The Geometry of the Reachability Set for Linear ... - Semantic Scholar
Proceedings of the Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2004), Katholieke Universiteit Leuven, Belgium, July 5-9, 2004
The Geometry of the Reachability Set for Linear Discrete–time Systems with Positive Controls Luca Benvenuti and Lorenzo Farina Dipartimento di Informatica e Sistemistica ”A. Ruberti” Universit`a degli Studi di Roma ”La Sapienza” Via Eudossiana 18, 00184 Roma, Italy May 8, 2004
Abstract
where the sum of two cones, as proved in [8], Theorem 3.8, coincides with the set of all finite nonnegative In this paper we study the geometrical properties of combinations of vectors belonging to the two cones. the set of reachable states of a single input discrete– Since the reachability set R (F, g) ⊆ IRn is a cone, time LTI system with positive controls. This set is a then it can be obviously written as cone and it can be expressed as the direct sum of a R (F, g) = S (F, g) ⊕ K (F, g) linear subspace and a proper cone. In order to give a complete geometrical characterization of the reachwhere S (F, g) is the maximal linear subspace conable set, we provide a formula to evaluate the dimentained in R (F, g) and K (F, g) is a proper cone consion of the largest reachable subspace and conditions c tained in the subspace S (F, g) complementary to for polyhedrality of the proper cone. S (F, g) in IRn . The problem of characterizing the geometrical properties of the reachable set R (F, g) of linear sys1 Introduction tem has been studied by Evans and Murthy and Son In this paper we study the geometrical properties in [6, 11] for discrete–time systems and by Brammer, of the set of reachable states xk of a single input Saperstone and Yorke, and Ohta et al. in [4, 10, 7] for continuous–time systems. Evans and Murthy, and discrete–time LTI system of the form: Brammer derived conditions for complete controllak = 0, 1, . . . (1) xk+1 = F xk + g uk bility, i.e. R (F, g) = S (F, g) = IRn for discrete with F ∈ IRn×n , g ∈ IRn when the input function and continuous–time respectively. Ohta et al. prouk is nonnegative for all times k. This situation is vided a simple formula to evaluate the dimension of frequently encountered, for example, in medical, eco- the largest reachable subspace, i.e. the dimension logical, chemical and economical applications where of S (F, g) for single–input continuous–time systems. the controls have a unidirectional influence [2]. More- Saperstone and Yorke, and Son consider complete over, this may also occur in electro-mechanical appli- controllability in the case of bounded inputs for concations (see the examples discussed in [10]). tinuous and discrete–time systems, respectively. It is worth noting that nonnegativity of the input In this paper we deal with single–input discrete– implies that the reachable set is a convex cone. In time systems and provide a complete geometrical fact, the set of states reachable in k steps can be characterization of the reachable set R (F, g), i.e. of written as both S (F, g) and K (F, g). More precisely, we give the k−1 dimension of the largest reachable subspace S (F, g) F k−i−1 g u(i), u(i) ≥ 0 = and provide conditions for polyhedrality of K (F, g). Rk (F, g) = x : x = Some preliminary results have appeared in [5]. Proofs i=0 of theorems are omitted hereafter for the sake of cone g, F g, . . . F k−1 g brevity. In what follows, we will consider the geometrical properties of the reachable set R (F, g) of a reachable pair (F, g) defined as 2 Definitions ∞ A set K ⊂ Rm is said to be a cone provided that Rk (F, g) = R (F, g) = cl (2) αK ⊆ K for all α ≥ 0. If a cone K ⊆ IRm contains an k=1 2 open ball of IRm then it is said to be solid and if K ∩ = cl cone g, F g, F g . . . 1
{−K} = {0} it is said to be pointed. A cone which is Im closed, convex, solid and pointed is said a proper cone. λ5 A cone K is said to be polyhedral if it is expressible as the intersection of a finite family of closed half-spaces. λ1 λ6 The notation cone(v1 , . . . , vM ) indicates the convex λ7 λ2 λ4 λ3 cone consisting of all nonnegative linear combinations Re of vectors v1 , . . . , vM , with M possibly infinite. * Given a square matrix F , pF (λ) is its characterisλ6 λ*1 tic polynomial, σF denotes the set of its eigenvalues * and deg λi , with λi ∈ σF , is the size of the largest λ5 block containing λi in the Jordan canonical form of F . If the matrix F has at least one nonnegative real eigenvalue, then ωF equals the maximal nonnegative real eigenvalue of F ; otherwise ωF = 0. Using the Figure 1: The spectrum of the matrix considered in above definitions, the set σF can be partitioned in example 1. The double cross indicates an eigenvalue λ with deg λ = 2. the following disjoint subsets: (1)
σF = {λi ∈ σF : |λi | > ωF } (2) σF
= {λi ∈ σF : |λi | = ωF and deg λi > deg ωF }
(3) σF
= {λi ∈ σF : |λi | = ωF and deg λi ≤ deg ωF }
(4) σF
= {λi ∈ σF : |λi | < ωF }
so that σF := given a set of
The dimension of the matrix A is µ= deg λi + (deg λi − deg ωF ) (1)
and that of A is
(0) σF
(1) (2) (3) (4) = σF ∪σF ∪σF ∪σF . (k) eigenvalues σF , we define
(2)
λi ∈σF
χ=n−µ=
Moreover,
λi ∈σF
(2) λi ∈σF
(k)
ρ(σF ) = max {|λi |}
deg ωF +
deg λi
(3) (4) λi ∈σF ∪σF
where summation over the empty set is considered to be zero.
(k)
λi ∈σF
(k)
and every eigenvalue λi ∈ σF such that |λi | = (k) (k) ρ(σF ) will be called a dominant eigenvalue of σF . If F is nonderogatory, then w.l.o.g. we can assume the matrix to be in following pseudo–Jordan form ⎞ ⎛ (1) J σF 0 0 0 0 ⎟ ⎜ ⎟ ⎜ (2) 0 J σF ∗ 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (2) F =⎜ 0 0 0 0 J σF ⎟ ⎟ ⎜ ⎟ ⎜ (3) 0 0 0 0 J σF ⎟ ⎜ ⎝ ⎠ (4) 0 0 0 0 J σF (3) A ∗ b = , g= b 0 A
0
triangular matrix of the
.
..
.
Theorem 1 Let the pair (F, g) be reachable. Then the dimension of the largest reachable subspace S (F, g) is deg λi + (deg λi − deg ωF ) µ= (1)
(2)
λi ∈σF
λi ∈σF
Moreover,
σF = {λ1 , λ∗1 , λ2 } (1)
⎞
σF = {λ5 , λ∗5 }
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1 ⎠ λ
σF = {λ3 , λ4 }
0⎟ ..
As stated in the Introduction, we will first present a result which provides the dimension of the largest reachable subspace S (F, g) and how the cone K(F, g) can be generated.
Example 1 For the sake of illustration, consider a matrix F having a spectrum as in figure 1 and a vector g such that the pair (F, g) is reachable. In this case, ωF = λ3 so that
F
1 .. .
Main Results
K (F, g) = cl cone b, Ab, A2 b, . . .
where (k) J σF = diagλi ∈σ(k) (Jdeg λi (λi )) k = 1, 3, 4 F (2) J σF = diagλi ∈σ(2) (Jdeg λi −deg ωF (λi )) F (2) J σF = diagλi ∈σ(2) (Jdeg ωF (λi )) and Jk (λ) is a k × k upper form ⎛ λ 1 ⎜ λ ⎜ ⎜ ⎜ Jk (λ) = ⎜ ⎜ ⎜ ⎝
3
(2) (3)
σF = {λ6 , λ∗6 , λ7 } (4)
In view of the above theorem, the dimension of the largest reachable subspace S (F, g) is µ = 5. 2
It is worth stating the following corollaries which directly follow from the above theorem and characterize the two special cases of R (F, g) = S (F, g) = IRn and R (F, g) = K (F, g).
(0) . . . A4b A2b v∞
Ab
(1) A3b . . . v∞
(0) . . . A 4b A 2b v∞
b Ab
b
(1) A3b . . . v∞
Corollary 2 [6] Let the pair (F, g) be reachable. Then, R (F, g) = IRn , that is µ = n, if and only if the matrix F has no real nonnegative eigenvalues. In this case F = A and g = b .
(0) ˆ A2 , b +v∞ ˆ A2 , Ab + Figure 2: The cones K and K (1) Corollary 3 Let the pair (F, g) be reachable. Then, v∞ (left) and the cone K(A, b) (right). R (F, g) is a proper cone, that is µ = 0, if and only if (2) (1) Example 2 In order to illustrate the previous theoσF σF = ∅ rem, consider the matrices or equivalently if and only if ρ(σF ) ∈ σF and ⎛ ⎞ 1 deg ρ(σF ) ≥ deg λi for each λi such that |λi | = ρ(σF ). ⎜ 1 ⎟ In this case F = A and g = b. ⎟ F = diag (−2, 1, −1, −0.8) , g = ⎜ ⎝ 1 ⎠ Secondly, we present hereafter a Lemma which pro1 vides conditions for polyhedrality of K (F, g). Define In this case we have 2 i−1 Ki (A, b) := cone b, Ab, ⎛ ⎞ A b, . . . ,c A b = 1 S (F, g) = Ri (F, g) A = diag (1, −1, −0.8) , b = ⎝ 1 ⎠ 1 and ˆ (A, b) := K
∞
Ki (A, b) = cone b, Ab, A2 b, . . .
and, for r = 2 the following limits exist ⎛ ⎞ 1 A2k b (0) ⎝ 1 ⎠ lim = v = ∞ k→∞ A2k b 0 ⎛ ⎞ 1 A2k+1 b (1) lim = v∞ = ⎝ −1 ⎠ k→∞ A2k+1 b 0
i=1
so that we have ˆ (A, b) =: K (A, b) K (F, g) = cl K
(4)
Moreover, by definition, K1 (A, b) ⊆ K2 (A, b) ⊆ K3 (A, b) ⊆ . . .
and for N = 1 equalities (5) holds for h = 0, 1 as figand if KN (A, b) = KN +1 (A, b) then K (F, g) = ure 2 clearly shows. Moreover, the figure also makes KN (A, b) = Ki (A, b) ∀i ≥ N . clear that, as shown in the proof of Theorem 1, any First note that if K(F, g) = {0} and ωF = 0 then (0) ˆ A2 , b + v∞ extremal vectors of the form Ai b of K χ the matrix A is nilpotent and A = 0. Hence 2 (1) ˆ A , Ab + v∞ is also an extremal vector of and of K ˆ (A, b) = Kχ (A, b) = K (F, g) = K K(A, b). = cone b, Ab, . . . , Aχ−1 b In what follows we provide the main result of the is polyhedral. Consequently, w.l.o.g. in the sequel we paper, that is a spectral characterization of polyhewill assume ωF > 0. drality of the cone K(F, g). Lemma 1 Let the pair (F, g) be reachable and ωF > 0. Then, K (F, g) is a polyhedral proper cone if and Theorem 4 Let the pair (F, g) be reachable and only if there exists a finite positive integer r such that ωF > 0. Then K (F, g) is a polyhedral proper cone if and only if one of the following sets of conditions the following limits holds: rk+h A b (h) = v∞ = 0 h = 0, . . . , r − 1 lim a1. deg ωF ≤ 2; k→∞ Ark+h b (2) (3) (i) (j) σF are among the r–th in σF exist with v∞ = v∞ for i = j, and there exists a a2. the eigenvalues r for some positive integer r; roots of ω F finite value N such that a3. taking the minimal value of r, no nonzero eigen(h) KN +1 (Ar , Ah b) + cone v∞ = (4) value in σF has an argument which is an integer (5) (h) r h = KN (A , A b) + cone v∞ multiple of 2π/r. for every h = 0, . . . , r − 1.
or 3
b b
A4b
A2b
b
A2b
A4b (0)
A2vb∞ 5
Ab Ab
...
(0) v∞ ...
A4b A6b
(0) v∞
... A5b
3
Ab
A3b
A5b
Ab
A3b
Ab
Figure 3: Planar section of the cone K(A, b) with the plane x1 = 0 for the three cases considered in example 3. b1. deg ωF = 1
phase equal to 2π. Moreover, also condition b4 fails (4) since the dominant eigenvalue of σF is λ2 = −0.9 (4) b2. the dominant eigenvalues of σF are simple; so that s = 2, r˜ = 2 and λ3 = 0.8 has a phase which (2) (3) is an integer multiple of π. Hence, the cone K(A, b) σF are among the r–th b3. the eigenvalues in σF is not polyhedral as shown in the right hand side of r roots of ωF for some positive integer r and the figure 3. (4) dominant eigenvalues of σF are among the s–th (4) roots of ρ(σF )s for some positive integer s; Note that, when χ = 2 the cone K (F, g) is always
polyhedral since obviously any cone in IR2 is polyhedral. In fact, in this case, the conditions of the theorem are always met as one can easily check. From the proof of the above Theorem, immediately follows the next corollaries which provide a geometrical and the corresponding spectral characterization Example 3 In order to illustrate the above theorem, of systems for which the cone K (F, g) is reachable consider the following pair in a finite number of steps. This property is clearly ˆ (A, b), or ⎛ ⎞ equivalent to requiring polyhedrality of K 1 ˆ that the condition K (F, g) = K (A, b) holds. F = diag (1, λ2 , λ3 ) , g=⎝ 1 ⎠ 1 Corollary 5 Let the pair (F, g) be reachable and ˆ (A, b) is a polyhedral proper cone if with λ2 , λ3 real and such that |λ3 | < |λ2 | < 1. Hence, ωF > 0. Then, K A = F , b = g, ωF = 1 and deg ωF = 1. Furthermore, and only if there exists a finite value N such that condition a1 holds and condition a2 holds with r = 1. KN +1 (A, b) = KN (A, b) Lastly, conditions b1, b2 hold and condition b3 holds with r = 1 and s ≤ 2. When λ2 = −0.9 and λ3 = −0.6, then also con- Corollary 6 Let the pair (F, g) be reachable and ˆ (A, b) is a polyhedral proper cone if dition a3 holds. Moreover, as expected, condition b4 ωF > 0. Then, K (4) fails since the dominant eigenvalue of σF is λ2 = and only if the following conditions hold: −0.9 so that s = 2, r˜ = 2 and λ3 = −0.6 has a phase 1. deg ωF = 1 equal to π. Hence, the cone K(A, b) is polyhedral as shown on the left hand side of figure 3. (2) (3) 2. the eigenvalues in σF σF are among the r–th When λ2 = 0.9 and λ3 = −0.8, then condition a3 roots of ωFr for some positive integer r; fails since the eigenvalue λ2 = 0.9 has a phase equal to 2π. By contrast, condition b3 holds with s = 1 3. taking the minimal value of r, no nonzero eigenso that r˜ = 1 and consequently condition b4 holds (4) value in σF has an argument which is an integer since λ3 = −0.8 has a phase which is not an integer multiple of 2π/r. multiple of 2π. Hence, the cone K(A, b) is polyhedral as shown in the middle picture of figure 3. Finally, we conclude the paper, with the special Finally, when λ2 = −0.9 and λ3 = 0.8, then condition a3 fails since the eigenvalue λ3 = 0.8 has a case of K (F, g) simplicial. b4. taking the minimal value of r and s, then no (4) nonzero non dominant eigenvalue of σF has an argument which is an integer multiple of 2π/ r, where r is the least common multiple between r and s.
4
Theorem 7 Let the pair (F, g) be reachable and χ > [12] C. Wende and L. Daming. Nonnegative realiza0. Then K (F, g) is a polyhedral proper cone with χ tions of systems over nonnegative quasi-fields. extremal vectors (simplicial) if and only if the polyActa Mathematicae Applicatae Sinica, 5:252– nomial 261, 1989. (λ − λi ) p(λ) := (3)
(4)
λi ∈σA ∪σA
has all nonpositive coefficients.
References [1] G. P. Barker and R. E. L. Turner. Some observations on the spectra of cone preserving maps. Linear Algebra and its Applications, 6:149–153, 1973. [2] L. Benvenuti, A. De Santis and L. Farina (Eds), Positive Systems. Proceedings of the First Multidisciplinary International Symposium on Positive Systems: Theory and Applications, Rome, Italy, 2003. [3] A. Berman and R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences.Academic Press, New York, 1979. [4] R. F. Brammer. Controllability in linear autonomous systems with positive controllers. SIAM Journal on Control, 10(2):339–353, 1972. [5] L. Farina and L. Benvenuti. Polyhedral reachable set with positive control. Mathematics of Control, Signals, and Systems, 10(4):364–380, 1997. [6] M. E. Evans and D. N. P. Murthy.. Controllability of discrete–time systems with positive controls. IEEE Transactions on Automatic Control, 22(6):942–945, 1977. [7] Y. Ohta, H. Maeda, and S. Kodama. Reachability, observability and realizability of continuoustime positive systems. SIAM Journal on Control and Optimization, 22(2):171–180, 1984. [8] R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, New Jersey, 1970. [9] M. Roitman and Z. Rubinstein. On linear recursion with nonnegative coefficients. Linear Algebra and its Applications, 167:151–155, 1992. [10] S. H. Saperstone and J. A. Yorke. Controllability of linear oscillatory systems using positive controls. SIAM Journal on Control, 9(2):253–262, 1971. [11] N. K. Son. Controllability of linear discrete– time systems with constrained controls in Banach spaces. Control and Cybernetics, 10(1– 2):5–16, 1981. 5
The Geometry of the Reachability Set for Linear Discrete–time Systems with Positive Controls Luca Benvenuti and Lorenzo Farina Dipartimento di Informatica e Sistemistica ”A. Ruberti” Universit`a degli Studi di Roma ”La Sapienza” Via Eudossiana 18, 00184 Roma, Italy May 8, 2004
Abstract
where the sum of two cones, as proved in [8], Theorem 3.8, coincides with the set of all finite nonnegative In this paper we study the geometrical properties of combinations of vectors belonging to the two cones. the set of reachable states of a single input discrete– Since the reachability set R (F, g) ⊆ IRn is a cone, time LTI system with positive controls. This set is a then it can be obviously written as cone and it can be expressed as the direct sum of a R (F, g) = S (F, g) ⊕ K (F, g) linear subspace and a proper cone. In order to give a complete geometrical characterization of the reachwhere S (F, g) is the maximal linear subspace conable set, we provide a formula to evaluate the dimentained in R (F, g) and K (F, g) is a proper cone consion of the largest reachable subspace and conditions c tained in the subspace S (F, g) complementary to for polyhedrality of the proper cone. S (F, g) in IRn . The problem of characterizing the geometrical properties of the reachable set R (F, g) of linear sys1 Introduction tem has been studied by Evans and Murthy and Son In this paper we study the geometrical properties in [6, 11] for discrete–time systems and by Brammer, of the set of reachable states xk of a single input Saperstone and Yorke, and Ohta et al. in [4, 10, 7] for continuous–time systems. Evans and Murthy, and discrete–time LTI system of the form: Brammer derived conditions for complete controllak = 0, 1, . . . (1) xk+1 = F xk + g uk bility, i.e. R (F, g) = S (F, g) = IRn for discrete with F ∈ IRn×n , g ∈ IRn when the input function and continuous–time respectively. Ohta et al. prouk is nonnegative for all times k. This situation is vided a simple formula to evaluate the dimension of frequently encountered, for example, in medical, eco- the largest reachable subspace, i.e. the dimension logical, chemical and economical applications where of S (F, g) for single–input continuous–time systems. the controls have a unidirectional influence [2]. More- Saperstone and Yorke, and Son consider complete over, this may also occur in electro-mechanical appli- controllability in the case of bounded inputs for concations (see the examples discussed in [10]). tinuous and discrete–time systems, respectively. It is worth noting that nonnegativity of the input In this paper we deal with single–input discrete– implies that the reachable set is a convex cone. In time systems and provide a complete geometrical fact, the set of states reachable in k steps can be characterization of the reachable set R (F, g), i.e. of written as both S (F, g) and K (F, g). More precisely, we give the k−1 dimension of the largest reachable subspace S (F, g) F k−i−1 g u(i), u(i) ≥ 0 = and provide conditions for polyhedrality of K (F, g). Rk (F, g) = x : x = Some preliminary results have appeared in [5]. Proofs i=0 of theorems are omitted hereafter for the sake of cone g, F g, . . . F k−1 g brevity. In what follows, we will consider the geometrical properties of the reachable set R (F, g) of a reachable pair (F, g) defined as 2 Definitions ∞ A set K ⊂ Rm is said to be a cone provided that Rk (F, g) = R (F, g) = cl (2) αK ⊆ K for all α ≥ 0. If a cone K ⊆ IRm contains an k=1 2 open ball of IRm then it is said to be solid and if K ∩ = cl cone g, F g, F g . . . 1
{−K} = {0} it is said to be pointed. A cone which is Im closed, convex, solid and pointed is said a proper cone. λ5 A cone K is said to be polyhedral if it is expressible as the intersection of a finite family of closed half-spaces. λ1 λ6 The notation cone(v1 , . . . , vM ) indicates the convex λ7 λ2 λ4 λ3 cone consisting of all nonnegative linear combinations Re of vectors v1 , . . . , vM , with M possibly infinite. * Given a square matrix F , pF (λ) is its characterisλ6 λ*1 tic polynomial, σF denotes the set of its eigenvalues * and deg λi , with λi ∈ σF , is the size of the largest λ5 block containing λi in the Jordan canonical form of F . If the matrix F has at least one nonnegative real eigenvalue, then ωF equals the maximal nonnegative real eigenvalue of F ; otherwise ωF = 0. Using the Figure 1: The spectrum of the matrix considered in above definitions, the set σF can be partitioned in example 1. The double cross indicates an eigenvalue λ with deg λ = 2. the following disjoint subsets: (1)
σF = {λi ∈ σF : |λi | > ωF } (2) σF
= {λi ∈ σF : |λi | = ωF and deg λi > deg ωF }
(3) σF
= {λi ∈ σF : |λi | = ωF and deg λi ≤ deg ωF }
(4) σF
= {λi ∈ σF : |λi | < ωF }
so that σF := given a set of
The dimension of the matrix A is µ= deg λi + (deg λi − deg ωF ) (1)
and that of A is
(0) σF
(1) (2) (3) (4) = σF ∪σF ∪σF ∪σF . (k) eigenvalues σF , we define
(2)
λi ∈σF
χ=n−µ=
Moreover,
λi ∈σF
(2) λi ∈σF
(k)
ρ(σF ) = max {|λi |}
deg ωF +
deg λi
(3) (4) λi ∈σF ∪σF
where summation over the empty set is considered to be zero.
(k)
λi ∈σF
(k)
and every eigenvalue λi ∈ σF such that |λi | = (k) (k) ρ(σF ) will be called a dominant eigenvalue of σF . If F is nonderogatory, then w.l.o.g. we can assume the matrix to be in following pseudo–Jordan form ⎞ ⎛ (1) J σF 0 0 0 0 ⎟ ⎜ ⎟ ⎜ (2) 0 J σF ∗ 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (2) F =⎜ 0 0 0 0 J σF ⎟ ⎟ ⎜ ⎟ ⎜ (3) 0 0 0 0 J σF ⎟ ⎜ ⎝ ⎠ (4) 0 0 0 0 J σF (3) A ∗ b = , g= b 0 A
0
triangular matrix of the
.
..
.
Theorem 1 Let the pair (F, g) be reachable. Then the dimension of the largest reachable subspace S (F, g) is deg λi + (deg λi − deg ωF ) µ= (1)
(2)
λi ∈σF
λi ∈σF
Moreover,
σF = {λ1 , λ∗1 , λ2 } (1)
⎞
σF = {λ5 , λ∗5 }
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 1 ⎠ λ
σF = {λ3 , λ4 }
0⎟ ..
As stated in the Introduction, we will first present a result which provides the dimension of the largest reachable subspace S (F, g) and how the cone K(F, g) can be generated.
Example 1 For the sake of illustration, consider a matrix F having a spectrum as in figure 1 and a vector g such that the pair (F, g) is reachable. In this case, ωF = λ3 so that
F
1 .. .
Main Results
K (F, g) = cl cone b, Ab, A2 b, . . .
where (k) J σF = diagλi ∈σ(k) (Jdeg λi (λi )) k = 1, 3, 4 F (2) J σF = diagλi ∈σ(2) (Jdeg λi −deg ωF (λi )) F (2) J σF = diagλi ∈σ(2) (Jdeg ωF (λi )) and Jk (λ) is a k × k upper form ⎛ λ 1 ⎜ λ ⎜ ⎜ ⎜ Jk (λ) = ⎜ ⎜ ⎜ ⎝
3
(2) (3)
σF = {λ6 , λ∗6 , λ7 } (4)
In view of the above theorem, the dimension of the largest reachable subspace S (F, g) is µ = 5. 2
It is worth stating the following corollaries which directly follow from the above theorem and characterize the two special cases of R (F, g) = S (F, g) = IRn and R (F, g) = K (F, g).
(0) . . . A4b A2b v∞
Ab
(1) A3b . . . v∞
(0) . . . A 4b A 2b v∞
b Ab
b
(1) A3b . . . v∞
Corollary 2 [6] Let the pair (F, g) be reachable. Then, R (F, g) = IRn , that is µ = n, if and only if the matrix F has no real nonnegative eigenvalues. In this case F = A and g = b .
(0) ˆ A2 , b +v∞ ˆ A2 , Ab + Figure 2: The cones K and K (1) Corollary 3 Let the pair (F, g) be reachable. Then, v∞ (left) and the cone K(A, b) (right). R (F, g) is a proper cone, that is µ = 0, if and only if (2) (1) Example 2 In order to illustrate the previous theoσF σF = ∅ rem, consider the matrices or equivalently if and only if ρ(σF ) ∈ σF and ⎛ ⎞ 1 deg ρ(σF ) ≥ deg λi for each λi such that |λi | = ρ(σF ). ⎜ 1 ⎟ In this case F = A and g = b. ⎟ F = diag (−2, 1, −1, −0.8) , g = ⎜ ⎝ 1 ⎠ Secondly, we present hereafter a Lemma which pro1 vides conditions for polyhedrality of K (F, g). Define In this case we have 2 i−1 Ki (A, b) := cone b, Ab, ⎛ ⎞ A b, . . . ,c A b = 1 S (F, g) = Ri (F, g) A = diag (1, −1, −0.8) , b = ⎝ 1 ⎠ 1 and ˆ (A, b) := K
∞
Ki (A, b) = cone b, Ab, A2 b, . . .
and, for r = 2 the following limits exist ⎛ ⎞ 1 A2k b (0) ⎝ 1 ⎠ lim = v = ∞ k→∞ A2k b 0 ⎛ ⎞ 1 A2k+1 b (1) lim = v∞ = ⎝ −1 ⎠ k→∞ A2k+1 b 0
i=1
so that we have ˆ (A, b) =: K (A, b) K (F, g) = cl K
(4)
Moreover, by definition, K1 (A, b) ⊆ K2 (A, b) ⊆ K3 (A, b) ⊆ . . .
and for N = 1 equalities (5) holds for h = 0, 1 as figand if KN (A, b) = KN +1 (A, b) then K (F, g) = ure 2 clearly shows. Moreover, the figure also makes KN (A, b) = Ki (A, b) ∀i ≥ N . clear that, as shown in the proof of Theorem 1, any First note that if K(F, g) = {0} and ωF = 0 then (0) ˆ A2 , b + v∞ extremal vectors of the form Ai b of K χ the matrix A is nilpotent and A = 0. Hence 2 (1) ˆ A , Ab + v∞ is also an extremal vector of and of K ˆ (A, b) = Kχ (A, b) = K (F, g) = K K(A, b). = cone b, Ab, . . . , Aχ−1 b In what follows we provide the main result of the is polyhedral. Consequently, w.l.o.g. in the sequel we paper, that is a spectral characterization of polyhewill assume ωF > 0. drality of the cone K(F, g). Lemma 1 Let the pair (F, g) be reachable and ωF > 0. Then, K (F, g) is a polyhedral proper cone if and Theorem 4 Let the pair (F, g) be reachable and only if there exists a finite positive integer r such that ωF > 0. Then K (F, g) is a polyhedral proper cone if and only if one of the following sets of conditions the following limits holds: rk+h A b (h) = v∞ = 0 h = 0, . . . , r − 1 lim a1. deg ωF ≤ 2; k→∞ Ark+h b (2) (3) (i) (j) σF are among the r–th in σF exist with v∞ = v∞ for i = j, and there exists a a2. the eigenvalues r for some positive integer r; roots of ω F finite value N such that a3. taking the minimal value of r, no nonzero eigen(h) KN +1 (Ar , Ah b) + cone v∞ = (4) value in σF has an argument which is an integer (5) (h) r h = KN (A , A b) + cone v∞ multiple of 2π/r. for every h = 0, . . . , r − 1.
or 3
b b
A4b
A2b
b
A2b
A4b (0)
A2vb∞ 5
Ab Ab
...
(0) v∞ ...
A4b A6b
(0) v∞
... A5b
3
Ab
A3b
A5b
Ab
A3b
Ab
Figure 3: Planar section of the cone K(A, b) with the plane x1 = 0 for the three cases considered in example 3. b1. deg ωF = 1
phase equal to 2π. Moreover, also condition b4 fails (4) since the dominant eigenvalue of σF is λ2 = −0.9 (4) b2. the dominant eigenvalues of σF are simple; so that s = 2, r˜ = 2 and λ3 = 0.8 has a phase which (2) (3) is an integer multiple of π. Hence, the cone K(A, b) σF are among the r–th b3. the eigenvalues in σF is not polyhedral as shown in the right hand side of r roots of ωF for some positive integer r and the figure 3. (4) dominant eigenvalues of σF are among the s–th (4) roots of ρ(σF )s for some positive integer s; Note that, when χ = 2 the cone K (F, g) is always
polyhedral since obviously any cone in IR2 is polyhedral. In fact, in this case, the conditions of the theorem are always met as one can easily check. From the proof of the above Theorem, immediately follows the next corollaries which provide a geometrical and the corresponding spectral characterization Example 3 In order to illustrate the above theorem, of systems for which the cone K (F, g) is reachable consider the following pair in a finite number of steps. This property is clearly ˆ (A, b), or ⎛ ⎞ equivalent to requiring polyhedrality of K 1 ˆ that the condition K (F, g) = K (A, b) holds. F = diag (1, λ2 , λ3 ) , g=⎝ 1 ⎠ 1 Corollary 5 Let the pair (F, g) be reachable and ˆ (A, b) is a polyhedral proper cone if with λ2 , λ3 real and such that |λ3 | < |λ2 | < 1. Hence, ωF > 0. Then, K A = F , b = g, ωF = 1 and deg ωF = 1. Furthermore, and only if there exists a finite value N such that condition a1 holds and condition a2 holds with r = 1. KN +1 (A, b) = KN (A, b) Lastly, conditions b1, b2 hold and condition b3 holds with r = 1 and s ≤ 2. When λ2 = −0.9 and λ3 = −0.6, then also con- Corollary 6 Let the pair (F, g) be reachable and ˆ (A, b) is a polyhedral proper cone if dition a3 holds. Moreover, as expected, condition b4 ωF > 0. Then, K (4) fails since the dominant eigenvalue of σF is λ2 = and only if the following conditions hold: −0.9 so that s = 2, r˜ = 2 and λ3 = −0.6 has a phase 1. deg ωF = 1 equal to π. Hence, the cone K(A, b) is polyhedral as shown on the left hand side of figure 3. (2) (3) 2. the eigenvalues in σF σF are among the r–th When λ2 = 0.9 and λ3 = −0.8, then condition a3 roots of ωFr for some positive integer r; fails since the eigenvalue λ2 = 0.9 has a phase equal to 2π. By contrast, condition b3 holds with s = 1 3. taking the minimal value of r, no nonzero eigenso that r˜ = 1 and consequently condition b4 holds (4) value in σF has an argument which is an integer since λ3 = −0.8 has a phase which is not an integer multiple of 2π/r. multiple of 2π. Hence, the cone K(A, b) is polyhedral as shown in the middle picture of figure 3. Finally, we conclude the paper, with the special Finally, when λ2 = −0.9 and λ3 = 0.8, then condition a3 fails since the eigenvalue λ3 = 0.8 has a case of K (F, g) simplicial. b4. taking the minimal value of r and s, then no (4) nonzero non dominant eigenvalue of σF has an argument which is an integer multiple of 2π/ r, where r is the least common multiple between r and s.
4
Theorem 7 Let the pair (F, g) be reachable and χ > [12] C. Wende and L. Daming. Nonnegative realiza0. Then K (F, g) is a polyhedral proper cone with χ tions of systems over nonnegative quasi-fields. extremal vectors (simplicial) if and only if the polyActa Mathematicae Applicatae Sinica, 5:252– nomial 261, 1989. (λ − λi ) p(λ) := (3)
(4)
λi ∈σA ∪σA
has all nonpositive coefficients.
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