IMA Journal of Mathematical Control and Information (2010) 27, 145 164 doi:10.1093/imamci/dnq005 Advance Access publication on April 9, 2010 Robust invariance in uncertain discrete event systems with applications to transportation networks YING SHANG Department of Electrical and Computer Engineering, School of Engineering, Southern Illinois University Edwardsville, Campus Box 1801, Edwardsville, IL 62026, USA yshang@siue.edu [Received on 17 February 2009; revised on 1 March 2010; accepted on 6 March 2010] This paper studies a class of uncertain discrete event systems over the max plus algebra, where system matrices are unknown but are convex combinations of known matrices. These systems model a wide range of applications, e.g. transportation systems with varying vehicle travel time and queueing networks with uncertain arrival and queuing time. This paper presents the computational methods for different robust invariant sets of such systems. A recursive algorithm is given to compute the supremal robust invariant sub-semimodule in a given sub-semimodule. The algorithm converges to a fixed point in a finite number of iterations under proper assumptions on the state semi-module. This paper also presents computational methods for positively robust invariant polyhedral sets. A search algorithm is presented for the positively robust invariant polyhedral sets. The main results are applied to the time table design of a public transportation network. Keywords: robust controlled invariance; discrete event systems; max plus algebra. 1. Introduction Semirings can be understood as a set of objects without inverses with respect to the corresponding operators. A semimodule over a semiring can be analogized to a linear space over a field. Systems over a semiring are systems evolving with variables taking values in semimodules over the semiring. Intuitively, such systems are not equipped with additive inverses. The max plus algebra (Golan, 1999) is a special semiring consisting of real numbers embedded with max and plus operations. Systems over the max plus algebra are often used to model discrete event systems, such as queueing systems (Cassandras, 1993), transportation systems (Baccelli et al., 1992) and communication networks (Le Boudec & Thiran, 2002). In the study of linear systems over a field or a ring, controlled invariant subspaces (or sets) play a key role in the fundamental problems of the geometric control theory (Conte & Perdon, 1995, 1998; Hautus, 1982; Inaba et al., 1988; Wonham, 1979), e.g. the disturbance decoupling problem, the model matching problem and the block decoupling problem. The geometric approach for systems over a semiring, however, is still a new research direction (Cohen et al., 1996, 1999) compared to the geometric theory for traditional linear systems over a field (Wonham, 1979). The goal of this paper is to study different controlled invariant sets for a class of discrete event systems with parameter uncertainties, which are modelled as uncertain linear systems over the max plus algebra. The ultimate impact of these results is to establish a foundation for the geometric control theory of uncertain linear systems over semirings. Researchers have been studying different computational methods for controlled invariant spaces (or sets) for discrete event systems over the max plus algebra (Katz, 2007; Bitsoris, 1988; Truffet, 2004). c The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
146 Y. SHANG However, there are many uncertain factors in the discrete event system modelling. This paper focuses on a class of discrete event systems over the max plus algebra in which system matrices are uncertain but can be written as linear combinations of known matrices. These systems model a class of discrete event systems, such as transportation systems with varying travel time and queuing networks with uncertain arrival time and queueing time. Most of these uncertainties can be characterized by max plus convex sets in Nitica & Singer (2007). For such uncertain discrete event systems, there are some research results on the model predictive control (Necoara et al., 2006a; Van den Boom & De Schutter, 2002) and worstcase optimal control (Necoara et al., 2006b). This paper, on the other hand, makes contributions in the analysis of robust invariant sets, which can be used in the geometric control problems. The main results in this paper are the computational methods for different robust controlled invariant sub-semimodules (and sets). A recursive algorithm is given to compute the supremal robust invariant sub-semimodule in a given sub-semimodule of the state semimodule. The algorithm converges to a fixed point in a finite number of iterations under proper assumptions on the state semimodule. The fixed point is the supremal robust controlled invariant sub-semimodule of the given sub-semimodule. The main challenge for systems over a semiring (or a ring) is that the (A, B)-invariant sub-semimodule (or submodule) does not coincide with ( A, B)-invariant sub-semimodule (or submodule) of feedback type (Conte & Perdon, 1995, 1998; Hautus, 1982). A computational method is introduced for robust controlled invariant sub-semimodule of feedback type in a given sub-semimodule. Moreover, this paper presents computational methods for positively robust invariant polyhedral sets in two cases, timeinvariant and time-varying polyhedral sets. The reason to focus on time-varying polyhedral invariant sets is that there are many quantities in discrete event systems varying with time. For instance, in transportation networks, the goal is to control the departure time of each vehicle such that it is contained in a desired time constraint. These constraints usually vary over time and can be used to design proper time tables for transportation systems. A search algorithm for time-varying positively robust invariant sets is presented and is illustrated using a public transportation network with varying vehicle travelling time. This paper is organized as follows. Section 2 introduces some mathematical preliminaries and discrete event systems over the max plus algebra. Section 3 presents computational methods for robust controlled invariant sub-semimodules. Section 4 presents the computational methods for positively robust invariant sets. A public transportation network is used to illustrate the main results. Section 5 concludes this paper with future research. 2. Mathematical preliminaries 2.1 Semiring and semimodule A monoid R is a semigroup (R, ) with an identity element e R with respect to the binary operation. The term semiring means a set, R = (R,, e R,, 1 R ) with two binary associative operations, and, such that (R,, e R ) is a commutative monoid and (R,, 1 R ) is a monoid, which are connected by a two-sided distributive law of over. Moreover, e R r = r e R = e R, for all r in R. R = (R,, e R,, 1 R ) is a semifield if and only if (R\{e R },, 1 R ) is a group, i.e. all of its elements have inverse elements with respect to the operator. A semifield R = (R,, e R,, 1 R ) is called an idempotent semifield if a a = a for all a R. The max plus algebra is an idempotent semifield, where the traditional addition and multiplication are replaced by the max operation and the plus operation, i.e. Addition : a b max{a, b}, Multiplication : a b a + b.
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 147 The max plus algebra is denoted as R Max =(R {ɛ},, ɛ,, e), or simply R Max, where R denotes the set of real numbers, ɛ = and e = 0. Similarly, Z Max = (Z {ɛ},, ɛ,, e) denotes the integer max plus algebra, where Z is the set of integers. Let (R,, e R,, 1 R ) be a semiring and (M, M, e M ) be a commutative monoid, where the subscript denotes the corresponding monoid for the operator. 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(m 1 M m 2 ) = rm 1 M rm 2, 2. (r 1 r 2 )m = r 1 m M r 2 m, 3. r 1 (r 2 m) = (r 1 r 2 )m, 4. 1 R m = m, 5. re M = e M = e R m, for any r, r 1, r 2 R and m, m 1, m 2 M. In this paper, e denotes the unit semimodule. A subsemimodule K of M is a submonoid of M with rk K, for all r R with k K. An R-morphism between two semimodules (M, M, e M ) and (N, N, e N ) is a map f : M N satisfying 1. f (m 1 M m 2 ) = f (m 1 ) N f (m 2 ); 2. f (rm) = r f (m), for all m, m 1, m 2 M and r R. Let N be a subset of an R-semimodule (M, M, e M ). We denote N 0 as the set of all elements of the form Mλ i n i, where n i N, λ i R and I is the index set. The sub-semimodule N 0 is said to i I be generated by N, and N is called a system of generators of N 0. The subset N of an R-semimodule M is called linearly independent if Mλ i n i = Mβ i n i implies λ i = β i for all i I. An R- i i semimodule M is called a free R-semimodule if it has a linearly independent subset N of M which generates M and then N is called a basis of M. If N has a finite number of elements, M is called a finitely generated R-semimodule, denoted as Span N. Katz (2007) defined the concept of volume for a semimodule. Let K Z n Max be a semimodule, the volume of K, denoted as vol(k), is the cardinality of the set K {x K x 1 x 2 x n = 0}. Also, if K Z n p Max, the volume of the semimodule K = ImK is denoted as vol(k ) = vol(imk ) = vol(k). REMARK. In Wagneur (1991, 2005), independence and weakly independence are defined differently. It can be shown that if the semimodule is linearly independent according to the definition in this paper, then it is also independent and weakly independent according to the definitions in Wagneur (1991, 2005). 2.2 Uncertain discrete event systems over the max plus algebra A class of uncertain discrete event system over the max plus algebra R Max is described by the following equation: x(k) = Ãx(k 1) Bu(k), (2.1) where the state semimodule X = R n Max and the input semimodule U = R r Max are free, A: X X and B: U X are R-semimodule morphisms. Although we assume the state semimodule, the input
148 Y. SHANG semimodule and the output semimodule are free for convenience, the main results in this paper did not require the free assumption. The system s state matrix morphism Ã: X X is unknown, but it can be represented as the linear combination of m known matrix morphisms A 1, A 2,..., A m, i.e. à = (λ i A i ) with λ i = e, where the parameters λ i are unknown. A set P is convex if (a x) (b y) P for any x, y P and a b = 1 R, where a, b R. Therefore, à is in the convex hull of A 1,..., A m, denoted as à co{a 1,..., A m }. The geometric concepts of different invariant subspaces for systems over a field can be generalized to systems over a semiring. Given a system of the form (2.1) over the max plus algebra R Max, a sub-semimodule V of the state semimodule X is called (Ã, B)-invariant or robust controlled invariant if and only if, for all x 0 V and an arbitrary matrix morphism à co{a 1,..., A m }, there exists a sequence of control inputs, u = {u 1, u 2,...}, such that every component in the state trajectory produced by this input, x(x 0 ; u) = {x 0, x 1,...}, remains inside of V. called (Ã, B)-invariant of feedback type or (à B F)-invariant, if and only if there exists a state feedback F: X U such that (à B F)V V, for an arbitrary matrix morphism à co{a 1,..., A m }. Unlike systems over a field, ( A, B)-invariant sub-semimodules of feedback type are not same as ( A, B)-invariant sub-semimodules for systems over a semiring or even a ring. Conte & Perdon (1998) proved that, for systems over a ring, if an (A, B)-invariant submodule is a direct summand of the free state semimodule X, then it is ( A, B)-invariant of feedback type. However, direct summand in semimodules is far more complicated than modules, the same result in Conte & Perdon (1998) is not true any more. This paper also generalizes positively invariant sets for deterministic systems to uncertain systems of the form (2.1) over the max plus algebra R Max. A non-empty subset P of R n Max is said to be positively invariant or A-positively invariant for the systems of the form x(k + 1) = Ax(k), if, for every initial condition x(0) P, the trajectory x(k) remains in P for all k 0. If a non-empty subset P of R n Max is said to be robust positively invariant or Ã-positively invariant for systems of the form x(k + 1) = Ãx(k), if, for every initial condition x(0) P, the state trajectory x(k) remains in P with respect to any uncertain matrix morphism à co{a 1,..., A n }. A non-empty subset P of R n Max is said to be a controlled positively invariant set for systems of the form x(k + 1) = Ax(k) Bu(k) if, for every initial condition x(0) P, there exists a control input u such that the trajectory x(k) remains in P with respect to any uncertain matrix morphism Ã. A non-empty subset P of R n Max is said to be a controlled robust positively invariant set for systems of the form (2.1) if, for every initial condition x(0) P, there exists a control input u such that the trajectory x(k) remains in P with respect to any uncertain matrix morphism à co{a 1,..., A n }. 3. Robust invariant sub-semimodules This section presents the computational methods for (Ã, B)-invariant sub-semimodules and (à B F)- invariant sub-semimodules in a given sub-semimodule. These computational methods are generalizations of deterministic discrete event systems in Katz (2007) to uncertain discrete event systems of the form (2.1) with parameter uncertainties.
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 149 3.1 (Ã, B)-invariant sub-semimodules A sub-semimodule V of the state semimodule is (Ã, B)-invariant if and only if V = V à 1 (V B), (3.1) where B = ImB and à 1 (V B) = {x R n Max u U, s.t. Ãx Bu V, à co{a 1,..., A m }}. (3.2) REMARK. Equation (3.1) is not a definition of (Ã, B)-invariant sub-semimodules, but it is a direct generalization from the definition of the (Ã, B)-invariant sub-semimodules in the last section. Moreover, it is often used in the calculations of the robust controlled invariant sub-semimodules. LEMMA 3.1 For an uncertain discrete event system of the form (2.1) over the max plus algebra, a subsemimodule V in the state semimodule is (Ã, B)-invariant if and only if V is (A i, B)-invariant for any i {1, 2,..., m}. Proof of Lemma 3.1. =. V is (Ã, B)-invariant, then, for all x V, Ãx Bu V, i.e. ( m λ i A i )x Bu is also in V. If λ i = e and λ j = ɛ for j i and i, j {1,..., m}, then m λ i A i will just produce the matrix morphism A i for any i {1, 2,..., m}. Therefore, A i x Bu V is true. Therefore, V is (A i, B)-invariant for any i {1, 2,..., m}. =. If V is (A i, B)-invariant for any i {1, 2,..., m}, then, for all x V, there exists a control input u i U such that A i x Bu i V. Because the sub-semimodule V is closed under the addition operation, m λ i A i x B m λ i u i = Ãx Bu V, where u = m λ i u i. Therefore, V is (Ã, B)-invariant. LEMMA 3.2 For an uncertain discrete event system of the form (2.1) over the max plus algebra, the following equality holds à 1 (V B) = m A 1 i (V B) for any sub-semimodule V of the state semimodule X. Proof of Lemma 3.2.. For any x à 1 (V B), there exists a control input u U such that Ãx Bu = m λ i A i x Bu V. If λ i = e and λ j = ɛ for j i and i, j {1,..., m}, then A i x Bu V holds. Therefore, x m Ai 1 (V B).. For any x m Ai 1 (V B), there exists a control input u i U such that A i x Bu i V for all i {1,..., m}. Therefore, m λ i A i x B m λ i u i = Ãx Bu V, where u = m λ i u i. Hence, x à 1 (V B). For a linear system over a field, the supremal (A, B)-invariant subspace V in a given subspace K of the state space is computed using the following algorithm (Wonham, 1979), V 1 = K, V k+1 = V k A 1 (V k + B), k N, (3.3)
150 Y. SHANG where A 1 (V i + B) = {x R n Ax V i + B}. This recursion converges to a fixed point after a finite number of iterations. The fixed point is the supremal controlled invariant subspace in K. However, for systems over a semiring or even a ring, this algorithm does not guarantee to terminate in a finite number of steps. For uncertain linear systems over the max plus algebra, to calculate the supremal controlled invariant sub-semimodule of a given sub-semimodule K, the algorithm in (3.3) is modified as follows by using Lemma 3.2: V 1 = K, V k+1 = V k à 1 (V k B), m = V k Ai 1 (V k B), k N. (3.4) LEMMA 3.3 Let {V k } k 0 be the family of sub-semimodules defined by the algorithm in (3.4). If there exists a well-defined non-empty semimodule k N V k, then any (Ã, B)-invariant sub-semimodule of K is contained in k N V k, namely the supremal controlled invariant sub-semimodule V is also contained in k N V k. Moreover, if the algorithm in (3.4) terminates in r steps, then V = V r. Proof of Lemma 3.3. V k V k 1, then The sequence of sub-semimodules {V k } k N is decreasing: clearly V 2 V 1, if V k+1 = V k à 1 (V k B) V k 1 à 1 (V k 1 B) = V k 1 m A 1 (V k 1 B) = V k. The next step is to show that any (A, B)-invariant sub-semimodule V in K is contained in k N V k. This conclusion will imply that the supremal controlled invariant sub-semimodule V is contained in k N V k. Clearly, V is contained in V 0 = K, suppose V is also contained in V k, then, V = V à 1 (V B) V k à 1 (V k B) = V k m A 1 i (V k B) = V k+1. Hence, V V k for all k N, namely V k N V k. Therefore, the supremal controlled invariant sub-semimodule V is contained in k N V k. If the algorithm in (3.4) terminates in r steps, then k N V k = V r. Because V r = V r à 1 (V r B), V r is (Ã, B)-invariant, namely V r V. On the other hand, V k N V k = V r from the previous step. Therefore, the equality holds for V = V r. The preceding lemma states and proves that the algorithm in (3.4) can be used to calculate the supremal controlled invariant sub-semimodule in K. Note that the algorithm in (3.4) does not always terminate in a finite number of steps. However, if restricting to a finite-volume semimodule in the integer max plus algebra Z Max, the algorithm in (3.4) guarantees to terminate in finite steps. PROPOSITION 3.1 Given a sub-semimodule K in Z n Max with a finite volume. The supremal (Ã, B)- invariant sub-semimodule of K under the dynamics of the system (2.1) is finitely generated. Moreover,
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 151 the sequence {V k } k N by the algorithm in (3.4) terminates in a finite number r of steps, i.e. V = V r for some r vol (K) + 1. Proof of Proposition 3.1. The proof is a direct generalization of Theorem 2 for the deterministic discrete event systems in Katz (2007). First of all, let us recall that the set of robust controlled invariant subsemimodules in a sub-semimodule K of X is a upper semilattice relative to sub-semimodule inclusion and operation. Therefore, there exists the supremal element V in the family of robust controlled invariant sub-semimodules, a sub-semimodule K in X. Because every semimodule in Z n Max with a finite volume is finitely generated and the supremal (Ã, B)-invariant sub-semimodule V is contained in K, we have vol(v ) vol(k) <. Therefore, V is also finitely generated. Moreover, the sequence {V k } k N is a decreasing sequence, so the sequence of the volumes of semimodules {V k } k N is a decreasing sequence of non-negative integers. There exists the smallest non-negative integer r vol(k) + 1, such that vol(v r ) = vol(v r+1 ) <. Hence, V r = V r+1 for some r vol(k) + 1. The following example illustrates the termination of the algorithm in (3.4) under the assumptions in Proposition 3.1. EXAMPLE 3.1 Consider an uncertain discrete event system over Z Max with system matrix morphism à co{a 1, A 2 }, where the two matrix morphisms A 1 : X X and A 2 : X X are given as A 1 = [ ] 1, A 0 2 = [ ] [ 2 0, B =. 0 0] We need to calculate the supremal (Ã, B)-invariant sub-semimodule in K = {(x, y) Z 2 Max x + 1 y x + 4}. The set K is K = {(x, y) K x y = 0} = {( 1, 0), ( 2, 0), ( 3, 0), ( 4, 0) }. The volume of K is 4. Then the algorithm in (3.4) to calculate the supremal controlled invariant subsemimodule will terminate at most vol(k) + 1 = 5 steps. We verify it by computing the algorithm in (3.4): V 1 = K, V 2 = V 1 2 A 1 i (V 1 B) = {(x, y) Z 2 Max x + 1 y x + 4} {(x, y) Z 2 Max x + 2 y x + 5} {(x, y) Z 2 Max x + 3 y x + 6} = {(x, y) Z 2 Max x + 3 y x + 4}, V 3 = V 2 2 A 1 i (V 2 B)
152 Y. SHANG = {(x, y) Z 2 Max x + 3 y x + 4} {(x, y) Z 2 Max x + 4 y x + 5} {(x, y) Z 2 Max x + 5 y x + 6} = {(, ) }, V 4 = V 3,. V k+1 = V 3,. The algorithm terminates in three steps and the supremal ( Ã, B)-invariant sub-semimodule in K is V = V 3 = {(, ) }. 3.2 (Ã, B)-invariant sub-semimodules of feedback type A sub-semimodule V is (à B F)-invariant if and only if there exists a state feedback F: X U for any à co{a 1,..., A m } such that (à B F)V V. For systems over a semiring or even a ring, a controlled invariant sub-semimodule is not identical with a control invariant sub-semimodule of feedback type, therefore, the computational method in the algorithm in (3.4) cannot be used to calculate (Ã, B)-invariant sub-semimodules of feedback type. LEMMA 3.4 A sub-semimodule V is ( Ã, B)-invariant of feedback type if and only if the sub-semimodule V is (A i, B)-invariant of feedback type, for any A i {A 1,..., A m } and any à co{a 1,..., A m }. Proof of Lemma 3.4. =. If a sub-semimodule V is (Ã, B)-invariant of feedback type for all à co{a 1,..., A m }, then there exists a state feedback F: X U such that (à B F)V V, which implies (A i B F)V V for any i {1,..., m}. Therefore, the sub-semimodule V is (A i, B)-invariant of feedback type. =. If the sub-semimodule V is (A i, B)-invariant of feedback type, for any A i {A 1,..., A m }, there there exists an F i : X U such that (A i B F i )V V. Therefore, ( m ) λ i (A i B F i ) V V = (à B F)V V, where F co{f 1,..., F m }. Hence, the sub-semimodule V is (Ã, B)-invariant of feedback type for all à co{a 1,..., A m }. For linear systems over the integer max plus algebra Z Max, if a given sub-semimodule V = ImQ, where Q Z n r Max, then V is (à B F)-invariant if and only if there exists matrices F Zq n Max and G Z r r Max such that (à B F)Q = QG (3.5) holds for all à co{a 1,..., A m }.
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 153 LEMMA 3.5 Given a sub-semimodule V = ImQ, where Q Z n r Max, then V is (Ã B F)-invariant if and only if there exists matrices F i Z q n Max and G i Z r r Max such that the equality (A i B F i )Q = QG i, i {1,..., m} (3.6) is true. Proof of Lemma 3.5. =. If (A i B F i )Q = QG i is true for any i {1,..., m}, then λ i (A i B F i )Q = λ i QG i implies ( m λ i A i B ) λ i F i Q = Q λ i G i implies (Ã B F)Q = Q G, where F co{f 1,..., F m } and G co{g 1,..., G m }. Then V is (Ã B F)-invariant. =. V is (Ã B F)-invariant, then (Ã B F)Q = QG, Ã co{a 1,..., A m }. Hence, (A i B F i )Q = QG i, for all i {1,..., m}. In order to find each pair of (F i, G i ) satisfying (3.6), the elimination method in Katz (2007) and the residuation theory in Baccelli et al. (1992) can be used by adding a variable t i to form the homogenous linear equation over the integer max plus algebra R Max : (A i t i B F i )Q = QG i, for i {1,..., m}. (3.7) Equation (3.6) has at least one set of solutions (F i, G i ) if and only if (3.7) has at least one set of solutions (t i, F i, G i ) with t i. This implies that the semimodule X = ImQ is (A i, B)-invariant of feedback type with F i = ( t i )F i. The necessary and sufficient condition for (t i, F i, G i ) to be a solution of (3.7) is t i (A i Q)\(QG i ), F i B\(QG i )/Q, G i Q\((A i t i B F)Q), for i {1,..., m}. The operations \ and / are defined for M Z n p Max M\N = {X Z p r Max M X N}, D/C = {X Z p r Max X C D}, and N Zn r Max, and for C Zr n Max and D Zp n Max. 4. Positively robust invariant polyhedral Sets This section presents the necessary and sufficient conditions for the polyhedral sets to be positively robust invariant under the dynamics of an uncertain linear system of the form (2.1) over an idempotent semiring R. Because the max plus algebra is a special idempotent semiring, the results in this section are applicable to systems of the form (2.1) over the max plus algebra.
154 Y. SHANG 4.1 Time-invariant polyhedral sets Considering the following time-invariant polyhedral set P(F, φ, ψ) = {x R n φ F x ψ}, φ, ψ R p and F R p n. For deterministic linear systems over an idempotent semiring, the necessary and sufficient condition for positively invariance of the set P(I n, φ, ψ) was established in Truffet (2004). LEMMA 4.1 (Truffet, 2004) Assume n = p and F = I n, where I n denotes the n n identity matrix. P(I n, φ, ψ) is positively invariant under the dynamics of the system over an idempotent semiring R, if and only if the condition x(k) = Ax(k 1), A R n n, (4.1) is true, where denotes the logic AND. (A ψ ψ) (φ A φ) (4.2) For uncertain discrete event systems of the form (2.1) over an idempotent semiring R, similar results can obtained if the input signals are zero. PROPOSITION 4.1 Assume n = p and F = I n, where I n denotes the n n identity matrix. P(I n, φ, ψ) is positively robust invariant under the dynamics of the system (2.1) over an idempotent semiring R and the inputs are zero, i.e. u = e U, if and only if for all i {1,..., m}. (A i ψ ψ) (φ A i φ), (4.3) Proof of Proposition 4.1. =. If P(I n, φ, ψ) is positively invariant under the dynamics of the system (2.1) over an idempotent and with zero inputs, then for any x R n such that φ x ψ, the condition φ à x ψ holds for any possible choice of à co{a 1,..., A m }. Therefore, the inequality φ A i x ψ is true, for any A i, where i {1,..., m}. Using Lemma 4.1, (A i ψ ψ) (φ A i φ) holds for all i {1,..., m}. =. Because (4.3) holds for all i {1,..., m}, we have ( m λ i A i ψ ) ( m λ i ψ λ i φ ) λ i A i φ. Since m λ i = 1 R and m λ i A i = Ã, the above condition becomes (à ψ ψ) (φ à φ). Using Lemma 4.1, P(I n, φ, ψ) is positively robust invariant for the uncertain discrete event system (2.1) over an idempotent semiring and with zero inputs.
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 155 THEOREM 4.1 Given E R n p satisfying { E F In, φ F E φ. The set P(F, φ, ψ) is positively robust invariant under the dynamics of the system (2.1) over an idempotent semiring R by x à x with zero inputs if and only if the condition (A i (F\ψ) (F\ψ)) (E φ A i E φ) (4.4) is true for all i {1,..., m}. The symbol F\ψ denotes {x X F x ψ}. Proof of Theorem 4.1. =. If P(F, φ, ψ) is positively invariant under the dynamics of the system (2.1) over an idempotent semiring R by x à x with zero inputs, then, for any x R n such that φ x ψ, we have φ à x ψ, for any possible choice of à co{a 1,..., A m }. Therefore, using Theorem 3.1 in Truffet (2004), the following condition holds: Therefore, for all i {1,..., m}, the equality [à (F\ψ) (F\ψ)] [E φ à E φ]. [A i (F\ψ) (F\ψ)] [E φ A i E φ] is true. =. Because (4.4) holds for all i {1,..., m}, we have λ i A i (F\ψ) λ i E φ λ i (F\ψ), λ i A i E φ. Since m λ i = 1 R and m λ i A i = Ã, the above condition becomes [à (F\ψ) (F\ψ)] [E φ à E φ]. (4.5) Using Theorem 3.1 in Truffet (2004), P(F, φ, ψ) is positively robust invariant for the discrete event system (2.1) over an idempotent semiring R by x Ãx with zero inputs. PROPOSITION 4.2 Assume n = p, P(F, φ, ψ) is positively invariant under the dynamics x Ax of the system (4.1) if there exists a matrix H R p p such that F A = H F and (H ψ ψ) (φ H φ). (4.6) The second condition means that P(I n, φ, ψ) is H-positively invariant.
156 Y. SHANG Proof of Proposition 4.2. We need to prove for any φ F x ψ, the condition φ F A x ψ holds. Since F A = H F, we only need to show that φ H F x ψ. Since P(I n, φ, ψ) is H-positively invariant, for all x R n, φ x ψ, we have φ H x ψ. Therefore, F x P(I n, φ, ψ). We have φ H F x ψ. Thus, P(F, φ, ψ) is positively invariant under the dynamics of the system (4.1). The preceding proposition is a sufficient condition for a polyhedral set, P(F, φ, ψ), to be positively invariant under the dynamics of the system (4.1). The following proposition states a sufficient condition for a polyhedral set P(F, φ, ψ) to be positively robust invariant under the dynamic of the system (2.1) over an idempotent semiring R and with zero inputs. PROPOSITION 4.3 Assume n = p, P(F, φ, ψ) is positively robust invariant under the dynamics of the system (2.1) over an idempotent semiring R and with zero inputs, i.e. x Ãx, if there exists a matrix H i R p p such that F A i = H i F and (H i ψ ψ) (φ H i φ). (4.7) The second condition means that P(I n, φ, ψ) is H i -positively invariant, for all i {1,..., m}. Proof of Proposition 4.3. We need to prove that, for any φ F x ψ, the condition φ F à x ψ holds for any arbitrary à co{a 1,..., A m }. Since F A i = H i F and P(I n, φ, ψ) is H i -positively invariant, for all x R n, φ x ψ, we have φ H i x ψ. Therefore, F x P(I n, φ, ψ). We have φ H i F x = F A i x ψ, for all i {1,..., m}. Combining them together to obtain φ = λ i φ (λ i F A i x) λ i ψ = ψ, where m (λ i F A i x) = F ( m λ i A i ) x = F à x. Therefore, we have φ F à x ψ for all φ F x ψ. Thus, P(F, φ, ψ) is positively invariant under the dynamics of the system (2.1) over an idempotent semiring R and with zero inputs. 4.2 Time-variant polyhedral sets Consider the following time-variant polyhedral sets: P(F, φ(k), ψ(k)) = {x(k) R n φ(k) F x(k) ψ(k)}, φ(k) = K φ φ(k 1), ψ(k) = K ψ ψ(k 1), where φ, ψ R p, k Z + and F R p n and x(k) is governed by system (2.1) or system (4.1). Timevariant polyhedral sets are polyhedrons with time-variant boundaries, conditions that are very common in applications such as public transportation networks with time-varying time tables and queuing networks with time-varying arrival or service time.
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 157 LEMMA 4.2 Assume n = p and F = I n, where I n denotes the n n identity matrix. P(I n, φ(k), ψ(k)) is positively robust invariant under the dynamics of system (4.1) by x Ax if and only if for all k Z +. (A ψ(k) K ψ ψ(k)) (K φ φ(k) A φ(k)), (4.8) Proof of Lemma 4.2. =. If P(I n, φ(k), ψ(k)) is positively invariant under the dynamics of the system (4.1), then for any x R n such that φ(k) x(k) ψ(k), we have φ(k +1) = K φ φ(k) A x(k) K ψ ψ(k) = ψ(k + 1). Since φ(k) and ψ(k) P(I n, φ(k), ψ(k)), we have (A ψ(k) K ψ ψ(k)) (K φ φ(k) A φ(k)). =. If the condition (A ψ(k) K ψ ψ(k)) (K φ φ(k) A φ(k)) holds, then for any x P(I n, φ(k), ψ(k)), i.e. φ(k) x(k) ψ(k), hence the condition K φ φ(k) A φ(k) A x(k) A ψ(k) K ψ ψ(k) implies φ(k + 1) x(k + 1) ψ(k + 1). Therefore, P(I n, φ(k), ψ(k)) is positively invariant under the dynamics of the system (4.1). Lemma 4.2 presents the necessary and sufficient conditions for a time-variant polyhedral set P(I n, φ(k), ψ(k)) to be positively invariant for the system (4.1). If we are given K φ and K ψ, the searching method for φ(k) ψ(k), such that P(I n, φ(k), ψ(k)), k 1, is positively invariant for the system (4.1), can be performed as follows. 1. Solve the following equations for φ ψ, A ψ = K ψ ψ, A φ = K φ φ. 2. For any x(0) R n such that let φ(0) = φ and ψ(0) = ψ and φ(0) x(0) ψ(0), we have A φ(0) A x(0) A ψ(0) = K φ φ(0) A x(0) K ψ ψ(0) = φ(1) x(1) ψ(1). Continuing this process, we are able to show that P(I n, φ(k), ψ(k)) is positively invariant with respect to system (4.1). Therefore, φ and ψ generate the possible boundaries for a positively invariant polyhedral set. PROPOSITION 4.4 Assume n = p and F = I n, where I n denotes the n n identity matrix. P(I n, φ(k), ψ(k)) is positively robust invariant under the dynamics of the system (2.1) over an idempotent
158 Y. SHANG semiring R and with zero inputs by x Ãx if and only if for all i {1,..., m} and k Z +. (A i ψ(k) K ψ ψ(k)) (K φ φ(k) A i φ(k)), (4.9) Proof of Proposition 4.4. =. If P(I n, φ(k), ψ(k)) is positively robust invariant under the dynamics of the system (2.1) over an idempotent semiring R and with zero inputs, then, for any x R n such that φ(k) x(k) ψ(k), we have φ(k + 1) = K φ φ(k) à x(k) K ψ ψ(k) = ψ(k + 1), for an arbitrary à co{a 1,..., A m }. Since φ(k) and ψ(k) P(I n, φ(k), ψ(k)), we have (à ψ(k) K ψ ψ(k)) (K φ φ(k) à φ(k)) = (A i ψ(k) K ψ ψ(k)) (K φ φ(k) A i φ(k)) for all i {1,..., m}. =. If we have (A i ψ(k) K ψ ψ(k)) (K φ φ(k) A i φ(k)) for all i {1,..., m}, then P(I n, φ(k), ψ(k)) is A i -positively invariant. Therefore, for any x P(I n, φ(k), ψ(k)), i.e. φ(k) x(k) ψ(k), we have K φ φ(k) A i x(k) K ψ ψ(k) = λ i K φ φ(k) λ i A i x(k) λ i K ψ ψ(k). Combining them together for all i {1,..., m} to obtain λ i K φ φ(k) λ i A i x(k) λ i K ψ ψ(k). Since m λ i = 1 R, the condition K φ φ(k) à x(k) K ψ ψ(k) holds for arbitrary à co{a 1,..., A m }. Therefore, P(I n, φ(k), ψ(k)) is positively robust invariant under the dynamics of the system (2.1) over an idempotent semiring R and with zero inputs. The preceding proposition presents a necessary and sufficient condition for P(I n, φ(k), ψ(k)) to be positively robust invariant under the dynamics of the system (2.1) with zero inputs. If we are given K φ and K ψ, the searching method for φ(k) ψ(k), such that P(I n, φ(k), ψ(k)), k 1, is positively robust invariant for the system (2.1) with zero inputs, is similar to the positively invariant case. 1. Solve the following equations for φ ψ, A i ψ = K ψ ψ, A i φ = K φ φ, for all possible A i, i {1,..., m}. 2. For any x(0) R n such that let φ(0) = φ and ψ(0) = ψ and φ(0) x(0) ψ(0), we have A i φ(0) A i x(0) A i ψ(0) = K φ φ(0) A i x(0) K ψ ψ(0) = φ(1) x(1) ψ(1),
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 159 for all i {1,..., m}. Continuing this process, we can show that P(I n, φ(k), ψ(k)) is positively robust invariant with respect to system (2.1) with zero inputs. Therefore, φ and ψ generate the possible boundaries for a positively robust invariant polyhedral set. The following proposition is a sufficient condition for a polyhedral set P(F, φ(k), ψ(k)) to be positively invariant under the dynamic of the system (4.1). PROPOSITION 4.5 Assume n = p, P(F, φ(k), ψ(k)) is positively invariant under the dynamics of the system (4.1) if there exists a matrix H R p p such that F A = H F and (H ψ(k) K ψ ψ(k)) (K ψ φ(k) H φ(k)). The second condition means that P(I n, φ(k), ψ(k)) is H-positively invariant. Proof of Proposition 4.5. We need to prove that, for any φ(k) F x(k) ψ(k), the condition K φ φ(k) = φ(k + 1) F A x(k) ψ(k + 1) = K ψ ψ(k) is true. Since F A = H F, we only need to show that K φ φ(k) H F x(k) K ψ ψ(k). Since P(I n, φ(k), ψ(k)) is H-positively invariant, for all x R n, φ(k) x(k) ψ(k), we have φ(k +1) H x(k) ψ(k +1). Therefore, F x(k) P(I n, φ(k), ψ(k)). We have φ(k + 1) H F x(k) ψ(k + 1). Thus, P(F, φ(k), ψ(k)) is positively invariant under the dynamics of the system (4.1). The following proposition states a sufficient condition for a polyhedral set P(F, φ(k), ψ(k)) to be positively robust invariant. PROPOSITION 4.6 Assume n = p, P(F, φ(k), ψ(k)) is positively robust invariant under the dynamics of the system (2.1) over an idempotent semiring R and with zero inputs if there exists a matrix H R p p such that F A i = H i F and (H i ψ(k) K ψ ψ(k)) (K φ φ(k) H i φ(k)). The second condition means that P(I n, φ(k), ψ(k)) is H i -positively invariant, where i {1,..., m} and k Z +. Proof of Proposition 4.6. We need to prove that, for any φ(k) F x(k) ψ(k), the condition φ(k+1) F à x(k) ψ(k+1) holds for an arbitrary à co{a 1,..., A m }. Since F A i = H i F and P(I n, φ(k), ψ(k)) is H i -positively invariant, for all x R n, φ(k) x(k) ψ(k), we have φ(k + 1) H i x(k) ψ(k + 1). Therefore, F x(k) P(I n, φ, ψ). We have φ(k + 1) H i F x(k) = F A i x(k) ψ(k + 1), for all i {1,..., m}. Adding them together to obtain φ(k + 1) (λ i F A i x(k)) ψ(k + 1), where m (λ i F A i x(k)) = F ( m λ i A i ) x(k) = F à x(k). Therefore, we have φ(k + 1) F à x(k) ψ(k + 1) for all φ(k) F x(k) ψ(k). Thus, P(F, φ(k), ψ(k))
160 Y. SHANG is positively invariant under the dynamics of the system (2.1) over an idempotent semiring R and with zero inputs. 4.3 Example: a public transportation network In this section, a public transportation network (De Vries et al., 1998) is modelled by an uncertain linear system over the max plus algebra, whose system matrices are uncertain. In a small public transportation network shown in Fig. 1, there are train services from P via Q to S and back and from Q to R and back. Trains from P to S have to stop at Q for the connection to trains with destination R and vice versa. If x i ( ) denotes the departure time of the train in the direction i, i {1,..., 4}. The train which is about to leave in the direction i for the kth time cannot leave if the train has not arrived yet. This condition can be represented as x i (k) a i j x j (k 1), (4.10) where x i (k) denotes the kth departure time in direction i and a i j is the travelling time from direction j to i, including the loading time of passengers. Another condition is that the train needs to wait for the possible connecting trains, i.e. x i (k) a il x l (k 1), (4.11) where l is the possible connecting direction and a il denotes the travelling time from direction l to i, also including the loading time of passengers. For the simple transportation network in Fig. 1, the system equation is the following: x 1 (k) = a 2 x 2 (k 1), x 2 (k) = a 3 x 3 (k 1) a 4 x 4 (k 1), x 3 (k) = a 1 x 1 (k 1) a 3 x 3 (k 1), a 4 x 4 (k 1), x 4 (k) = a 1 x 1 (k 1) a 3 x 3 (k 1), where a i denotes the travelling time on direction i {1,..., 4}. The state equation can be rewritten in the matrix form ɛ a 2 ɛ ɛ ɛ ɛ a x(k) = Ax(k 1), where A = 3 a 4 a 1 ɛ a 3 a 4. a 1 ɛ a 3 ɛ If the travelling time a i is deterministic, then the linear system is a deterministic discrete event system. In reality, however, the travelling time usually varies due to traffic and other emergency situations. For instance, assume a 1 [10, 20] minutes, a 2 [15, 25] minutes, a 3 [10, 20] minutes and a 4 [15, 25] minutes, then the system becomes an uncertain linear system x(k) = Ãx(k 1), where à co{a 1, A 2 }
ROBUST INVARIANCE IN UNCERTAIN DISCRETE EVENT SYSTEMS 161 FIG. 1. A small public transportation network De Vries et al. (1998). and ɛ 15 ɛ ɛ ɛ 25 ɛ ɛ ɛ ɛ 10 15 A 1 = 10 ɛ 10 15 and A ɛ ɛ 20 25 2 = 20 ɛ 20 25. 10 ɛ 10 ɛ 20 ɛ 20 ɛ Considering a time-varying polyhedral set where K φ = 13.3333, K ψ = 23.3333 and P(I 4, φ(k), ψ(k)) = {x R φ(k) x(k) ψ(k)}, φ(k) = K φ φ(k 1) and ψ(k) = K ψ ψ(k 1). Firstly, we solve the following equations for φ ψ, A i ψ = K ψ ψ, A i φ = K φ φ, for all possible A i, i {1, 2}. Because K φ and K ψ are eigenvalues for A 1 and A 2, respectively, we obtain that φ(0) = φ = [30, 28.3333, 28.3333, 26.6667] and ψ(0) = ψ = [50, 48.3333, 48.3333, 46.6667], which are the eigenvectors for both A i matrices, for i = 1, 2. Using Proposition 4.4, we can verify that (A i ψ(k) K ψ ψ(k)) (K φ φ(k) A i φ(k)), (4.12) for all i {1,..., m} and k Z +. Therefore, the given polyhedral set is positively robust invariant with respect to the public transportation network for arbitrary choice of matrix à co{a 1, A 2 }. After computing the system trajectories, the four states, x i (k), are in the set, P(I 4, φ(k), ψ(k)), as shown in Fig. 2. The values for φ(k) and ψ(k) can be used as time-variant time tables for the train station because the departure time in each direction is constrained in this positively robust invariant set.
162 Y. SHANG FIG. 2. The travelling time in direction 1, 2, 3, 4. 4.4 Positively invariant under the state feedback control Considering uncertain discrete event systems of the form (2.1) over an idempotent semiring R and with non-zero control input signals, the necessary and sufficient condition is stated in the following theorem for the polyhedral set P(I n, φ, ψ) to be positively robust invariant under a state feedback controller F: X U. THEOREM 4.2 There exists a state feedback controller F: X U for the system described by (2.1) over an idempotent semiring R such that the polyhedral set P(I n, φ, ψ) is positively robust invariant if and only if for all A i {A 1,..., A m }. [φ (A i B(B\(ψ/ψ))) φ] (A i ψ/ψ), (4.13) Proof of Theorem 4.2. If the polyhedral set P(I n, φ, ψ) is positively robust invariant under a state feedback controller F: X U for the system described by(2.1) over an idempotent semiring R, then for any x P(I n, φ, ψ), φ (Ã B F)x ψ φ (A i B F)x ψ, for i {1,..., m}. Therefore, P(I n, φ, ψ) is positively invariant under x (A i B F)x. By Lemma 4.1, this is equivalent to F (φ (A i B F) φ) (A i B F) ψ ψ. (4.14) The condition F (A i B F) ψ ψ is equivalent to (A i ψ/ψ) ( F F B\(ψ/ψ)). Therefore, the condition in (4.14) is equivalent to (4.13), i.e. for all A i {A 1,..., A m }. [φ (A i B(B\(ψ/ψ))) φ] (A i ψ/ψ),
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