a bounded continuous function u the output y := H ◦ (uY ∗1 ) with support in Rr>0 is also bounded. Proof. Since P H = Q in ChziZ we conclude that P ◦ (H ◦ u) = (P H) ◦ u = Q ◦ u. Since P =
X
aµ sµ =
µ∈Nr
X
aµ z −µ ∈ C[s] ⊂ C[z, z −1] = C[z]Z ,
µ∈Nr
substitution of Y for z yields X X P (Y ) = aµ Y ∗(−µ) = aµ δ (µ) and µ∈Nr
P ◦y =
X
µ∈Nr
aµ y (µ) =
µ∈Nr
X
µ∈Nr
aµ δ (µ) ∗ y = P (Y ) ∗ y.
′ Since D+ (Rr ), ∗ is an integral domain and P (Y ) 6= 0 the homogeneous equation P ◦y = 0 has only the zero solution. This implies that y := H ◦u is the unique solution ′ of P ◦ y = Q ◦ u in D+ (Rr ). Corollary 4. The equation H ◦ Y ∗1 = H(Y ) ∗ Y ∗1 =
X
Hµ Y ∗µ ∗ Y ∗1
µ∈Nr
=
X
Hµ Y ∗(µ+1) =
µ∈Nr
X
µ∈Nr
Hµ
tµ ∗1 ∗1 b Y = H(t)Y µ!
b shows that BIBO stability of H implies the boundedness of H(t) on Rr>0 .
Remark 5. The difficult injectivity and therefore divisibility of the C[s]-module D′ (Rr ) of distributions according to Ehrenpreis, Malgrange and Palamodov (see also [17]) implies that for every rational function H = Q P ∈ C(s) and every distribution u the differential equation P ◦ y = Q ◦ u has a distributional solution y, in particular P ◦ h = Q ◦ δ. However, in general and in contrast to Corollary 3 the convolution h ∗ u and the equation P ◦(h∗u) = Q◦(δ∗u) make no sense for u with non-compact support and cannot be used to solve P ◦ y = Q ◦ u. The search for fundamental solutions with good properties is one of the basic tasks in the theory of partial differential equations ([11], [12]). Remark 6 (Comparison with the usual approach to multidimensional stabilisation). As mentioned in the introduction most papers on multidimensional stabilisation and especially those of A. Quadrat ([20] and its predecessors) consider an integral domain S of SISO (single input / single output) stable plants with a quotient field K. A SISO system is described therein by a transfer function H ∈ K and a MIMO (multiple input / multiple output) system by a transfer matrix with coefficients in K. Here we restrict the discussion to r-dimensional continuous systems where K is the field C(s) = C(s1 , . . . , sr ) of r-variate complex rational functions.
10
For a systems theoretic interpretation of H ∈ C(s) as input / output map which assigns an output signal y := H ◦ u to an input signal u one has to introduce a suitable class of signals u and then define the action H ◦ u. Otherwise one can only discuss the algebraic or module theoretic part of the systems as it has been done, for instance, in Quadrat’s paper [20] and its predecessors. Theorem 2 of the present paper shows that the action or scalar multiplication H ◦ u = h ∗ u, h := H(Y ) = H ◦ δ, is well-defined if u is a distribution with left bounded support and if H is proper. Thus this theorem is a helpful complement to the algebraic work of Quadrat et al. 1 In contrast, the expression (s1 + s2 )−1 ◦ u (note that s1 +s is not proper) is not 2 naturally defined and shows that not all transfer functions in K = C(s) can be interpreted as input / output maps. It seems that for multidimensional continuous systems theory the description of a plant by an arbitrary rational matrix as input / output map is too general. These problems do not arise if one considers arbitrary IO equations P ◦ y = Q ◦ u, P, Q ∈ C[s], P 6= 0, H = P −1 Q, where H may be an arbitrary rational function. For arbitrary distributional input u there exists a distributional output y with P ◦ y = Q ◦ u. However, an input u with left bounded support does not imply that y has left bounded support too and in general y is not unique, i.e, u 7−→ y is not a map. The preceding considerations suggested the stabilisation theory of IO behaviours instead of IO maps in [18]. Example 7. For α = (α1 , . . . , αr ) ∈ Cr and 1 6cw µ := (µ1 , . . . , µr ) ∈ Nr consider the proper rational function H := (s − α)−µ =
r Y
(sρ − αρ )−µρ =
r Y
zρµρ (1 − αρ zρ )−µρ .
ρ=1
ρ=1
For any k > 0 and a ∈ C the binomial series furnishes −k
(s1 − a)
=
z1k (1
−k
− az1 )
∞ ∞ X X (k + i − 1)! i k+i −k a z1 (−a)i z1k+i = = i!(k − 1)! i i=0 i=0
and hence the representations −k
(s1 − a)
(Y ) =
∞ X (k + i − 1)! i=0
i!(k − 1)!
ai
tk+i−1 tk−1 at1 1 Y1 = 1 e Y1 (k + i − 1)! (k − 1)!
and tµr −1 αr tr tµ1 1 −1 α1 t1 Y1 ∗ · · · ∗ r Yr e e (µ1 − 1)! (µr − 1)! tµ−1 α•t ∗1 e Y with α • t = α1 t1 + · · · + αr tr . = (µ − 1)!
(s − α)−µ (Y ) =
If ℜ(αρ ) < 0 for all ρ the polynomial (s − α)µ is stable or strictly Hurwitz and the rational function H is obviously BIBO stable. 11
The preceding example and partial fraction decomposition imply the well-known and important fact that each univariate proper stable rational function is BIBO stable. Necessary and sufficient conditions for BIBO stability can be derived from the disjoint decomposition ] ] {µ ∈ Nr ; supp(µ) = S} = (1S + NS ), Nr = S
S⊆{1,...,r}
which induces the direct decomposition CJzK = ⊕S CJzS KzS1S . Hence any H ∈ CJzK has a unique finite sum representation X X HS (zS ), where (11) H(z) = Hµ z µ = µ∈Nr
HS (zS ) := µ∈Nr ;
S⊆{1,...,r}
X
Hµ zSµS ∈ CJzS K.
(12)
supp(µ)=S
If H is convergent and u is continuous on Rr this implies X e S (tS ) ⊗ρ∈S Y (tρ ) ⊗ ⊗ρ∈S ′ δ(tρ ) H(Y ) = H S
with
e S (tS ) := (s1S ◦ H b S )(tS ) = H S
X
Hµ
µ∈Nr ; supp(µ)=S
(H ◦ (uY ∗1 ))(t) = (H(Y ) ∗ (uY ∗1 ))(t) =
XZ S
tµSS −1S ∈ O(CS , exp) and (µS − 1S )! tS
0S
e S (τS )u(tS − τS , tS ′ ) dτS . H
Corollary 8. A proper rational function H ∈ P ⊂ Chzi is BIBO stable if the 2r e S (tS )Y ∗1S , S ⊆ {1, . . . , r}, are absolutely integrable over RS . If, in functions H >0 particular, X P (s) := sd + Pµ sµ , dρ > 0 for ρ = 1, . . . , r, and µcw 1
is absolutely integrable, i.e.,
R∞ 0
···
R∞ 0
e )| dτ1 · · · dτr < ∞. |H(τ 12
With the notations of the preceding corollary we derive integral representations of e b the functions H(t) and H(t) as in [13, Equation (4.5.7)]. Let Γρ , ρ = 1, . . . , r, be simple, piecewise smooth closed curves around the origin in C and Γ := Γ1 × · · · × Γr . Then the standard equations Z 1 sµ ds = δ−1,µ , (2πi)r Γ where ds := ds1 · · · dsr , µ ∈ Zr and δ is the Kronecker-δ, hence Z 1 tµ = et•s s−µ−1 ds, t ∈ Cr , µ! (2πi)r Γ hold. Now choose R0 ∈ Rr>0 such that the proper rational function H satisfies X X Q(s) = H(z) = Hµ s−µ ∈ B(R0 )−1 , i.e., |Hµ |(R0 )−µ < ∞, P (s) r r µ∈N
µ∈N
and the curves Γρ such that Γρ ⊂ {sρ ∈ C; |sρ | > Rρ0 } for all ρ. P −µ is strictly proper as in the Corollary 9. 1. If H(s) = Q(s) µ∈Nr Hµ s P (s) = preceding corollary and if the curves Γρ satisfy the conditions above then the integral representations Z X 1 tµ−1 e = et•s H(s) ds and H(t) = Hµ (µ − 1)! (2πi)r Γ µ>cw 1 Z (13) X tµ 1 t•s H(s) b H(t) = Hµ ds = e µ! (2πi)r Γ s1 · · · sr µ>cw 1
are valid.
2. If Q(s) = P (s)
H(s) =
X
S⊆{1,...,r}
X
Hµ s−µ
r
µ∈N , supp(µ)=S
is a (not necessarily strictly) proper rational function and if the conditions on the curves Γρ are satisfied, then for each subset S ⊆ {1, . . . , r} with |S| elements: e S (tS ) = H =
X
Hµ
µ∈Nr ; supp(µ)=S
1 (2πi)|S|
Z
Q
ρ∈S
Γρ
tµSS −1S (µS − 1S )!
etS •sS
X
µ∈Nr , supp(µ)=S
Hµ s−µ dsS .
3 Jury’s Conjecture for Strictly Proper Rational Functions e The preceding integral representation of H(t) permits to prove the generalisation of Jury’s conjecture to bivariate strictly proper rational functions with arbitrary numerators. Let r = 2 in this section. 13
Definition 10. A bivariate polynomial X X X b2j (s1 )sj2 ∈ C[s] = C[s1 , s2 ] P (s1 , s2 ) = b1i (s2 )si1 = Pµ sµ = µ6cw d
j6d2
i6d1
of degree d = (d1 , d2 ) := deg(P ) ∈ N2 is called a very strict Hurwitz polynomial (VSHP for short) if it is cw-unital, strict Hurwitz, and if the two univariate polynomials b1d1 and b2d2 are also strict Hurwitz, i.e., have no zeros in C+ . Theorem 11 (Generalisation of Jury’s conjecture [14, Th. 23 and Remark]). If H(s) = Q(s) P (s) is a bivariate strictly proper rational function with a very strict Hurwitz polynomial P then there are a vector a ∈ R2>0 and a constant C ∈ R>0 such that e |H(t)| 6 Ce−a•t = Ce−(a1 t1 +a2 t2 ) for t ∈ R2>0 .
e In particular, H(t) is absolutely integrable over R2>0 and H is BIBO stable.
Proof. We are going to apply Cauchy’s theorem to the integral representations of Corollary 9. 1. First we have to choose some suitable numbers and sets. (a) Choose R0 ∈ R2>0 such that X P (s) = sd Pµ z d−µ 6= 0 Pd + |{z} µ Rρ0 , ρ = 1, 2 .
(b) Since b1d1 has no zeros in C+ it is possible to choose b a2 , R11 ∈ R>0 with 1 0 R1 > R1 such that X b1i (s2 )z1d1 −i 6= 0 P (s1 , s2 ) = sd11 b1d1 (s2 ) + i R11 , |s2 | 6 R20 , ℜ(s2 ) > −b a2 .
(c) Similarly, since b2d2 has no zeros in C+ we can choose b a1 , R21 ∈ R>0 with 1 0 R2 > R2 such that X b2j (s1 )z2d2 −j 6= 0 P (s1 , s2 ) = sd22 b2d2 (s1 ) + j −b a1 , |s2 | > R21 .
(d) Since P = 6 0 on the compact set (s1 , s2 ) ∈ C2 ; |sρ | 6 Rρ1 , ℜ(s1 ) > 0, ρ = 1, 2 ⊆ Λ2
there are a1 and a2 with 0 < aρ < b aρ , ρ = 1, 2, such that P 6= 0 on M4 := (s1 , s2 ) ∈ C2 ; |sρ | 6 Rρ1 , ℜ(sρ ) > −aρ , ρ = 1, 2 . 14
2. Using these numbers we define several curves. If γ1 and γ2 are two parametrised piecewise smooth curves in C such that the endpoint of γ1 is the initial point of γ2 , their concatenation is denoted by γ1 ∨ γ2 . The following curves are circle arcs, line segments, and concatenations thereof. The orientation of the circle arcs is counter-clockwise and those of the line segments is upwards. A minus sign in front of a curve changes its orientation. For ρ = 1, 2 define Lρ L− ρ L+ ρ L± ρ Γ− ρ Γ+ ρ Γρ ∆ρ Γ ∆
:= {sρ ∈ C, ℜ(sρ ) = −aρ , |sρ | 6 Rρ0 }, := {sρ ∈ C, ℜ(sρ ) = −aρ , Rρ0 6 |sρ | 6 Rρ1 , ℑ(sρ ) < 0}, := {sρ ∈ C, ℜ(sρ ) = −aρ , Rρ0 6 |sρ | 6 Rρ1 , ℑ(sρ ) > 0}, + := L− ρ ∪ Lρ , := {sρ ∈ C, |sρ | = Rρ1 , ℜ(sρ ) 6 −aρ }, := {sρ ∈ C, |sρ | = Rρ0 , ℜ(sρ ) > −aρ }, − + + − ± + := Γ− ρ ∨ L ρ ∨ Γρ ∨ L ρ = Γρ ∪ L ρ ∪ Γρ , − − + − ± := Γρ ∨ Lρ ∨ Lρ ∨ Lρ = Γρ ∪ Lρ ∪ Lρ for ρ = 1, 2, and := Γ1 × Γ2 , := ∆1 × ∆2 ,
(14)
see Figure 1 for a draft. Since Γ is contained in M1 , it is a product of curves as needed in Corollary 9.
+
Lρ
− Γρ
R 1ρ
Lρ
R 0ρ
−a ρ +
Γρ Lρ−
Figure 1: A draft of the curves defined in (14). We write g(s) = g(s1 , s2 ) :=
1 Q(s) 1 et•s H(s) = et•s (2πi)2 (2πi)2 P (s)
for brevity. For fixed t the function g is holomorphic in s where P 6= 0, and in particular in M1 ∪ M2 ∪ M3 ∪ M4 . The compact sets Γ = Γ1 × Γ2 and ∆ are contained in this union and therefore Z Z Z e g(s)ds = H(t) = g(s1 , s2 )ds1 ds2 and Γ1 Γ2 Γ Z Z Z g(s)ds = g(s1 , s2 )ds1 ds2 ∆
∆1
15
∆2
are well-defined. We are going to show that these two integrals coincide by a twofold application of Cauchy’s one-dimensional integral theorem. The integral R g(s)ds decomposes into a sum of integrals over Γ − Γ− 1 × Γ2 ,
± Γ− 1 × L2 ,
+ Γ− 1 × Γ2 ,
− L± 1 × Γ2 ,
± ± + L± 1 × L 2 , L 1 × Γ2 ,
− Γ+ 1 × Γ2 ,
± + Γ+ Γ+ 1 × L2 , 1 × Γ2 . R An analogous decomposition holds for ∆ g(s)ds, where in the integration the curves Γ+ ρ are replaced by Lρ , ρ = 1, 2. We are going to show that the corresponding summands indeed coincide.
• For fixed s1 ∈ Γ− 1 {s1 } × X2 := {s1 } × {s2 ∈ C; |s2 | 6 R20 , ℜ(s2 ) > −a2 } ⊆ M2 and hence g(s1 , s2 ) as a function of s2 on X2 is holomorphic. Since Γ+ 2 and L2 are homotopic curves in X2 Cauchy’s integral theorem implies Z Z g(s1 , s2 )ds2 , g(s1 , s2 )ds2 = G1 (s1 ) := Γ+ 2
L2
and this function is continuous on Γ− 1 due to the continuous dependence on the parameter s1 . Hence Z
+ Γ− 1 ×Γ2
g(s1 , s2 )ds1 ds2 = =
Z
Γ− 1
Z
L2
Z
Γ− 1
Z
Γ+ 2
g(s1 , s2 )ds2
Z g(s1 , s2 )ds2 ds1 =
Γ− 1 ×L2
!
ds1
g(s1 , s2 )ds1 ds2 .
Interchanging the roles of s1 and s2 we likewise obtain Z Z g(s)ds. g(s)ds = L1 ×Γ− 2
− Γ+ 1 ×Γ2
• We apply a similar argument to L± 1
R
+ L± 1 ×Γ2
g(s)ds. Since g(s) is holomorphic
on M4 , a fixed s1 ∈ gives rise to the holomorphic function g(s1 , s2 ) in s2 ∈ X2 . The argument from above also gives Z Z g(s)ds and g(s)ds = L± 1 ×L2
+ L± 1 ×Γ2
Z
± Γ+ 1 ×L2
g(s)ds =
Z
L1 ×L± 2
g(s)ds.
• We use again that g is holomorphic on M4 which, as before, gives Z Z g(s1 , s2 )ds2 for g(s1 , s2 )ds2 = G2 (s1 ) := Γ+ 2
L2
s1 ∈ X1 := {s1 ∈ C; |s1 | 6 16
R11 ,
ℜ(s1 ) > −a1 }.
The function G2 (s1 ) is holomorphic on X1 due to the holomorphic dependence on the parameter s1 ∈ X1 . Since Γ+ 1 and L1 are homotopic in X1 Cauchy’s theorem again implies Z Z G(s1 )ds1 g(s1 , s2 )ds1 ds2 = + Γ+ 1 ×Γ2
=
Γ+ 1
Z
G(s1 )ds1 =
L1
Z
g(s1 , s2 )ds1 ds2 .
L1 ×L2
3. The preceding part of the proof shows that Z Z Z Z 1 Q(s) e H(t) = g(s)ds = g(s)ds = ds1 ds2 . et•s 2 (2πi) P (s) Γ ∆ ∆1 ∆2 The paths ∆ρ lie in {sρ ∈ C; ℜ(sρ ) 6 −aρ }, hence t•s e 6 e−t1 a1 −t2 a2 if t > 0 and s ∈ ∆.
We conclude e H(t) 6
1 (2π)2
Z
∆1
Z
for t = (t1 , t2 ) ∈ R2>0 with C :=
∆2
t•s Q(s) −t•a e P (s) ds1 ds2 = Ce
1 (2π)2
R Q(s) ds. ∆ P (s)
Example 12. The rational functions H1 (s) := (s1 s2 + s1 + 2s2 + 1)−1 ∈ C[s1 , s2 ] and H2 (s) := (s1 s2 + s1 + s2 + 1)−1 = (s1 + 1)−1 (s2 + 1)−1 ∈ C[s1 , s2 ] satisfy the assumptions of the preceding theorem. BIBO stability of H2 follows from Example 7 too. Various tests for Hurwitz polynomials were described already in [14]. That quantifier elimination and cylindrical algebraic decomposition are useful for such tests was observed already thirty years ago in [2, §3]. Consult [5] for a recent survey. A. Dolzmann in Passau has developed the program REDLOG (Reduce Logic) on the basis of REDUCE, which can be used for this purpose.
4 Jury’s Conjecture for Proper Rational Functions In this section we prove Jury’s conjecture for arbitrary proper rational functions in dimension 2. More generally, we derive BIBO stability of a class of proper rational functions once it has been shown for strictly proper rational functions in the class. In dimension 2 this needed preparation was given in the preceding section. ′
Definition and Lemma 13. For S ⊆ {1, . . . , r}, µS ′ ∈ NS and P ∈ C[s] with ′ µ deg(P ) =: d = (dS , dS ′ ) ∈ Nr = NS × NS let coeff(P, sSS′ ′ ) be the polynomial coefficient of P such that the identity X µ µ P = coeff(P, sSS′ ′ )sSS′ ′ ∈ C[sS ][sS ′ ] = C[s] µS ′ ∈NS ′
17
holds. Here we consider P as a polynomial in sS ′ with coefficients in the integral domain A := C[sS ] as in the beginning of Section 2. We define the leading coefficient with respect to sS ′ , d lcS (P ) := coeff(P, sSS′ ′ ) ∈ C[sS ]. It satisfies lcS (P1 P2 ) = lcS (P1 )lcS (P2 ) for P1 , P2 ∈ C[s]. Proof. Let di = deg(Pi ), i = 1, 2. The identity follows from the standard formula for the product of two polynomials, P1 P2 =
X
′
ν∈NS ν6cw d1S ′ +d2S ′
X
′
µ1 ,µ2 ∈NS µ1 +µ2 =ν
1 2 coeff(P1 , sµS ′ )coeff(P2 , sµS ′ ) sνS ′ .
For the transfer of desired BIBO properties from strictly proper to proper rational functions we make the following Assumption 14. Let V ⊆ C[s] be a set of polynomials with the following properties: 1. 1 ∈ V and V is multiplicatively closed and saturated, i.e., P1 |P ∈ V implies P1 ∈ V. 2. For S ⊆ {1, . . . , r} and P ∈ V the leading coefficient lcS (P ) ∈ C[sS ] lies in V too. P µ S) 3. If HS = Q(s µ∈NS Hµ zS ∈ C(sS ) ∩ CJzS K is strictly proper and P (sS ) = P (sS ) ∈ V then the function e S) = H(t
X
µ∈NS ; µ>cw 1S
Hµ
S tµ−1 S (µ − 1S )!
is absolutely integrable over RS>0 . Example 15. Let r = 2. The set V of very strict Hurwitz polynomials, i.e., ( d2 d1 X X b2j (s1 )sj2 ∈ C[s1 , s2 ] = C[s] b1i (s2 )si1 = V = P (s) = i=0
j=0
2
where d = (d1 , d2 ) := deg(P ); P (s) 6= 0 for s ∈ Λ2 = C+ , ) b1d1 (s2 ) 6= 0 for s2 ∈ C+ , and b2d2 (s1 ) 6= 0 for s1 ∈ C+ ,
meets the requirements of the assumption above, in detail: 1. A product of polynomials has no zeros on a set if and only if none of its factors has any zeros there. Together with the observation that lc{1} (P ) = b2d2 , lc{2} (P ) = b1d1 , and the identity of Definition 13, this shows that V is multiplicatively closed and saturated. 1 ∈ V is obvious. 18
2. If f ∈ C[s1 ] ⊆ C[s1 , s2 ] is a strict Hurwitz polynomial in s1 then so it is as a polynomial in s1 and s2 . 3. We proved the third condition in Theorem 11. The multiplicatively closed set V gives rise to the quotient ring C[s]V and the intersection Q Q SBIBO := C[s]V ∩ CJzK = ∈ C(s); proper, P ∈ V P P Q ∈ C(s); deg(Q) 6cw deg(P ), P cw-unital, P ∈ V . = P Theorem 16. Under Assumption 14 each rational function in SBIBO is BIBO stable. P µ Proof. Let H = Q µ∈Nr Hµ z ∈ SBIBO with deg(Q) 6cw deg(P ) =: d, P P = cw-unital and P ∈ V. 1. First we prove H(zS , 0S ′ ) ∈ CJzS K is in SBIBO for S ⊆ {1, . . . , r}. We write P ν ν Q(s) ν6 d ′ coeff(Q, sS ′ )sS ′ = P cw S H(s) = ν ν P (s) ν6cw dS ′ coeff(P, sS ′ )sS ′ P dS ′ −ν ν d z S′ ′ Q ν6 d ′ coeff(Q, sS ′ )zS ′ = Sd ′ = P cw S dS ′ −ν ν zSS′ P ν6cw dS ′ coeff(P, sS ′ )zS ′ with d = (dS , dS ′ ). Thus,
d
d
H(zS , 0S ′ ) =
coeff(Q, sSS′ ′ ) d
coeff(P, sSS′ ′ )
=
coeff(Q, sSS′ ′ ) . lcS (P )
With P also lcS (P ) is cw-unital of degree dS = (dS , 0S ′ ) and d
deg(coeff(Q, sSS′ ′ )) 6cw dS = (dS , 0S ′ ). By the second condition of Assumption 14 the polynomial lcS (P ) belongs to V. This shows H(zS , 0S ′ ) ∈ SBIBO . 2. Consider the representation X HT (zT ), where HT (zT ) = H(z) = T ⊆{1,...,r}
X
Hµ z µ
µ∈Nr ; supp(µ)=T
from (11) and (12). If S ⊆ {1, . . . , r} is arbitrary, zS ′ = 0 and T 6⊆ S then z µ = 0 for all µ with supp(µ) = T and hence X H(zS , 0S ′ ) = HT (zT ). T ⊆S
Möbius inversion according to [1, Theorem 4.18 and p. 153] furnishes X HS (zS ) = (−1)|S\T | H(zT , 0T ′ ). T ⊆S
Since SBIBO is a ring which contains all H(zT , 0T ′ ) by the first part of this proof HS (zS ) ∈ SBIBO for all S ⊆ {1, . . . , r} too. 19
3. By definition HS (zS ) ∈ CJzS K of Assumption 14 implies e S (tS ) = H
Q
ρ∈S
X
zρ is strictly proper. The third condition
Hµ
µ∈Nr ; supp(µ)=S
tSµS −1S ∈ L1 (RS>0 ) (µS − 1S )!
for all S ⊆ {1, . . . , r}. Corollary 8 finally yields that H is BIBO stable.
Theorem 17. A proper bivariate rational function H = Q P ∈ C(s1 , s2 ) with a very strict Hurwitz polynomial P is BIBO stable. Jury’s conjecture states this for Q = 1. Proof. Theorem 11 of the preceding section proved this for strictly proper H = Q P with a VSHP P . Example 15 implies that Theorem 16 is applicable, and this shows the assertion.
5 Discrete BIBO Stability In this section we will discuss BIBO stability in the standard case of partial difference equations, i.e., using the stability decomposition Cr = Λ1 ⊎ Λ2 , Λ2 = {w ∈ C; |w| > 1}r .
(15)
Let T denote the set of all stable polynomials with respect to this decomposition. We define the ring of proper stable rational functions, S := P ∩ C[s]T .
(16)
In [18] the ring C[s]T of stable rational function was used in context with the stabilisation of input/output behaviours. Using the characterisation of proper rational functions (2) on page 5 we can write S as f S= ∈ C[s]; t(λ) 6= 0 for λ ∈ Λ2 , t cw-unital, deg(f ) 6cw deg(t) . t We are going to show that the ring S coincides precisely with the ring of structurally stable rational functions as discussed, for instance, in [10, Def. 3.47] and [16, p. 60]. As in Section 2 we use various rings of power series and the abbreviation −1 zρ = s−1 K = CJzK = CJz1 , . . . , zr K. ρ , hence CJs
In the following considerations we apply Laurent series in r variables and need the following preparations. For vectors R, R− , R+ ∈ Rr>0 with 0 6 R− < R+ , i.e., 0 6 Rρ− < Rρ+ , ρ = 1, . . . , r we consider the open resp. closed polydiscs U (R) := U (R) :=
r Y
ρ=1 r Y
{zρ ∈ C; |zρ | < Rρ } ⊂ {zρ ∈ C; |zρ | 6 Rρ } ,
ρ=1
20
in particular U (1) ⊂ U (1) with 1 = (1, . . . , 1), and the polyrings U (R− , R+ ) := U (R− , R+ ) :=
r Y
ρ=1 r Y
ρ=1
zρ ∈ C; Rρ− < |zρ | < Rρ+ ⊂ zρ ∈ C; Rρ− 6 |zρ | 6 Rρ+ .
If X is an arbitrary subset of Cr a function f is called holomorphic in X if it is defined in an open neighbourhood of X and is holomorphic there. If X is open, f need, of course, be defined on X only. The algebra O(X) of all holomorphic functions on X is a subalgebra of the algebra C 0 (X) of continuous functions P on X. If f is holomorphic on U (R), R ∈ Rr>0 , it has a Taylor series f (z) = µ∈Nr aµ z µ , which is compactly convergent in U (R), i.e., uniformly convergent on compact subsets of U (R). The Banach algebras BR from (3), in particular X X ℓ1 (Nr ) := B1 = f = aµ z µ ∈ CJzK; kf k1 = |aµ | < ∞ r r µ∈N
µ∈N
satisfy
BR ⊂ O(U (R)) ∩ C 0 (U (R)).
Let ℓ∞ (Nr ) denote the Banach space of all bounded multisequences, i.e., X r ℓ∞ := u = (uµ )µ∈Nr = uµ z µ ∈ CN ; ∃M > 0 ∀µ ∈ Nr : |uµ | 6 M . r µ∈N
Corollary 18. The algebra B1 = ℓ1 (Nr ) and the space ℓ∞ (Nr ) are B1 -submodules of CJzK with respect to the convolution multiplication ∗ of CJzK. In other terms if H ∈ ℓ1 (Nr ), u = (uµ )µ∈Nr ∈ ℓi (Nr ) then H ∗ u ∈ ℓi (Nr ), i = 1, ∞. The first resp. second of these properties is called the ℓ1 , ℓ1 - resp. BIBO stability of the transfer function H. Let Sss denote the ring of structurally stable, causal rational functions according to [10, Def. 3.47], i.e., a(z) Sss = ∈ C(z); ∀z ∈ U (1) : b(z) 6= 0 ⊂ C[z]m , b(z) Pr with the maximal ideal m := ρ=1 C[z]zρ .
Remark 19. A causal structurally stable rational function H=
a(z) with b(z) 6= 0 for all z ∈ U (1) and a, b relatively prime b(z)
belongs to B1 = ℓ1 (Nr ) ⊂ O(U (1)) ∩ C 0 (U (1)), i.e., is BIBO stable. 21
If a = 1 the converse is true as is easily seen. In general, BIBO stability of H = implies that b(z) has no zeros in U (1) and that any zero of b in U (1) is also one
a(z) b(z)
of a. It may happen that there are infinitely many such zeros in U (1) \ U (1). Consult [6, Section 1.2, Problem 1 on p. 245], [4, Section 1.4.1], [7, pp. 190–198], [14], [24, pp. 492–496], [23] and the original papers quoted there for further discussions of multidimensional discrete BIBO stable transfer functions and various stability tests. Our next goal is to prove BIBO stability for the ring S = P ∩ C[s]T = C[z]m ∩ C[s]T ⊂ C(s) = C(z)
(17)
of proper stable rational functions (16) with respect to the stabilitiy decomposition (15). It is surprising that the multidimensional Laurent series of the next result are not mentioned in various standard textbooks on functions in several complex variables. Result 20. [25, p. 38, 39] Let R− , R, R+ ∈ Rr>0 with 0 6 R− < R < R+ and let f : U (R− , R+ ) −→ C be a holomorphic function. Then f has a unique Laurent representation X f (z) = aµ z µ µ∈Zr
with compact convergence in U (R− , R+ ). The coefficients can be calculated via r Z Z f (w) 1 ··· dw. aµ = µ−1 2πi w |wr |=Rr |w1 |=R1
Corollary 21. Let f be holomorphic in U (1) such that f (0) 6= 0 and f (z) 6= 0 for 0 < |zρ | 6 1, ρ = 1, . . . , r. Then f (z) 6= 0 for z ∈ U(1) and g :=
1 ∈ O(U (1)) ⊂ B1 = ℓ1 (Nr ). f
Proof. Since f (0) 6= 0, f (z) is not zero for z in a suitable neighbourhood of 0, say z ∈ U (R) for some R ∈ Rr>0 , R < 1. Therefore g = f1 is holomorphic in U (R). Since f (z) 6= 0 for 0 < |zρ | 6 1, ρ = 1, . . . , r, it has no zeros in a slightly bigger open polyring U (0, R+ ) with R+ > 1. Hence g is holomorphic in U (0, R+ ); altogether g ∈ O(U (R)) ∩ O(U (0, R+ )). This implies that g has two compactly convergent representations P a z µ for z ∈ U (R) Pµ∈Nr µ µ g(z) = for z ∈ U (0, R+ ) µ∈Zr bµ z
which coincide on U (0, R) = U (R) ∩ U (0, R+ ) according to Result 20, i.e., bµ = 0 b µ = aµ
for µ ∈ Zr \ Nr and for µ ∈ Nr .
Hence g ∈ O(U (R+ )) ⊂ O(U (1)). The equation f (z)g(z) = 1 in U (0, 1) is also valid on U(1) by continuous extension, in particular f (z) 6= 0 for z ∈ U(1). 22
Theorem 22. The two rings of stable rational functions coincide, i.e., S = Sss , in particular S ⊂ O(U (1)) ⊂ B(1) = ℓ1 (Nr ). Therefore proper stable rational functions are ℓ1 , ℓ1 stable and BIBO stable. Proof. ⊇ : Let H =
a(z) b(z)
∈ Sss . By equation (17) H is proper. Let d ∈ Nr such that d
d >cw degz (a), degz (b). We rewrite H = s a(z) and observe that sd a(z) and t d −1 t := s b(z) are polynomials in s. For λ ∈ Λ2 the vector λ−1 = (λ−1 1 , . . . , λr ) d −1 is in U (1), thus t(λ) = λ b(λ ) 6= 0, therefore H is stable. ⊆ : Now let H = p(s) t(s) ∈ S, i.e., degs (H) 6cw 0, t cw-unital and t(λ) 6= 0 for λ ∈ Λ2 . Set d := degs (t) and write H=
a(z) where a(z) := z d p(s), b(z) := z d t(s) ∈ C[z]. b(z)
Then b(0) 6= 0 because of the unitality of t. Moreover, b(z) 6= 0 for z ∈ C with 0 < |zρ | 6 1, ρ = 1, . . . , r, since z −1 ∈ Λ2 for those z. By Corollary 21, b has no zeros on U (1), i.e., H ∈ Sss .
6 Conclusion We have generalised and proven Jury’s conjectur on continuous BIBO stability of certain bivariate complex rational functions. The extension of this result to higher dimensions is open in two respects: 1. It is not obvious how the set of VSHPs should be defined in dimensions greater than two. 2. For the generalisation which we already tried we could not extend the proof of Theorem 11 with its multiple application of Cauchy’s integral theorem, whereas the results of Section 4 are valid for arbitrary dimension. In [15, Theorem 13 and Remark 3] Kharitonov and Torres Muñoz characterise real bivariate VSHP as those proper and strict Hurwitz polynomials which preserve these properties under a small change of coefficients. In [15, Definition 4] the authors define a class of real multivariate stable polynomials by induction on the number r of indeterminates: The univariate polynomials are just the Hurwitz stable ones and the bivariate polynomials are the VSHP. They call these real polynomials just “stable”, whereas we will call them robustly stable here to make a distinction. Robustly stable polynomials P (s1 , . . . , sr ) of fixed degree deg(P ) = (n1 , . . . , nr ) remain so if their (n1 + 1) · · · (nr + 1) coefficients are subjected to arbitrary small changes [15, Theorem 19]. This suggests to call these polynomials also “structurally stable”. We propose the following
23
Conjecture 23. Let H = Q P ∈ R(s1 , . . . , sr ), r > 3, be a proper rational function whose denominator P is robustly stable in the sense of [15]. Then ′ ′ H◦ : D+ (Rr ) 7−→ D+ (Rr )
is BIBO stable in the sense of the present paper. In context with our proof in Section 3, Theorem 23 of [15] supports this conjecture: If P ∈ R[s1 , . . . , sr ] is robustly stable there is an ε > 0 such that P (s1 , . . . , sr ) 6= 0 for all s ∈ Cr with ℜ(si ) > −ε, i = 1, . . . , r.
References [1] M. Aigner. Combinatorial theory. Springer-Verlag, Berlin, 1979. [2] B. D. O. Anderson, N. K. Bose, and E. I. Jury. Output feedback stabilization and related problems – solution via decision methods. IEEE Trans. Automatic Control, 20:53–66, 1975. [3] N. K. Bose. Trends in multidimensional systems theory. In Bose et al. [6], pages 1–40. [4] N. K. Bose. Two decades of multidimensional systems research and future trends. In Gałkowski and Wood [8], pages 5–27. [5] N. K. Bose. Two decades (1985 – 2005) of Gröbner bases in multidimensional systems. to appear in Radon Series Comp. Appl. Math., 2007. [6] N. K. Bose, J. P. Guiver, E. W. Kamen, H. M. Valenzuela, and B. Buchberger, editors. Multidimensional Systems Theory, volume 16 of Mathematics and its Applications. D. Reidel Publishing Co., Dordrecht, 1985. [7] D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice-Hall, Englewood Cliffs, 1984. [8] K. Gałkowski and J. Wood, editors. Multidimensional Signals, Circuits and Systems. Taylor and Francis, London, 2001. [9] H. Grauert and R. Remmert. Analytische Stellenalgebren, volume 176 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 1971. [10] J. P. Guiver and N. K. Bose. Causal and weakly causal 2-D filters with application in stabilization. In Bose et al. [6], pages 52–100. [11] L. Hörmander. The Analysis of Linear Partial Differential Operators. I. SpringerVerlag, Berlin, 1983. [12] L. Hörmander. The Analysis of Linear Partial Differential Operators. II. SpringerVerlag, Berlin, 1983. [13] L. Hörmander. An Introduction to Complex Analysis in Several Variables. NorthHolland Publishing Co., Amsterdam, 1990.
24
[14] E. I. Jury. Stability of multidimensional systems and related problems. In S. G. Tzafestas, editor, Multidimensional Systems. Techniques and Applications, pages 89–159. Marcel Dekker, New York, 1986. [15] V. L. Kharitonov and J. A. Torres Muñoz. Robust stability of multivariate polynomials. Part 1: Small coefficient perturbations and Part 2: Polytopic coefficient variations. Multidimens. Systems Signal Process., 10(1):7–20 and 21–32, 1999. [16] Z. Lin. Output feedback stabilizability and stabilization of linear nD systems. In Gałkowski and Wood [8], pages 59–76. [17] U. Oberst. Multidimensional constant linear systems. Acta Appl. Math., 20:1– 175, 1990. [18] U. Oberst. Stability and stabilization of multidimensional input/output systems. SIAM Journal on Control and Optimization, 45(4):1467–1507, 2006. [19] U. Oberst and M. Scheicher. A survey of (BIBO) stability and (proper) stabilization of multidimensional input/output systems. to appear in Radon Series Comp. Appl. Math., 2007. [20] A. Quadrat. A lattice approach to analysis and synthesis problems. Math. Control Signals Systems, 18(2):147–186, 2006. [21] H. C. Reddy, P. K. Rojan, and G. S. Moschytz. Two-dimensional Hurwitz polynomials. In R. Jeltsch and M. Mansour, editors, Stability theory, volume 121 of International Series of Numerical Mathematics, pages 75–85. Birkhäuser Verlag, Basel, 1996. [22] L. Schwartz. Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Hermann, Paris, 1966. [23] A. T. Tomerlin and W. W. Edmonson. BIBO stability of d-dimensional filters. Multidimens. Systems Signal Process., 13(3):333–340, 2002. [24] R. Unbehauen. Systemtheorie. Oldenburg, München, 1990. [25] V. S. Vladimirov. Methods of the Theory of Functions of Many Complex Variables. The M.I.T. Press, Cambridge, Mass.-London, 1966.
25