Control and Cybernetics vol. 35 (2006) No. 4. Discrete-time control systems on homogeneous spaces: partition property 1 by. Jens Jordan. UniversitÃ¤t ...

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35 (2006) No. 4

Discrete-time control systems on homogeneous spaces: partition property 1 by Jens Jordan Universität Würzburg, Institut für Mathematik 97074 Würzburg, Germany Abstract: If the system semigroup of a control system is a group, the system has the partition property, i.e., the reachable sets form a disjoint partition in the state space. The converse is not true in general. In this work we give sufficient conditions for the partition property for a family of discrete-time control systems on homogeneous spaces. We apply our results to Inverse Iteration systems on flag manifolds, which are closely related to numerical algorithms. Keywords: discrete-time control systems, inverse iteration, partition property, system semigroup.

1.

Introduction

Given a discrete-time control system, the reachable set of an initial point is defined as the set of points one may obtain in finitely many iteration steps using arbitrary controls. In applications it is necessary to understand the structure of the reachable sets, since they provide fundamental limitations on possible convergence behavior under feedback laws. In general, the reachable sets of a discrete-time control system coincide with orbits of a certain semigroup action on the state space. Therefore, the reachable sets form a partition in the state space if this particular semigroup – the system semigroup – is a group. Unfortunately, in many important applications the system semigroup is not a group. We are interested in conditions, under which the state space is, nevertheless, a disjoint union of reachable sets. One sufficient condition for this is controllability of the system, i.e., the system semigroup acts transitively on the state space. For example San Martin and Mittenhuber showed necessary and sufficient conditions for transitive semigroup actions, see San Martin (1998) and Mittenhuber (2001) for more details. Unfortunately, 1 This work has been supported by the German Research Foundation Grant DFG HE 1858/10-1 “KONNEW”

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in many applications – for example for systems with fixed points – we have naturally more than one reachable set and therefore no controllability. For a family of discrete-time systems on homogeneous spaces, we will propose conditions for the partition property, which are weaker than transitivity of the semigroup action. This setting is motivated by a certain application concerning numerical algorithms. The idea is to interpret iterative algorithms as discrete-time dynamical systems. Interesting examples of such an approach can be found in the work by Ammar and Martin (1986), Batterson and Smillie (1989, 1990), and Shub and Vasquez (1987). In this context, topological and geometric structures naturally appear. Nevertheless, to prove our main result, we only use algebraic properties of the system. Therefore, we are able to state our result in a very general context. The paper is organized as follows. In Section 2 we introduce the partition property in the general context of discrete-time control systems. For a family of such systems on homogeneous spaces we give a sufficient condition for the partition property in Section 3. In Section 4 we will apply our previous results on Inverse Iteration systems on flag manifolds, which are closely related to iterative numerical algorithms, such as QR algorithm. In particular, we show that a certain matrix semigroup - related to the system semigroup - is a group if and only if the Inverse Iteration system has partition property.

2.

Partition property

Following the notation in Sontag (1998) we define a discrete-time control system as a triple Σ = (M, U, f ) containing of a state space M , a set of control parameters U and a transition map f : M × U → M . The system Σ describes the iteration x0 ∈ M,

xt+1 := f (ut , xt ),

ut ∈ U.

(1)

The reachable set R(x) of a point x is the set of all states to which one may steer from x in finitely many iterations, using arbitrary controls in each step. For T ∈ N we define recursively f1 = f and fT : U T × H → M by fT : (u0 , . . . uT −1 , x) 7→ f (uT −1 , fT −1 (u0 , . . . , uT −2 , x)).

(2)

By SΣ we denote the set of all maps one can generate in this way, i.e., SΣ = {F : M → M | ∃ T < ∞, ∃ u ∈ U T : F = fT (u, ·)}.

(3)

SΣ is a semigroup, the so called system semigroup of Σ. The reachable set of a point x ∈ M can be regarded as the orbit R(x) = SΣ · x of the semigroup action SΣ × M → M,

(s, m) → s · m := s(m).

Note that x ∈ R(y) implies R(x) ⊂ R(y).

(4)

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We say that a system Σ has the partition property if the reachable sets form a partition in the state space, i.e., for every x ∈ M there exists y ∈ M , such that x ∈ R(y) and for all x, y ∈ M it is either R(x) = R(y) or R(x) ∩ R(y) = ∅. Note that the partition property implies x ∈ R(x) for all x ∈ M , even if the identity is not an element of the system semigroup. Since the reachable sets are orbits of a semigroup action, a system Σ has the partition property whenever SΣ is a group. The following example shows that the partition property may hold even when the system semigroup is not a group. Example 1 Let M = R, U = R+ and ux x ≥ 0 f (x, u) = 2ux x < 0.

(5)

Note that f (·, u) is bijective for every u ∈ U . Every element of SΣ has the form ux x ≤ 0 F (x) = (6) 2k ux x < 0 with u ∈ U and k ∈ N. In particular, identity does not belong to SΣ and therefore SΣ is not a group. On the other hand, Σ has the partition property, since R(x) = R+ for every x > 0, R(0) = {0} and R(x) = R− for x < 0.

3.

Control systems on homogeneous spaces

This work was motivated by the analysis of the shifted eigenvector algorithms such as Inverse Iteration and QR algorithm. Both methods can be formulated as discrete-time control systems on related homogeneous spaces. It turns out that the reachable sets of both systems are semigroup orbits of the same semigroup. Moreover, for a given matrix the Inverse Iteration system has the partition property if and only if the QR algorithm has the partition property. The proof of these facts is purely algebraic and can be extended to a very general setting. Let G be a group and H be a subgroup of G. The set of cosets G/H := {gH |g ∈ G} is called homogeneous space. Every element x ∈ G/H can be represented in the form x = gH. Note that gH = g˜H if and only if g −1 g˜ ∈ H. Canonically, we define for any subsemigroup S˜ of G a product S˜ × G/H → G/H,

s · gH = sgH.

(7)

Now, let U be a set of control parameters and Φ : U → G be a map, which induces the transition map f : G/H × U → G/H by f (x, u) = Φ(u) · x. In the following we analyze the system Σ = (G/H, U, Φ), respectively the iteration x0 ∈ G/H,

xt+1 := Φ(ut ) · xt ,

ut ∈ U.

(8)

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In this setting the reachable sets are orbits of the semigroup (T ) Y ˜ S := Φ(ut ) | T ∈ N0 , ut ∈ U ,

(9)

t=0

˜ Typically, S˜ is easier to handle than the system i.e., R(x) = {Φ · x | Φ ∈ S}. ˜ semigroup SΣ since S is a subsemigroup of G. Obviously, system Σ has the partition property if S˜ is a group. In the following we show a weaker condition. For that purpose we introduce the normal core of a homogeneous space G/H defined as \ C := gHg −1 . (10) g∈G

Note that C is the largest normal subgroup of G contained in H. In particular, it is gC = Cg for all g ∈ G. Lemma 1 Let Σ = (G/H, U, Φ) be a system of Type (8). ˜ is a subsemigroup of G. a) C S˜ = {cs | c ∈ C, s ∈ S} b) If S˜ is a group then C S˜ is a group. c) For every point x ∈ G/H there holds R(x) = C S˜ · x. Proof. All statements follow from the fact that C is a normal subgroup of G. a) For all c1 s1 , c2 s2 ∈ C S˜ there exist c˜2 ∈ C, such that c1 s1 c2 s2 = c1 c˜2 s1 s2 ∈ ˜ Therefore C S˜ is a semigroup. C S. ˜ b) If S˜ is a group, then C S˜ is a group, since C S˜ = SC. c) Since C is a subgroup of H it is C · gH = gH for all g ∈ G and therefore ˜ R(gH) = S˜ · gH = SCgH = C S˜ · gH. Note that C S˜ is a group if and only if the system semigroup SΣ is a group. Nevertheless, for our applications it is more convenient to deal with C S˜ instead of SΣ . Lemma (1) shows, that in order to analyze the reachable sets of a system of Type (8) we can use more algebraic structure if we regard the reachable sets ˜ It follows that the partition property is already as orbits of C S˜ instead of S. given, if C S˜ is a group. In Section 4 we will give an example where C S˜ is a group but S˜ is not. In the following we give a sufficient condition, under which the partition property implies that C S˜ is a group. Recall that a subset S˜ ⊂ G of a group G generates the subgroup (N ) Y −1 ˜ := hSi si | N < ∞, si ∈ S˜ ∪ S˜ . (11) i=1

˜ is the smallest subgroup of G which contains S. ˜ Moreover, C S˜ is Note that hSi ˜ a group if and only if C S˜ = hC Si.

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˜ is a Theorem 1 Let Σ = (G/H, U, Φ) be a system of Type (8) where H ∩ hSi subgroup of C. The following statements are equivalent: (i) C S˜ is a group. ˜ · x holds. (ii) For all x ∈ G/H the equation R(x) = hSi (iii) System Σ has the partition property. Proof. The implications (i)⇒(iii) and (ii)⇒(iii) are obvious. (iii)⇒(ii) For any s˜ ∈ S˜−1 and x ∈ G/H it is x = s˜−1 s˜ · x ∈ R(˜ s · x). It follows that R(x) = R(˜ s ·x) and therefore s˜·x ∈ S˜ ·x. We conclude that for an arbitrary ˜ i.e., δ = QN s˜t with s˜t ∈ S˜ ∪ S˜−1 there exist s1 , . . . , sN such that δ ∈ hSi, t=1 QN ˜ · x. δ · x = t=1 st · x ∈ S˜ · x and therefore R(x) = S˜ · x = hSi ˜ \ C S. ˜ Since C is (iii)⇒(i) If C S˜ is not a group, then there exists a ∈ hC Si ˜ a normal subgroup of G we can factorize a = c˜δ with c˜ ∈ C and δ ∈ hSi. Supposing that the reachable sets form a partition in G/H, then for every gH ∈ G/H — and in particular for δ −1 H — there exists cs ∈ C S˜ such that a · δ −1 H = cs · δ −1 H.

(12)

Since C is a normal subgroup of G, with C ⊂ H, it follows that sδ −1 ∈ H. ˜ and H ∩ hSi ˜ ⊂ C we have c˘s = δ with c˘ ∈ C. Moreover, since s, δ ∈ hSi ˜ Hence, C S˜ is a group. Therefore, a = c˜c˘s, which is a contradiction to a ∈ / C S.

4.

Partition property of Inverse Iteration

In the following we want to apply our results of the previous section to shifted Inverse Iteration on flag manifolds. Let F be an arbitrary field. With F∗ we denote the multiplicative group of F. A flag V is an increasing sequence of F-linear subspaces {0} $ V1 $ V2 $ . . . $ Vk ⊂ Fn . The type of the flag V = (V1 , . . . , Vk ) is defined by the k-tuple d := (d1 , . . . , dk ) of dimensions di = dimF Vi , i = 1, . . . , k. For any such sequence of integers d = (d1 , . . . , dk ), 1 ≤ d1 < · · · < dk ≤ n, we denote the set of all flags of type d with Flag(d, Fn ). Note that every A ∈ GLn (F) acts on Flag(d, Fn ) via A(V1 , . . . , Vk ) = (AV1 , . . . , AVk ). Via the bijection TV : gHV 7→ (gV1 , . . . , gVk ) we can identify Flag(d, Fn ) with the homogeneous space Flag(d, Fn ) := GLn (F)/HV , where V = (V1 , . . . , Vk ) is any fixed reference flag of type d and HV := {g ∈ GLn (F) | gVi = Vi } is the stabilizer group of V. For a given matrix A ∈ Fn×n we define U = F \ Spec(A) and the map Φ : U → GLn (F), Φ(u) = (A − uI)−1 . Note that for every x ∈ GLn (F)/HV the equation TV (Φ(u) · x) = Φ(u)TV (x) holds.

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The Inverse Iteration system is given by ΣA,d = (Flag(d, Fn ), U, Φ). Obviously, S˜II =

(

N Y

−1

(A − ut I)

)

| N < ∞, ut ∈ F \ Spec(A) .

t=1

(13)

Note that Flag(d, Fn ) is the projective space PFn−1 for d = (1). In this case ΣA,(1) describes the well known Inverse Iteration algorithm. The fixed points of ΣA,(1) coincide with the eigenspaces of A. Another important special case is dc = (1, 2, . . . , n − 1), yielding the complete flag manifold. The fixed points of ΣA,dc coincide with the eigenflags E = (V1 , . . . , Vn−1 ) of A, i.e., AVi = Vi for i = 1, . . . , n − 1. It is known that the dynamics of Inverse Iteration on complete flag manifolds is closely related to the QR algorithm. We refer to Ammar, Martin (1986), Shub, Vasquez (1987), for more details. In applications as above, one is usually interested in the behavior of cyclic matrices, i.e., matrices A ∈ Fn×n , such that there exists a cyclic vector v ∈ Fn such that span(v, Av, . . . , An−1 v) = Fn . For F = R, C the set of cyclic matrices is open and dense in Fn×n . Therefore, cyclicity is a generic assumption on a matrix. The following lemma shows that the conditions of Theorem 1 are satisfied. Lemma 2 Let A ∈ Fn×n be cyclic and Flag(d, Fn ) a flag manifold of type d, such that d1 = 1. The Inverse Iteration system ΣA,d = (Flag(d, Fn ), U, Φ) satisfies the following conditions: a) C = F∗ I. b) HV ∩ hS˜II i ⊂ C for a reference flag V of type d. Proof. As above we identify the set Flag(d, Fn ) with the homogeneous space GLn (F)/HV . For that purpose we choose a reference flag V = (V1 , . . . , Vk ) of type d, such that a nonzero vector v ∈ V1 is a a cyclic vector. a) For any c ∈ C and for all g ∈ GLn (F) we have cgV1 = gV1 . Therefore, every one dimensional vector space is an eigenspace of c. It follows that c ∈ F∗ I. b) Obviously, hS˜II i is a subgroup of the centralizer Z(A) := {B ∈ GLn (F) | AB = BA}. Moreover, since A is cyclic, there is Z(A) = {p(A) | p ∈ F[t], p coprim χA , deg(p) ≤ n − 1}

(14)

where χA is the characteristic polynomial of A (see Fuhrmann, 1996, Proposition Pn−1 6.1.2). Therefore, if X = j=0 αj Aj ∈ HV ∩ hS˜II i and v ∈ V cyclic, then P i k−1 Xv = λv and (α0 − λ)v + n−1 v is a basis of j=1 αi A v = 0. Since v, Av, . . . , A Fn we conclude that X = λI. Applying Theorem 1 we have the following result:

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Theorem 2 Let A be cyclic and Flag(d, Fn ) a flag manifold with d1 = 1. Then the following statements are equivalent: (i) F∗ S˜II is a group. (ii) System Σd,A has the partition property. The following example shows that R∗ S˜II may be a group even if S˜II is not. Example 2 Let F = R and 0 −1 A := . 1 0

(15)

We show that S˜II is not a group. Let (N −1 Y −ut 1 −1 B := ∈ S˜II = 1 1 1 t=1

−1 −ut

) −1 N < ∞, ut ∈ R . (16)

AssumeQB −1 ∈ S˜II , i.e., there exist shift parameters u1 , . . . , uN ∈ R such that N B −1 = t=1 (A − ut I)−1 . Then −1 ! Y N N Y 1 −u −1 t −1 = det(B ) = det (17) 2 + 1 ≤ 1, 1 −ut u t=1 t t=1 which is a contradiction to det(B) = 12 . Hence, S˜II is not a group. On the other hand, the inverse of (A − uI)−1 ∈ S˜II is given by A − uI = (u2 + 1)A−1 A−1 (A + uI)−1 ∈ R∗ S˜II .

(18)

Therefore, R∗ S˜II is a group. Let ΣA,d be the Inverse Iteration system on a flag manifold Flag(d, Fn ) and (A − uI)−1 be an arbitrary element of S˜II . In order to find out if F∗ S˜II is a group one has to find elements α1 , . . . , αn−1 ∈ F \ Spec(A) and r ∈ F∗ such that r

n−1 Y

(A − αt I)−1 (A − uI)−1 = I.

(19)

t=1

In the case F = C this can always be done. Let χA be the characteristic polynomial of A. Since C[z] is an Euclidean ring and since u is not a zero of χA we have the identity χA = (t − u)k − r with k ∈ C[z] and r ∈ C \ {0}. Qn−1 The linear factorization of χA + r gives us χA + r = (t − u) t=1 (t − αt ) for α1 , . . . , αn−1 ∈ C \ Spec(A). Therefore, there holds n−1 Y 1 I = (A − uI) (A − αi I). r t=1

(20)

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We conclude that in the case F = C the Inverse Iteration system ΣA,d always has the partition property. In the case F = R the situation is much more complicated. The following example shows that we cannot expect to have the partition property for all matrices A ∈ Rn×n . Example 3 Let ΣA,d = (PR2 , R \ Spec(A), ΦII ) be the Inverse Iteration system for the cyclic matrix 0 −1 0 A := 1 0 0 . (21) 0 0 0

We show that the identity is not an element of the semigroup R∗ S˜II . By Theorem 2 it follows that ΣA,d fails to have the partition property. Suppose r ∈ R∗ and control parameters u1 , . . . , uN ∈ U such that QN there exist −1 I = r t=0 (A − ut I) . The block structure of A yields the equations

1 0

0 1

N Y −ut =r 1 t=1

−1 −ut

and

1=r

N Y

(−ut ).

(22)

t=0

Comparison of the determinants shows that r2

N Y

t=0

(u2t + 1) = r2

N Y

u2t ,

(23)

t=0

which is a contradiction to r 6= 0.

5.

Conclusion and remarks

We have shown a sufficient condition for partition property for a family of discrete-time control systems on homogeneous space. As an application we obtain a necessary and sufficient condition, for which Inverse Iteration on flag manifolds has the partition property. In particular, for a given matrix, Inverse Iteration on projective space has the partition property if and only if Inverse Iteration on complete flag manifolds – and therefore the QR algorithm – has the partition property. To analyze Inverse Iteration concerning its controllability properties and adherence structure of the reachable sets – as it is done in Helmke and Fuhrmann (2000), Helmke and Wirth (2001), Helmke and Jordan (2002) – it is appropriate to use the specific topological, geometric and algebraic structure of the system. Nevertheless, for the proof of our results stated in the paper we only need purely algebraic properties. It might be possible to apply and adapt those results to other applications of control theory with discrete state spaces, such as coding theory or cryptography.

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