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A Frequency-Domain Approach for Max-Plus Linear Systems
A Frequency-Domain Approach for Max-Plus Linear Systems Ying Shang Department of Electrical and Computer Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026 USA (e-mail: [email protected]). Abstract: This paper surveys recent investigations on a frequency-domain approach to study max-plus linear systems, which can be used to model queueing systems, communication networks, and manufacturing systems. The challenging problem for a well-developed frequencydomain theory of such systems is the lack of inverse operations. This paper proposes a frequencydomain approach by revisiting Kalman’s original realization theory for max-plus linear systems. The main advantage of Kalman’s theory is that the frequency-domain method and the statevariable method merge into a single framework. Moreover, it introduces the concepts of poles and zeros as semimodules instead of point poles and zeros, which cannot be traditionally defined without inverse operations. Moreover, the pole and zero semimodules can characterize the common Petri net components in the solutions to the model matching problem. Keywords: Frequency domains, state-space realization, Discrete-event systems, and model matching problem. 1. INTRODUCTION
components of the solutions to the model matching problem.
The max-plus algebra [Baccelli et al. (1992); Cohen et al. (1999); Heidergott et al. (2005)], the min-plus algebra [LeBoudec et al. (2001)], and the Boolean semiring [Golan (1999)] are characterized as a special algebraic structure, called semiring, which has no additive inverses. Maxplus linear systems are linear systems over the max-plus algebra to model communication networks [LeBoudec et al. (2001)], genetic regulatory networks [DeJong (2002)], transportation networks [DeVries et al. (1998)] and queueing systems [Baccelli et al. (1992)]. Max-plus linear systems evolve with variables taking values in the max-plus semimodules. The fundamental differences between maxplus linear systems and traditional linear systems slow down the progress of a well-established system theory for max-plus linear systems. Nowadays, researchers have studied some fundamental problems for the max-plus linear systems, for example, controllability [Prou et al. (1999)], observability [Hardouin et al. (2010)], and the model reference control problem [Maia et al. (2005)]. Most of these results focus on the time-domain analysis methodology, except for the transfer function representation [Baccelli et al. (1992)]. The challenge for a well-established frequencydomain theory is that defining poles and zeros in the frequency domain requires operation inverses.
2. MATHEMATICAL PRELIMINARIES
This paper surveys recent study on the frequency-domain approach for max-plus linear systems. First of all, poles and zeros are defined as semimodules instead of point poles and zeros. Second, Kalman’s realization theory is generalized to max-plus linear systems. Third, the model matching problem is studied using the pole and zero semimodules instead of residuation theory [Maia et al. (2005)]. Fixed pole/zero structures characterize common Petri net
A monoid R is a semigroup (R, ¢) with an identity element eR with respect to the binary operation ¢. The term semiring means a set, R = (R, ¢, eR , £, 1R ) with two binary associative operations, ¢ and £, such that (R, ¢, eR ) is a commutative monoid and (R, £, 1R ) is a monoid, which are connected by a two-sided distributive law of £ over ¢. Moreover, eR £ r = r £ eR = eR , for all r in R. R = (R, ¢, eR , £, 1R ) is a semifield if and only if (R\{eR }, £, 1R ) is a group, i.e. all of its elements have inverse elements with respect to the £ operator. An idempotent semifield R is a semifield satisfying a ¢ a = a for all a ∈ R. The common example for idempotent semifields is the max-plus algebra, which replaces the traditional addition and multiplication into the max operation and the plus operation, Addition : a ⊕ b ≡ max{a, b}, Multiplication : a ⊗ b ≡ a + b. In max-plus algebra literature, we usually denote it as RMax =(R ∪ {²}, ⊕, ², ⊗, e), where R is the set of real numbers, ² = −∞, and e = 0. Let (R, ¢, eR , £, 1R ) be a semiring, and (M, 2M , eM ) be a commutative monoid. M is called a left R-semimodule if there exists a map µ : R × M → M , denoted by µ(r, m) = rm, for all r ∈ R and m ∈ M , such that the following conditions are satisfied: (1) r(m1 2M m2 ) = rm1 2M rm2 , (2) (r1 ¢ r2 )m = r1 m 2M r2 m,
(3) r1 (r2 m) = (r1 £ r2 )m, (4) 1R m = m, (5) r eM = eM = eR m, for any r, r1 , r2 ∈ R and m, m1 , m2 ∈ M . In this paper, e denotes the unit semimodule. A sub-semimodule K of M is a submonoid of M with rk ∈ K, for all r ∈ R with k ∈ K. A sub-semimodule K of M is called subtractive if k ∈ K and k 2M m ∈ K imply m ∈ K, for m ∈ M . An R-morphism between two semimodules (M, 2M , eM ) and (N, 2N , eN ) is a map f : M → N satisfying (1) f (m1 2M m2 ) = f (m1 ) 2N f (m2 ), (2) f (rm) = rf (m), for all m, m1 , m2 ∈ M and r ∈ R. There are two different kernels for R-semimodule morphisms. The kernel of an R-semimodule morphism f : M → N is defined as Ker f = {x ∈ M |f (x) = eN }. The equivalence kernel is an equivalence relation defined by Kereq f = {(x1 , x2 ) ∈ M × M |f (x1 ) = f (x2 )}. There are two different images for R-semimodule morphisms [Takahashi (1981)]. Given an R-semimodule morphism f : M → N , one image is defined to be the set of all values f (m), m ∈ M , i.e. f (M ) = {n ∈ N |n = f (m), for some m ∈ M }. (1) It is called the proper image of an R-semimodule morphism f . The other image of f is defined as Im f = {n ∈ N |n 2N f (m) = f (m0 ) for some m, m0 ∈ M }. It is called the image of f to distinguish from the proper image. The two images coincide for the module case. For the semimodule case, if the two images are the same, the R-morphism of semimodules f : M → N is called i-regular or image-regular. The Bourne relation, introduced in [Golan (1999), pp. 164], is an equivalence relation for an R-semimodule. If K is a sub-semimodule of an R-semimodule M , then the Bourne relation is defined by setting m ≡K m0 if and only if there exist two elements k and k 0 of K such that m 2M k = m0 2M k 0 . The factor semimodule M/ ≡K induced by ≡K is also written as M/K. If K is equal to the kernel of an R-semimodule morphism f : M → N , then m ≡Ker f m0 if and only if there exist two elements k, k 0 of Ker f , such that m 2M k = m0 2M k 0 . Applying f on both sides, we obtain that f (m) = f (m0 ). Hence this special Bourne relation and the equivalence relation induced by the morphism f satisfy the partial order ≤, i.e. ≡Ker f ≤ ≡f . In general, we do not have ≡f ≤ ≡Ker f for an R-semimodule morphism f . If an R-semimodule morphism f : M → N satisfies ≡f ≤ ≡Ker f , then f is called a steady or k-regular R-semimodule morphism. Given two R-semimodule morphisms f : (A, 2A , eA ) → (B, 2B , eB ) and g : (B, 2B , eB ) → (C, 2C , eC ), the sequence f
g
(2) A −−−−→ B −−−−→ C is called an exact sequence if Im f = Ker g. Because Im f is not the same as the proper image f (A) for the
R-semimodule morphism f , the sequence is said to be a proper exact sequence if f (A) = Ker g. For the module case, Im f is always the same as the proper image f (A), so every exact sequence of modules is also proper exact. 3. MAX-PLUS LINEAR SYSTEMS Max-plus linear systems are described by the following equations: x(k + 1) = A x(k) ⊕ B u(k), y(k) = C x(k) ⊕ D u(k), (3) where x is in the state semimodule X ∼ = RnMax , y is in p the output semimodule Y ∼ = RMax , and u is in the input r . A : X → X, B : U → X, semimodule U ∼ R = Max C : X → Y and D : U → Y are four R-semimodule morphisms. RMax [z] denotes the polynomial semiring with coefficients in the max-plus algebra RMax , and ΩX, ΩU , and ΩY denote the polynomial RMax [z]-semimodules of states, inputs, and outputs, respectively. The operator ΩX is an alternative notation for the polynomial RMax [z]semimodules X[z] of states. For instance, given a sequence of states, {· · · , x(−2), x(−1), x(0), x(1), x(2), · · · , }, ΩX = x(0) ⊕ x(1)z ⊕ x(2)z 2 ⊕ · · · ⊕ x(k)z k , for a finite k ∈ N, is isomorphic to a sequence of states starting from the time instant 0 to the future. Let RMax (z) denote the set of formal Laurent series in z −1 , with coefficients in R. In this manner, let X(z) denote the set of formal Laurent series in z −1 , with coefficients in X. We define U (z) and Y (z) similar as X(z). The symbol z −1 is similar as the γ symbol in the dioid of formal power series in the event and time variables [Cohen et al. (1999)]. The transfer function G(z) : U (z) → Y (z) of the system in Eq. (3) is G(z) = C(z −1 A)∗ Bz −1 ⊕ D = CBz −1 ⊕ CABz −2 ⊕ · · · ⊕ D. (4) The star operator A∗ for an n × n matrix mapping A : X → X is defined as A∗ = In×n ⊕ A ⊕ · · · ⊕ An ⊕ · · · . (5) where In×n denotes the identity matrix mapping from X to X. Without loss of generality, we assume D = 0 for a given transfer function G(z) : U (z) → Y (z) in this paper. 4. POLE AND ZERO SEMIMODULES 4.1 Pole Semimodules This section defines two different pole semimodules, the pole semimodule of output type for the transfer function G(z) : U (z) → Y (z) as G(ΩU ) , G(ΩU ) ∩ ΩY and the pole semimodule of input type as XO (G(z)) =
(6)
ΩU . (7) G−1 (ΩY ) ∩ ΩU The pole semimodule of input type can be understood as inputs without poles producing outputs with poles. XI (G(z)) =
The pole semimodule of output type can be understood as outputs with poles produced by inputs without poles. The poles in the output come from the poles of the given plant, therefore, it helps the discoveries of the poles in the plant transfer function. Moreover, pole semimodules can generate the state matrix A for a system over a semiring R, which is related to the eigenvalues or point poles of the given system. In the module case, each pole module has been used by the preference of the researchers, because XI (G(z)) is isomorphic to XO (G(z)). However, for the semimodule case, there exists an R[z]-semiisomorphism between XI (G(z)) and XO (G(z)) instead, i.e. an unit kernel R[z]-semimodule epimorphism. Lemma 1. (Shang (2010)). Given a transfer function G(z) : U (z) → Y (z) and the pole semimodules of input and output type as shown in Eq. (6) and Eq. (7), there exists an R[z]-semimodule semiisomorphism G(z) from XI (G(z)) to XO (G(z)). 4.2 Zero Semimodules This section extends the concepts of the zero modules and the extended zero modules introduced by Wyman and Sain in [Wyman et al. (1981)] to the semimodule case. The zero semimodule of input type for the transfer function G(z) : U (z) → Y (z) is defined as ZI (G(z)) =
ΩY ∩ G(U (z)) . ΩY ∩ G(ΩU )
(8)
Intuitively, the zero semimodule of input type consists of the polynomial outputs produced by the inputs with poles. The poles of the inputs are canceled by the zeros of the plant, hence, the zero semimodule ZI (G(z)) leads to the discovery of the plant’s zeros. In the zero semimodule ZI (G(z)), the polynomial outputs produced by the inputs without poles are removed, because they cannot discover the plant’s zeros. The zero semimodule of output type for a given transfer function G(z) : U (z) → Y (z) is defined as G−1 (ΩY ) . (9) ZO (G(z)) = −1 G (ΩY ) ∩ (Ker G ⊕ ΩU ) Intuitively, the zero semimodule of output type consists of the inputs with poles, which produce outputs without poles. The poles in the inputs are canceled by the zeros of the plant, therefore, the zero semimodule ZO (G(z)) leads to the discoveries of the plant’s zeros. The zero semimodule of output type removes the kernel of the transfer function G(z) and the polynomial inputs because they will not help to discover the zeros of the plant. In the module case, the zero modules of input and output type are isomorphic forms of each other; however, that is not the case for the zero semimodules of input and output type. There exists an R[z]-semimodule epimorphism, that is, a surjective R[z]-morphism, instead of an R[z]isomorphism from the zero semimodule of output type to the zero semimodule of input type. Lemma 2. (Shang et al. (2009)). Given a transfer function G(z) : U (z) → Y (z) and the zero semimodules of input and output type as shown in Eq. (8) and Eq. (9),
G(z )
U (z )
p
i W U
Y (z )
G # (z )
GY
Fig. 1. The Kalman input/output map G# (z). there exists an R[z]-semimodule epimorphism from the zero semimodule of output type to the zero semimodule of input type. 4.3 The Γ-Zero and Ω-Zero Semimodules Given a transfer function G(z) : U (z) → Y (z), the Γ-zero semimodule is defined as ZΓ (G(z)) =
G−1 (ΩY ) . G−1 (ΩY ) ∩ ΩU
(10)
When Ker G is unit, ZΓ (G(z)) is identical to ZO (G(z)). The Ω-zero semimodule ZΩ (G(z)) of the transfer function G(z) : U (z) → Y (z) is given by ZΩ (G(z)) =
ΩY . ΩY ∩ G(ΩU )
(11)
This R[z]-semimodule is finitely generated, because ΩY is finitely generated. If the proper image G(U (z)) of G(z) is equal to Y (z), then the zero semimodule of input type ZI (G(z)) is equal to the Ω-zero semimodule of G(z). The Ω-zero semimodule differs from the zero semimodule of input type when the cokernel of G(z) is not a unit semimodule. In this case, ZI (G(z)) is a sub-semimodule of ZΩ (G(z)), and there exists a natural inclusion from ZI (G(z)) to ZΩ (G(z)), with the cokernel as Ω(G(z)) = ΩY /{ΩY ∩ G(U (z))}. 5. GENERALIZATION OF KALMAN’S REALIZATION THEORY This section generalizes the Kalman’s realization theory from systems over fields in [Kalman et al. (1969)] to systems over semirings. Kalman’s realization theory characterizes a canonical realization by the property that is reachable from the unit element and observable with respect to the unit input. A realization (A, B, C) of a transfer function G(z) is canonical if and only if it is both reachable from the unit element and observable with respect to the unit input. In other words, if a transfer function can be factored through X by an onto map g and a one-to-one map h, then X is a canonical realization for the given transfer function. If g is an onto map, but h is not a oneto-one map, then X is called a reachable realization. On the other hand, if h is a one-to-one map, but g is not an onto map, then X is called an observable realization. The Kalman input semimodule is defined as the set of polynomial inputs, ΩU = U [z]. The Kalman output semimodule is defined as the set of strictly proper outputs, denoted as a factor semimodule induced by the Bourne relation, ΓY = Y (z)/ΩY . The factor semimodule ΓY consists of equivalence classes in which two elements y1 (z) and y2 (z) are equivalent if and only if they contain the same
6. MODEL MATCHING PROBLEM
G # (z )
W U
GY
f
~ B
W U G - 1 (W Y ) I W U
g
~ C
l
h
X G (W U ) G (W U ) I W Y
The model matching problem (MMP) for max-plus linear systems can be described as follows. Given two transfer functions, T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z), one is to find another transfer function, M (z) : C(z) → U (z), or given T (z) and M (z), one is to find another transfer function P (z), such that the following model matching equation is satisfied, T (z) = P (z) M (z).
(12)
Fig. 2. The generalized Kalman realization diagram. 6.1 Fixed Poles of Model Matching Problem strictly proper component. The factor semimodule can be intuitively understood as Y (z) without the polynomial component ΩY . The Kalman input semimodule contains sequences of input signals starting at a finite negative time and continuing up to the time zero. The Kalman output semimodule contains sequences of output signals from the time instant 1 into the future. The Kalman input/output map G# (z) can be constructed from the commutative diagram shown in Figure 1, where i is the insertion and p is the natural projection induced by the Bourne relation. If we consider the Kalman input/output map G# (z) : ΩU → ΓY , then we can obtain the commutative diagram shown in Figure 2. X is the state semimodule. The e : ΩU → X and C e : X → ΓY are defined mappings B by e · u) = ABu; B(z Cxz −1 ⊕ C(Ax)z −2 ⊕ C(A2 x)z −3 ⊕ · · · e . C(x) = ΩY For the pole module of output type XO (G(z)), the mapG(u) ping h is defined by the action h : u 7→ G(ΩU )∩ΩY , where u ∈ ΩU . The mapping l is defined by the action G(u) G(u) l : G(ΩU )∩ΩY 7→ ΩY . Therefore, l is an induced map from the identity map Id. For the pole semimodule of input type XI (G(z)), the mapping f is a natural projection, i.e. u f : u 7→ G−1 (ΩY )∩ΩU for u ∈ ΩU . The mapping g is defined by the action g :
u G−1 (ΩY )∩ΩU
7→
G(u) ΩY .
From the Kalman’s realization diagram in Figure 2, the pole semimodule of output type XO (G(z)) is a reachable realization of the Kalman input/output map, because G# (z) is an onto map from ΩU to XO (G(z)). When the map l is one-to-one, the pole semimodule of output type becomes a canonical realization. The pole semimodule of input type XI (G(z)) is also a reachable but not observable realization of G# (z), because f is onto and g is not oneto-one. Using Lemma 1, the pole semimodule of input type becomes isomorphic to the pole semimodule of output type, i.e. G(z) becomes an isomorphism, if and only if the Kalman input/output map G# (z) is steady or k-regular. Therefore, Kalman’s original realization theory for systems over fields is generalized to max-plus linear systems, even there is no “subtraction” operation at present corresponding to the max operation.
We are given two transfer functions T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z). To study the solution M (z) to the model matching problem, an R[z]-semimodule P (T, P ) is defined as follows: P (T, P ) =
T (ΩC) ⊕ P (ΩU ) . P (ΩU )
(13)
The semimodule P (T, P ) in Eq. (13) is called the fixed pole semimodule for the solution M (z) to the model matching equation T (z) = P (z)M (z). The same terminology, fixed poles, is adopted from systems over a field in [Conte et al. (1988)], in which the fixed pole module behaves as a factor module. The R[z]-semimodule P (T, P ) is a fixed pole structure because it has a direct relation with the pole semimodule of output type for the solution M (z). Proposition 3. (Shang et al. (2009)). For the MMP with two known transfer functions T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z) and a solution to M (z) : C(z) → U (z) to the model matching problem, there exists a surjective R[z]-semimodule morphism, from the pole semimodule of output type XO (M ) of the solution M (z) to the R[z]semimodule P (T, P ). Theorem 4. (Shang et al. (2009)). Consider two transfer functions T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z), and a solution M (z) : C(z) → U (z) to the model matching problem, there exist R[z]-semimodules Z1 and P1 , and R[z]-semimodule morphisms α, β, φ, and ψ such that the following three sequences are exact: p
i
e → Z1 − → ZI (P ) − → ZI ([T P ]) → e; β
α
e → XO (P ) − → XO ([T P ]) − → P1 → e; φ
ψ
Z1 − → P (T, P ) − → P1 → e,
(14) (15) (16)
where Z1 and P1 are defined as (T (ΩC) ⊕ P (ΩU )) ∩ ΩY , P (ΩU ) ∩ ΩY T (ΩC) ⊕ P (ΩU ) . P1 = (T (ΩC) ⊕ P (ΩU )) ∩ ΩY ⊕ P (ΩU )
Z1 =
(17) (18)
e denotes the identity semimodule. The map φ has a unit kernel if P (ΩU ) is subtractive, and then the last sequence becomes φ
ψ
e → Z1 − → P (T, P ) − → P1 → e, which is a short exact sequence.
(19)
Proposition 3 states the relationship between the fixed pole semimodules and solutions to MMP. In particular, for the MMP with unknown controller M (z), an R[z]epimorphism P (z) : XO (M ) → P (T, P ) is established in the proof of Proposition 3. Therefore, we have the following short exact sequence
Proposition 5 states the relationships between the fixed zero semimodules and the extended zero semimodules of the solutions to the model matching problem. In particular, for the model matching problem with an unknown controller M (z), we have the exact sequence: βΩ (z)
i
e → C(M ) − → ZΩ (M (z)) −−−→ Z(T, P ) → e,
P (z)
α
e → C(M ) − → XO (M ) −−−→ P (T, P ) → e,
(20)
with C(M ) = Ker P (z), which is called the inessential pole semimodule of the solutions to the MMP. The fixed pole semimodule helps us discover essential components in the solutions to MMP, which will be illustrated later with a discrete-event system example. 6.2 Fixed Zeros of Model Matching Problem Define an R[z]-semimodule Z(T, P ) as follows: Z(T, P ) =
T (ΩC) ⊕ P (ΩU ) . T (ΩC)
(21)
The R[z]-semimodule Z(T, P ) is called the fixed zero semimodule for the solution M (z) to the model matching equation T (z) = P (z)M (z). In the next theorem, we will establish a relation between the fixed zero semimodule Z(T, P ) and the Ω-zero semimodule of the solution M (z) to the MMP. Proposition 5. (Shang et al. (2007)). If we are given two transfer functions T (z) : C(z) → Y (z), P (z) : U (z) → Y (z), and the solution M (z) : C(z) → U (z) satisfying the model matching equation T (z) = P (z)M (z), there exists an R[z]-epimorphism βΩ (z) between the Ω-zero semimodule of M (z), ZΩ (M (z)), and the fixed zero semimodule, Z(T, P ): βΩ (z)
ZΩ (M (z)) −−−−→ Z(T, P ) −−−−→ e.
(22)
Theorem 6. (Shang et al. (2007)). If we are given two transfer functions T (z) : C(z) → Y (z), P (z) : U (z) → Y (z), and the solution M (z) : C(z) → U (z) to the model matching equation T (z) = P (z)M (z), there exist R[z]semimodules Z1 and P1 , which are defined below, such that the following three sequences are exact: p
i
e → Z1 − → ZΩ (T (z)) − → ZΩ ([T (z) P (z)]) → e; (23)
where C(M ) = Ker βΩ (z), which is called the inessential zero semimodule of the solutions to the MMP. 7. A DISCRETE EVENT SYSTEM APPLICATION This section uses a discrete event system example to illustrate the fixed zero semimodule characterizes the common Petri net components in the solutions to the model matching problem. The Petri net models for the four-machine manufacturing system (top) and the twomachine manufacturing system (bottom) are given in Fig. 3. y2 y1 x1
u1
u1
x2
φ
ψ
where (T (ΩC) ⊕ P (ΩU )) ∩ ΩY , Z1 = T (ΩC) ∩ ΩY T (ΩC) ⊕ P (ΩU ) . P1 = (T (ΩC) ⊕ P (ΩU )) ∩ ΩY ⊕ T (ΩC) e denotes the identity semimodule. The morphism φ has a unit kernel if T (ΩC) is subtractive, in which the last sequence becomes the following short exact sequence, φ
ψ
e → Z1 − → Z(T, P ) − → P1 → e.
x1
y1
Fig. 3. The Petri net realizations for the two manufacturing systems. The control problem is to design a compensator such that the manufacturing system with four machines has the same processing time as the manufacturing system with two machines. The discrete event systems can be modeled as a system over the max-plus algebra. The plant transfer function P (z) : U (z) → Y (z) is · P (z) =
¸ z −3 ⊕ z −4 ² . z −3 ²
The reference transfer function T (z) : C(z) → Y (z) is obtained as the following form: ·
e → XO (T (z)) − → XO ([T (z) P (z)]) − → P1 → e; (24) Z1 − → Z(T, P ) − → P1 → e, (25)
x4
y2
β
α
x3
x2
T (z) =
¸ z −1 ⊕ z −2 ² . z −1 ²
In the semimodule case of the fixed pole structure, the equivalence classes cannot be constructed by simply removing T (ΩC) from T (ΩC) ⊕ P (ΩU ) because of the Bourne equivalent relation. Instead, two elements y1 = T (z)cp ⊕ P (z)up and y2 = T (z)˜ cp ⊕ P (z)˜ up in T (ΩC) ⊕ P (ΩU ) are equivalent, if and only if y1 ⊕l1 = y2 ⊕l2 , where l1 and l2 in T (ΩC). By tedious mathematical manipulations, one can show that y1 and y2 are equivalent if and only if the first rows in u1p and u ˜1p have the same coefficients 0 1 for z and z . Due to limited space, the detail derivations are omitted here. In other words, the representative of the equivalent classes in the fixed zero semimodule is
· P (z)up = P (z)
¸ a0 ⊕ a1 z ⊕ wp1 , , wp2 ,
where wp1 is an arbitrary polynomial element in ΩRMax with the order of z higher than 1 and wp2 is an arbitrary polynomial element in ΩRMax . In order to obtain the fixed zero semimodule, the representative in the quotient equivalence classes are P (z)up by removing vectors with the first element which has the order of z greater than −2 and with the second element which has the order of z greater than −1. The fixed zero semimodule Z(T, P ) of Eq. (21) characterize a matrix Af as · ¸ ² ² Af = . e² This matrix is connected to the state space realizations of the solutions to the model matching problem. For examples, any two solutions M1 (z) and M2 (z) to the model matching equation P (z)M (z) = T (z) and construct the realization matrices A1 and A2 for them. For instance, two solutions M1 (z) and M2 (z) are given as follows: · 2 ¸ · 2 ¸ z ² z ² M1 (z) = , M2 (z) = . ² z z2 z2 Therefore, the Ω-zero semimodules of M1 (z) and M2 (z) generate two A matrices for these two controllers: " A1 =
# ² ² ² ² e ² ² ² , A2 = ² ²e² ²
² ² ² ²
² ² . ² ²
² ² ² e
Moreover, we can model these two systems by Petri nets shown in Fig. 4. We find out that two Petri net realizations f1 (z) = z −3 M1 (z) and M f2 (z) = z −3 M2 (z) contain for M the same set of components marked in the dashed boxes, which are generated by the essential matrix Afixed . ~ Realization for M 1 ( z ) : u1
x1
y1
u2
x2
x3
y2
y1 u ~ Realization for M 2 ( z ) : 1
x1
x2 y2
u2
x3
x4
f1 (z) and M f2 (z). Fig. 4. The Petri net realizations for M 8. CONCLUSION This paper is a summary of the frequency-domain analysis for max-plus linear systems, including the concepts of poles and zeros as semimodules instead of points, a generalization of Kalman realization theory, and the model matching problem in controller synthesis. The main advantage of Kalman’s theory is that the frequency-domain method and the state-variable method merge into a single framework. Moreover, it introduces the concepts of poles and zeros
as semimodules instead of point poles and zeros, which cannot be traditionally defined without inverse operations. Moreover, the pole and zero semimodules can characterize the common Petri net components in the solutions to the model matching problem. These results in this paper can provide an alternative approach in the frequency-domain analysis of max-plus linear systems. REFERENCES Baccelli, F.L., Cohen, G., Olsder,G.J., and Quadrat, J.P. (1992) Synchronization and Linearity: An Algebra for Discrete Event Systems. John Wiley and Sons, New York. Cohen, G., Gaubert, S., Quadrat, J.P. (1999) Max-plus algebra and system theory: where we are and where to go now. Annual Reviews in Control, 23, 207-219. Conte, G., Perdon, A.M., and Wyman, B.F. (1988) Fixed poles in transfer function equations. SIAM J. Control and Optimization, 26, 2, 356-368. De Jong, H. (2002) Modeling and simulation of genetic regulatory systems: a literature review, Journal of Computational Biology, 9, 1, 67-103. De Vries, R., De Schutter, B., and De Moor, B. (1998) On max-algebraic models for transportation networks. Proceedings of the International Workshop on Discrete Event Systems, 457-462. Golan, J.S. (1999) Semirings and Their Applications. Kluwer Academic Publishers, Boston. Hardouin, L., Maia, C.A., Cottenceau, B, and Lhommeau, M. (2010) Observer Design for (max,plus) Linear Systems, IEEE Transactions on Automatic Control, 55, 2, 538-543. Kalman, R.E., Falb, P.L., and Arbib, M.A. (1969) Topics in Mathematical System Theory. McGraw-Hill, New York, 1st edition. Heidergott, B., Jan Olsder, G. and Van Der Woude, J. (2005) Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications, Princeton University Press, Princeton, New Jersey, 1st edition. Le Boudec, J.-Y. & Thiran, P. (2002) Network Calculus. Springer-Verlag, New York., 1st edition. Maia, C.A, Hardouin,L., Santos-Mendes, R. and Cottenceau, B.(2005) On the Model Reference Control for Max-Plus Linear Systems. Proceedings of the 44th IEEE Conference on Decision and Control, pp. 7799-7803. Prou, J.-M., and Wagneur, E. (1999) Controllability in the max-algebra. KYBERNETIKA. 35, 1, 13-24. Shang, Y and Sain, M.K. (2009) Fixed poles in the model matching problem for systems over semirings, Linear Algebra and its Applications, 430, 8, 2368-2388. Shang, Y. and Sain, M.K. (2007) Fixed zeros in the model matching problem for systems over a semiring. Proceedings of American Control Conference, pages 3381-3386. Shang, Y. (2010) Kalman’s Realization and Controlled Invariance for Systems over Semirings, IMA Journal of Control and Information, under review. Takahashi, M. (1981) On the Bordism categories II: elementary properties of semimodules. Mathematics Seminar Notes, 9, 495-530. Wyman B.F. and Sain, M.K. (1981) The zero module and essential inverse systems, IEEE Transactions on Circuits and Systems, 8, 2, 112-126.
components of the solutions to the model matching problem.
The max-plus algebra [Baccelli et al. (1992); Cohen et al. (1999); Heidergott et al. (2005)], the min-plus algebra [LeBoudec et al. (2001)], and the Boolean semiring [Golan (1999)] are characterized as a special algebraic structure, called semiring, which has no additive inverses. Maxplus linear systems are linear systems over the max-plus algebra to model communication networks [LeBoudec et al. (2001)], genetic regulatory networks [DeJong (2002)], transportation networks [DeVries et al. (1998)] and queueing systems [Baccelli et al. (1992)]. Max-plus linear systems evolve with variables taking values in the max-plus semimodules. The fundamental differences between maxplus linear systems and traditional linear systems slow down the progress of a well-established system theory for max-plus linear systems. Nowadays, researchers have studied some fundamental problems for the max-plus linear systems, for example, controllability [Prou et al. (1999)], observability [Hardouin et al. (2010)], and the model reference control problem [Maia et al. (2005)]. Most of these results focus on the time-domain analysis methodology, except for the transfer function representation [Baccelli et al. (1992)]. The challenge for a well-established frequencydomain theory is that defining poles and zeros in the frequency domain requires operation inverses.
2. MATHEMATICAL PRELIMINARIES
This paper surveys recent study on the frequency-domain approach for max-plus linear systems. First of all, poles and zeros are defined as semimodules instead of point poles and zeros. Second, Kalman’s realization theory is generalized to max-plus linear systems. Third, the model matching problem is studied using the pole and zero semimodules instead of residuation theory [Maia et al. (2005)]. Fixed pole/zero structures characterize common Petri net
A monoid R is a semigroup (R, ¢) with an identity element eR with respect to the binary operation ¢. The term semiring means a set, R = (R, ¢, eR , £, 1R ) with two binary associative operations, ¢ and £, such that (R, ¢, eR ) is a commutative monoid and (R, £, 1R ) is a monoid, which are connected by a two-sided distributive law of £ over ¢. Moreover, eR £ r = r £ eR = eR , for all r in R. R = (R, ¢, eR , £, 1R ) is a semifield if and only if (R\{eR }, £, 1R ) is a group, i.e. all of its elements have inverse elements with respect to the £ operator. An idempotent semifield R is a semifield satisfying a ¢ a = a for all a ∈ R. The common example for idempotent semifields is the max-plus algebra, which replaces the traditional addition and multiplication into the max operation and the plus operation, Addition : a ⊕ b ≡ max{a, b}, Multiplication : a ⊗ b ≡ a + b. In max-plus algebra literature, we usually denote it as RMax =(R ∪ {²}, ⊕, ², ⊗, e), where R is the set of real numbers, ² = −∞, and e = 0. Let (R, ¢, eR , £, 1R ) be a semiring, and (M, 2M , eM ) be a commutative monoid. M is called a left R-semimodule if there exists a map µ : R × M → M , denoted by µ(r, m) = rm, for all r ∈ R and m ∈ M , such that the following conditions are satisfied: (1) r(m1 2M m2 ) = rm1 2M rm2 , (2) (r1 ¢ r2 )m = r1 m 2M r2 m,
(3) r1 (r2 m) = (r1 £ r2 )m, (4) 1R m = m, (5) r eM = eM = eR m, for any r, r1 , r2 ∈ R and m, m1 , m2 ∈ M . In this paper, e denotes the unit semimodule. A sub-semimodule K of M is a submonoid of M with rk ∈ K, for all r ∈ R with k ∈ K. A sub-semimodule K of M is called subtractive if k ∈ K and k 2M m ∈ K imply m ∈ K, for m ∈ M . An R-morphism between two semimodules (M, 2M , eM ) and (N, 2N , eN ) is a map f : M → N satisfying (1) f (m1 2M m2 ) = f (m1 ) 2N f (m2 ), (2) f (rm) = rf (m), for all m, m1 , m2 ∈ M and r ∈ R. There are two different kernels for R-semimodule morphisms. The kernel of an R-semimodule morphism f : M → N is defined as Ker f = {x ∈ M |f (x) = eN }. The equivalence kernel is an equivalence relation defined by Kereq f = {(x1 , x2 ) ∈ M × M |f (x1 ) = f (x2 )}. There are two different images for R-semimodule morphisms [Takahashi (1981)]. Given an R-semimodule morphism f : M → N , one image is defined to be the set of all values f (m), m ∈ M , i.e. f (M ) = {n ∈ N |n = f (m), for some m ∈ M }. (1) It is called the proper image of an R-semimodule morphism f . The other image of f is defined as Im f = {n ∈ N |n 2N f (m) = f (m0 ) for some m, m0 ∈ M }. It is called the image of f to distinguish from the proper image. The two images coincide for the module case. For the semimodule case, if the two images are the same, the R-morphism of semimodules f : M → N is called i-regular or image-regular. The Bourne relation, introduced in [Golan (1999), pp. 164], is an equivalence relation for an R-semimodule. If K is a sub-semimodule of an R-semimodule M , then the Bourne relation is defined by setting m ≡K m0 if and only if there exist two elements k and k 0 of K such that m 2M k = m0 2M k 0 . The factor semimodule M/ ≡K induced by ≡K is also written as M/K. If K is equal to the kernel of an R-semimodule morphism f : M → N , then m ≡Ker f m0 if and only if there exist two elements k, k 0 of Ker f , such that m 2M k = m0 2M k 0 . Applying f on both sides, we obtain that f (m) = f (m0 ). Hence this special Bourne relation and the equivalence relation induced by the morphism f satisfy the partial order ≤, i.e. ≡Ker f ≤ ≡f . In general, we do not have ≡f ≤ ≡Ker f for an R-semimodule morphism f . If an R-semimodule morphism f : M → N satisfies ≡f ≤ ≡Ker f , then f is called a steady or k-regular R-semimodule morphism. Given two R-semimodule morphisms f : (A, 2A , eA ) → (B, 2B , eB ) and g : (B, 2B , eB ) → (C, 2C , eC ), the sequence f
g
(2) A −−−−→ B −−−−→ C is called an exact sequence if Im f = Ker g. Because Im f is not the same as the proper image f (A) for the
R-semimodule morphism f , the sequence is said to be a proper exact sequence if f (A) = Ker g. For the module case, Im f is always the same as the proper image f (A), so every exact sequence of modules is also proper exact. 3. MAX-PLUS LINEAR SYSTEMS Max-plus linear systems are described by the following equations: x(k + 1) = A x(k) ⊕ B u(k), y(k) = C x(k) ⊕ D u(k), (3) where x is in the state semimodule X ∼ = RnMax , y is in p the output semimodule Y ∼ = RMax , and u is in the input r . A : X → X, B : U → X, semimodule U ∼ R = Max C : X → Y and D : U → Y are four R-semimodule morphisms. RMax [z] denotes the polynomial semiring with coefficients in the max-plus algebra RMax , and ΩX, ΩU , and ΩY denote the polynomial RMax [z]-semimodules of states, inputs, and outputs, respectively. The operator ΩX is an alternative notation for the polynomial RMax [z]semimodules X[z] of states. For instance, given a sequence of states, {· · · , x(−2), x(−1), x(0), x(1), x(2), · · · , }, ΩX = x(0) ⊕ x(1)z ⊕ x(2)z 2 ⊕ · · · ⊕ x(k)z k , for a finite k ∈ N, is isomorphic to a sequence of states starting from the time instant 0 to the future. Let RMax (z) denote the set of formal Laurent series in z −1 , with coefficients in R. In this manner, let X(z) denote the set of formal Laurent series in z −1 , with coefficients in X. We define U (z) and Y (z) similar as X(z). The symbol z −1 is similar as the γ symbol in the dioid of formal power series in the event and time variables [Cohen et al. (1999)]. The transfer function G(z) : U (z) → Y (z) of the system in Eq. (3) is G(z) = C(z −1 A)∗ Bz −1 ⊕ D = CBz −1 ⊕ CABz −2 ⊕ · · · ⊕ D. (4) The star operator A∗ for an n × n matrix mapping A : X → X is defined as A∗ = In×n ⊕ A ⊕ · · · ⊕ An ⊕ · · · . (5) where In×n denotes the identity matrix mapping from X to X. Without loss of generality, we assume D = 0 for a given transfer function G(z) : U (z) → Y (z) in this paper. 4. POLE AND ZERO SEMIMODULES 4.1 Pole Semimodules This section defines two different pole semimodules, the pole semimodule of output type for the transfer function G(z) : U (z) → Y (z) as G(ΩU ) , G(ΩU ) ∩ ΩY and the pole semimodule of input type as XO (G(z)) =
(6)
ΩU . (7) G−1 (ΩY ) ∩ ΩU The pole semimodule of input type can be understood as inputs without poles producing outputs with poles. XI (G(z)) =
The pole semimodule of output type can be understood as outputs with poles produced by inputs without poles. The poles in the output come from the poles of the given plant, therefore, it helps the discoveries of the poles in the plant transfer function. Moreover, pole semimodules can generate the state matrix A for a system over a semiring R, which is related to the eigenvalues or point poles of the given system. In the module case, each pole module has been used by the preference of the researchers, because XI (G(z)) is isomorphic to XO (G(z)). However, for the semimodule case, there exists an R[z]-semiisomorphism between XI (G(z)) and XO (G(z)) instead, i.e. an unit kernel R[z]-semimodule epimorphism. Lemma 1. (Shang (2010)). Given a transfer function G(z) : U (z) → Y (z) and the pole semimodules of input and output type as shown in Eq. (6) and Eq. (7), there exists an R[z]-semimodule semiisomorphism G(z) from XI (G(z)) to XO (G(z)). 4.2 Zero Semimodules This section extends the concepts of the zero modules and the extended zero modules introduced by Wyman and Sain in [Wyman et al. (1981)] to the semimodule case. The zero semimodule of input type for the transfer function G(z) : U (z) → Y (z) is defined as ZI (G(z)) =
ΩY ∩ G(U (z)) . ΩY ∩ G(ΩU )
(8)
Intuitively, the zero semimodule of input type consists of the polynomial outputs produced by the inputs with poles. The poles of the inputs are canceled by the zeros of the plant, hence, the zero semimodule ZI (G(z)) leads to the discovery of the plant’s zeros. In the zero semimodule ZI (G(z)), the polynomial outputs produced by the inputs without poles are removed, because they cannot discover the plant’s zeros. The zero semimodule of output type for a given transfer function G(z) : U (z) → Y (z) is defined as G−1 (ΩY ) . (9) ZO (G(z)) = −1 G (ΩY ) ∩ (Ker G ⊕ ΩU ) Intuitively, the zero semimodule of output type consists of the inputs with poles, which produce outputs without poles. The poles in the inputs are canceled by the zeros of the plant, therefore, the zero semimodule ZO (G(z)) leads to the discoveries of the plant’s zeros. The zero semimodule of output type removes the kernel of the transfer function G(z) and the polynomial inputs because they will not help to discover the zeros of the plant. In the module case, the zero modules of input and output type are isomorphic forms of each other; however, that is not the case for the zero semimodules of input and output type. There exists an R[z]-semimodule epimorphism, that is, a surjective R[z]-morphism, instead of an R[z]isomorphism from the zero semimodule of output type to the zero semimodule of input type. Lemma 2. (Shang et al. (2009)). Given a transfer function G(z) : U (z) → Y (z) and the zero semimodules of input and output type as shown in Eq. (8) and Eq. (9),
G(z )
U (z )
p
i W U
Y (z )
G # (z )
GY
Fig. 1. The Kalman input/output map G# (z). there exists an R[z]-semimodule epimorphism from the zero semimodule of output type to the zero semimodule of input type. 4.3 The Γ-Zero and Ω-Zero Semimodules Given a transfer function G(z) : U (z) → Y (z), the Γ-zero semimodule is defined as ZΓ (G(z)) =
G−1 (ΩY ) . G−1 (ΩY ) ∩ ΩU
(10)
When Ker G is unit, ZΓ (G(z)) is identical to ZO (G(z)). The Ω-zero semimodule ZΩ (G(z)) of the transfer function G(z) : U (z) → Y (z) is given by ZΩ (G(z)) =
ΩY . ΩY ∩ G(ΩU )
(11)
This R[z]-semimodule is finitely generated, because ΩY is finitely generated. If the proper image G(U (z)) of G(z) is equal to Y (z), then the zero semimodule of input type ZI (G(z)) is equal to the Ω-zero semimodule of G(z). The Ω-zero semimodule differs from the zero semimodule of input type when the cokernel of G(z) is not a unit semimodule. In this case, ZI (G(z)) is a sub-semimodule of ZΩ (G(z)), and there exists a natural inclusion from ZI (G(z)) to ZΩ (G(z)), with the cokernel as Ω(G(z)) = ΩY /{ΩY ∩ G(U (z))}. 5. GENERALIZATION OF KALMAN’S REALIZATION THEORY This section generalizes the Kalman’s realization theory from systems over fields in [Kalman et al. (1969)] to systems over semirings. Kalman’s realization theory characterizes a canonical realization by the property that is reachable from the unit element and observable with respect to the unit input. A realization (A, B, C) of a transfer function G(z) is canonical if and only if it is both reachable from the unit element and observable with respect to the unit input. In other words, if a transfer function can be factored through X by an onto map g and a one-to-one map h, then X is a canonical realization for the given transfer function. If g is an onto map, but h is not a oneto-one map, then X is called a reachable realization. On the other hand, if h is a one-to-one map, but g is not an onto map, then X is called an observable realization. The Kalman input semimodule is defined as the set of polynomial inputs, ΩU = U [z]. The Kalman output semimodule is defined as the set of strictly proper outputs, denoted as a factor semimodule induced by the Bourne relation, ΓY = Y (z)/ΩY . The factor semimodule ΓY consists of equivalence classes in which two elements y1 (z) and y2 (z) are equivalent if and only if they contain the same
6. MODEL MATCHING PROBLEM
G # (z )
W U
GY
f
~ B
W U G - 1 (W Y ) I W U
g
~ C
l
h
X G (W U ) G (W U ) I W Y
The model matching problem (MMP) for max-plus linear systems can be described as follows. Given two transfer functions, T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z), one is to find another transfer function, M (z) : C(z) → U (z), or given T (z) and M (z), one is to find another transfer function P (z), such that the following model matching equation is satisfied, T (z) = P (z) M (z).
(12)
Fig. 2. The generalized Kalman realization diagram. 6.1 Fixed Poles of Model Matching Problem strictly proper component. The factor semimodule can be intuitively understood as Y (z) without the polynomial component ΩY . The Kalman input semimodule contains sequences of input signals starting at a finite negative time and continuing up to the time zero. The Kalman output semimodule contains sequences of output signals from the time instant 1 into the future. The Kalman input/output map G# (z) can be constructed from the commutative diagram shown in Figure 1, where i is the insertion and p is the natural projection induced by the Bourne relation. If we consider the Kalman input/output map G# (z) : ΩU → ΓY , then we can obtain the commutative diagram shown in Figure 2. X is the state semimodule. The e : ΩU → X and C e : X → ΓY are defined mappings B by e · u) = ABu; B(z Cxz −1 ⊕ C(Ax)z −2 ⊕ C(A2 x)z −3 ⊕ · · · e . C(x) = ΩY For the pole module of output type XO (G(z)), the mapG(u) ping h is defined by the action h : u 7→ G(ΩU )∩ΩY , where u ∈ ΩU . The mapping l is defined by the action G(u) G(u) l : G(ΩU )∩ΩY 7→ ΩY . Therefore, l is an induced map from the identity map Id. For the pole semimodule of input type XI (G(z)), the mapping f is a natural projection, i.e. u f : u 7→ G−1 (ΩY )∩ΩU for u ∈ ΩU . The mapping g is defined by the action g :
u G−1 (ΩY )∩ΩU
7→
G(u) ΩY .
From the Kalman’s realization diagram in Figure 2, the pole semimodule of output type XO (G(z)) is a reachable realization of the Kalman input/output map, because G# (z) is an onto map from ΩU to XO (G(z)). When the map l is one-to-one, the pole semimodule of output type becomes a canonical realization. The pole semimodule of input type XI (G(z)) is also a reachable but not observable realization of G# (z), because f is onto and g is not oneto-one. Using Lemma 1, the pole semimodule of input type becomes isomorphic to the pole semimodule of output type, i.e. G(z) becomes an isomorphism, if and only if the Kalman input/output map G# (z) is steady or k-regular. Therefore, Kalman’s original realization theory for systems over fields is generalized to max-plus linear systems, even there is no “subtraction” operation at present corresponding to the max operation.
We are given two transfer functions T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z). To study the solution M (z) to the model matching problem, an R[z]-semimodule P (T, P ) is defined as follows: P (T, P ) =
T (ΩC) ⊕ P (ΩU ) . P (ΩU )
(13)
The semimodule P (T, P ) in Eq. (13) is called the fixed pole semimodule for the solution M (z) to the model matching equation T (z) = P (z)M (z). The same terminology, fixed poles, is adopted from systems over a field in [Conte et al. (1988)], in which the fixed pole module behaves as a factor module. The R[z]-semimodule P (T, P ) is a fixed pole structure because it has a direct relation with the pole semimodule of output type for the solution M (z). Proposition 3. (Shang et al. (2009)). For the MMP with two known transfer functions T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z) and a solution to M (z) : C(z) → U (z) to the model matching problem, there exists a surjective R[z]-semimodule morphism, from the pole semimodule of output type XO (M ) of the solution M (z) to the R[z]semimodule P (T, P ). Theorem 4. (Shang et al. (2009)). Consider two transfer functions T (z) : C(z) → Y (z) and P (z) : U (z) → Y (z), and a solution M (z) : C(z) → U (z) to the model matching problem, there exist R[z]-semimodules Z1 and P1 , and R[z]-semimodule morphisms α, β, φ, and ψ such that the following three sequences are exact: p
i
e → Z1 − → ZI (P ) − → ZI ([T P ]) → e; β
α
e → XO (P ) − → XO ([T P ]) − → P1 → e; φ
ψ
Z1 − → P (T, P ) − → P1 → e,
(14) (15) (16)
where Z1 and P1 are defined as (T (ΩC) ⊕ P (ΩU )) ∩ ΩY , P (ΩU ) ∩ ΩY T (ΩC) ⊕ P (ΩU ) . P1 = (T (ΩC) ⊕ P (ΩU )) ∩ ΩY ⊕ P (ΩU )
Z1 =
(17) (18)
e denotes the identity semimodule. The map φ has a unit kernel if P (ΩU ) is subtractive, and then the last sequence becomes φ
ψ
e → Z1 − → P (T, P ) − → P1 → e, which is a short exact sequence.
(19)
Proposition 3 states the relationship between the fixed pole semimodules and solutions to MMP. In particular, for the MMP with unknown controller M (z), an R[z]epimorphism P (z) : XO (M ) → P (T, P ) is established in the proof of Proposition 3. Therefore, we have the following short exact sequence
Proposition 5 states the relationships between the fixed zero semimodules and the extended zero semimodules of the solutions to the model matching problem. In particular, for the model matching problem with an unknown controller M (z), we have the exact sequence: βΩ (z)
i
e → C(M ) − → ZΩ (M (z)) −−−→ Z(T, P ) → e,
P (z)
α
e → C(M ) − → XO (M ) −−−→ P (T, P ) → e,
(20)
with C(M ) = Ker P (z), which is called the inessential pole semimodule of the solutions to the MMP. The fixed pole semimodule helps us discover essential components in the solutions to MMP, which will be illustrated later with a discrete-event system example. 6.2 Fixed Zeros of Model Matching Problem Define an R[z]-semimodule Z(T, P ) as follows: Z(T, P ) =
T (ΩC) ⊕ P (ΩU ) . T (ΩC)
(21)
The R[z]-semimodule Z(T, P ) is called the fixed zero semimodule for the solution M (z) to the model matching equation T (z) = P (z)M (z). In the next theorem, we will establish a relation between the fixed zero semimodule Z(T, P ) and the Ω-zero semimodule of the solution M (z) to the MMP. Proposition 5. (Shang et al. (2007)). If we are given two transfer functions T (z) : C(z) → Y (z), P (z) : U (z) → Y (z), and the solution M (z) : C(z) → U (z) satisfying the model matching equation T (z) = P (z)M (z), there exists an R[z]-epimorphism βΩ (z) between the Ω-zero semimodule of M (z), ZΩ (M (z)), and the fixed zero semimodule, Z(T, P ): βΩ (z)
ZΩ (M (z)) −−−−→ Z(T, P ) −−−−→ e.
(22)
Theorem 6. (Shang et al. (2007)). If we are given two transfer functions T (z) : C(z) → Y (z), P (z) : U (z) → Y (z), and the solution M (z) : C(z) → U (z) to the model matching equation T (z) = P (z)M (z), there exist R[z]semimodules Z1 and P1 , which are defined below, such that the following three sequences are exact: p
i
e → Z1 − → ZΩ (T (z)) − → ZΩ ([T (z) P (z)]) → e; (23)
where C(M ) = Ker βΩ (z), which is called the inessential zero semimodule of the solutions to the MMP. 7. A DISCRETE EVENT SYSTEM APPLICATION This section uses a discrete event system example to illustrate the fixed zero semimodule characterizes the common Petri net components in the solutions to the model matching problem. The Petri net models for the four-machine manufacturing system (top) and the twomachine manufacturing system (bottom) are given in Fig. 3. y2 y1 x1
u1
u1
x2
φ
ψ
where (T (ΩC) ⊕ P (ΩU )) ∩ ΩY , Z1 = T (ΩC) ∩ ΩY T (ΩC) ⊕ P (ΩU ) . P1 = (T (ΩC) ⊕ P (ΩU )) ∩ ΩY ⊕ T (ΩC) e denotes the identity semimodule. The morphism φ has a unit kernel if T (ΩC) is subtractive, in which the last sequence becomes the following short exact sequence, φ
ψ
e → Z1 − → Z(T, P ) − → P1 → e.
x1
y1
Fig. 3. The Petri net realizations for the two manufacturing systems. The control problem is to design a compensator such that the manufacturing system with four machines has the same processing time as the manufacturing system with two machines. The discrete event systems can be modeled as a system over the max-plus algebra. The plant transfer function P (z) : U (z) → Y (z) is · P (z) =
¸ z −3 ⊕ z −4 ² . z −3 ²
The reference transfer function T (z) : C(z) → Y (z) is obtained as the following form: ·
e → XO (T (z)) − → XO ([T (z) P (z)]) − → P1 → e; (24) Z1 − → Z(T, P ) − → P1 → e, (25)
x4
y2
β
α
x3
x2
T (z) =
¸ z −1 ⊕ z −2 ² . z −1 ²
In the semimodule case of the fixed pole structure, the equivalence classes cannot be constructed by simply removing T (ΩC) from T (ΩC) ⊕ P (ΩU ) because of the Bourne equivalent relation. Instead, two elements y1 = T (z)cp ⊕ P (z)up and y2 = T (z)˜ cp ⊕ P (z)˜ up in T (ΩC) ⊕ P (ΩU ) are equivalent, if and only if y1 ⊕l1 = y2 ⊕l2 , where l1 and l2 in T (ΩC). By tedious mathematical manipulations, one can show that y1 and y2 are equivalent if and only if the first rows in u1p and u ˜1p have the same coefficients 0 1 for z and z . Due to limited space, the detail derivations are omitted here. In other words, the representative of the equivalent classes in the fixed zero semimodule is
· P (z)up = P (z)
¸ a0 ⊕ a1 z ⊕ wp1 , , wp2 ,
where wp1 is an arbitrary polynomial element in ΩRMax with the order of z higher than 1 and wp2 is an arbitrary polynomial element in ΩRMax . In order to obtain the fixed zero semimodule, the representative in the quotient equivalence classes are P (z)up by removing vectors with the first element which has the order of z greater than −2 and with the second element which has the order of z greater than −1. The fixed zero semimodule Z(T, P ) of Eq. (21) characterize a matrix Af as · ¸ ² ² Af = . e² This matrix is connected to the state space realizations of the solutions to the model matching problem. For examples, any two solutions M1 (z) and M2 (z) to the model matching equation P (z)M (z) = T (z) and construct the realization matrices A1 and A2 for them. For instance, two solutions M1 (z) and M2 (z) are given as follows: · 2 ¸ · 2 ¸ z ² z ² M1 (z) = , M2 (z) = . ² z z2 z2 Therefore, the Ω-zero semimodules of M1 (z) and M2 (z) generate two A matrices for these two controllers: " A1 =
# ² ² ² ² e ² ² ² , A2 = ² ²e² ²
² ² ² ²
² ² . ² ²
² ² ² e
Moreover, we can model these two systems by Petri nets shown in Fig. 4. We find out that two Petri net realizations f1 (z) = z −3 M1 (z) and M f2 (z) = z −3 M2 (z) contain for M the same set of components marked in the dashed boxes, which are generated by the essential matrix Afixed . ~ Realization for M 1 ( z ) : u1
x1
y1
u2
x2
x3
y2
y1 u ~ Realization for M 2 ( z ) : 1
x1
x2 y2
u2
x3
x4
f1 (z) and M f2 (z). Fig. 4. The Petri net realizations for M 8. CONCLUSION This paper is a summary of the frequency-domain analysis for max-plus linear systems, including the concepts of poles and zeros as semimodules instead of points, a generalization of Kalman realization theory, and the model matching problem in controller synthesis. The main advantage of Kalman’s theory is that the frequency-domain method and the state-variable method merge into a single framework. Moreover, it introduces the concepts of poles and zeros
as semimodules instead of point poles and zeros, which cannot be traditionally defined without inverse operations. Moreover, the pole and zero semimodules can characterize the common Petri net components in the solutions to the model matching problem. These results in this paper can provide an alternative approach in the frequency-domain analysis of max-plus linear systems. REFERENCES Baccelli, F.L., Cohen, G., Olsder,G.J., and Quadrat, J.P. (1992) Synchronization and Linearity: An Algebra for Discrete Event Systems. John Wiley and Sons, New York. Cohen, G., Gaubert, S., Quadrat, J.P. (1999) Max-plus algebra and system theory: where we are and where to go now. Annual Reviews in Control, 23, 207-219. Conte, G., Perdon, A.M., and Wyman, B.F. (1988) Fixed poles in transfer function equations. SIAM J. Control and Optimization, 26, 2, 356-368. De Jong, H. (2002) Modeling and simulation of genetic regulatory systems: a literature review, Journal of Computational Biology, 9, 1, 67-103. De Vries, R., De Schutter, B., and De Moor, B. (1998) On max-algebraic models for transportation networks. Proceedings of the International Workshop on Discrete Event Systems, 457-462. Golan, J.S. (1999) Semirings and Their Applications. Kluwer Academic Publishers, Boston. Hardouin, L., Maia, C.A., Cottenceau, B, and Lhommeau, M. (2010) Observer Design for (max,plus) Linear Systems, IEEE Transactions on Automatic Control, 55, 2, 538-543. Kalman, R.E., Falb, P.L., and Arbib, M.A. (1969) Topics in Mathematical System Theory. McGraw-Hill, New York, 1st edition. Heidergott, B., Jan Olsder, G. and Van Der Woude, J. (2005) Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications, Princeton University Press, Princeton, New Jersey, 1st edition. Le Boudec, J.-Y. & Thiran, P. (2002) Network Calculus. Springer-Verlag, New York., 1st edition. Maia, C.A, Hardouin,L., Santos-Mendes, R. and Cottenceau, B.(2005) On the Model Reference Control for Max-Plus Linear Systems. Proceedings of the 44th IEEE Conference on Decision and Control, pp. 7799-7803. Prou, J.-M., and Wagneur, E. (1999) Controllability in the max-algebra. KYBERNETIKA. 35, 1, 13-24. Shang, Y and Sain, M.K. (2009) Fixed poles in the model matching problem for systems over semirings, Linear Algebra and its Applications, 430, 8, 2368-2388. Shang, Y. and Sain, M.K. (2007) Fixed zeros in the model matching problem for systems over a semiring. Proceedings of American Control Conference, pages 3381-3386. Shang, Y. (2010) Kalman’s Realization and Controlled Invariance for Systems over Semirings, IMA Journal of Control and Information, under review. Takahashi, M. (1981) On the Bordism categories II: elementary properties of semimodules. Mathematics Seminar Notes, 9, 495-530. Wyman B.F. and Sain, M.K. (1981) The zero module and essential inverse systems, IEEE Transactions on Circuits and Systems, 8, 2, 112-126.