On the Computation of Backward Reachable Sets for Max-Plus Linear Systems with Disturbances

On the Computation of Backward Reachable Sets f or Max-Plus Linear Systems with Disturbances Y uda Li and Xiang Y in Abstract — This paper in vestigates one-step backward reach- ability for uncertain max-plus linear systems with additi ve disturbances. Given a tar get set, the problem is to compute the set of states from which there exists an admissible control input such that, for all admissible disturbances, the succes- sor state remains in the target set. This problem is closely related to safety analysis and is challenging due to the high computational complexity of existing approaches. T o address this issue, we develop a computational framework based on tropical polyhedra. W e assume that the target set, the control set, and the disturbance set are all represented as tropical polyhedra, and study the structural properties of the associated backward operators. In particular , we sho w that these operators preser ve the tropical-polyhedral structure, which enables the constructive computation of reachable sets within the same framework. The proposed approach provides an effective geo- metric and algebraic tool for reachability analysis of uncertain max-plus linear systems. Illustrative examples are included to demonstrate the proposed method. I . I N T R O D U C T I ON Max-plus linear systems (MPLSs) is an important class of discrete-ev ent systems formulated over the max-plus algebra [1], [2]. This algebraic setting is particularly well suited for describing synchronization and delay effects in timed event- driv en systems. MPLSs are closely connected with timed ev ent graphs [3], [4], which form a subclass of Petri nets characterized. Because of their effecti veness in representing ev ent timings under precedence relations, max-plus models hav e been widely used in applications such as scheduling [5], manufacturing [6], [7], [8], and transportation systems [9], [10], [11]. In practical applications, howe ver , max-plus systems are often affected by uncertainty arising from disturbances, mod- eling errors, or parameter variations. This makes reachability analysis and controller synthesis significantly more challeng- ing. Existing approaches for uncertain MPLSs can be broadly divided into two categories. The first models uncertainty probabilistically . For example, [12] studied formal verifica- tion for stochastic max-plus linear systems by treating the system matrix as a random variable, while [13] in vestigated model predictiv e control for max-plus linear systems with uncertain parameters endowed with probability distributions. These approaches provide a probabilistic characterization of This w ork was supported by the National Science and T echnology Major Project (2025ZD1600700) and the National Natural Science Foundation of China (62573291,62533017,62173226). Y uda Li and Xiang Y in are with School of Automation and Intelli- gent Sensing, Shanghai Jiao T ong Univ ersity , Shanghai 200240, China. { yuda.li, yinxiang } @sjtu.edu.cn . uncertainty and enable the use of stochastic verification and control techniques. A second line of work models uncertainty through deter - ministic ambiguity sets rather than probability distributions. In this direction, [14] studied reachability analysis for un- certain max-plus linear systems subject to bounded noise, disturbances, and modeling errors. There, the uncertainty set is described in the interval max-plus framew ork, and reachability is computed by transforming the system into a piecewise affine representation and then applying difference- bound matrices (DBMs). Howe ver , the associated computa- tions may become expensi ve as the system dimension and uncertainty structure grow . In this paper, we study the backward reachability problem for max-plus linear systems with disturbances. Given a target set, the objectiv e is to compute the set of states from which there exists an admissible control input such that, for all admissible disturbances, the successor state remains in the target set. This problem is closely connected to safety analysis and controlled in variance, since backward reachable sets provide the basic ingredient for determining safe sets and synthesizing safety-preserving controllers. While such problems can in principle be addressed using DBM-based techniques, the resulting computations may suffer from high complexity . Moreover , compared with the disturbance-free case studied in our recent work [15], the presence of dis- turbances introduces a substantial new difficulty through the univ ersal quantification over the uncertainty set. Our approach is motiv ated by recent developments in trop- ical con vexity , in particular the theory of tropical polyhedra [16], [17]. W e model the control and disturbance sets, as well as the target set, within a tropical polyhedral framework, which provides a flexible and expressi ve representation of uncertainty in the max-plus setting. Based on this represen- tation, we dev elop a computational procedure for backward reachability analysis of uncertain max-plus linear systems. The proposed framework accommodates a broader class of uncertainty structures than interval-based descriptions, while preserving a compact geometric representation amenable to computation. By exploiting structural properties of tropical polyhedra, we sho w that backward reachable sets can still be characterized and computed within the same framework. In this way , the paper offers a ne w geometric perspectiv e on the analysis of uncertain max-plus systems and pro vides an effecti ve tool for safety verification and control synthesis. I I . P R E L I M I N A RY In this section, we introduce the basic notation and pre- liminary results used throughout the paper . A. The Max-Plus Algebra R max The max plus algebra R max is the set of real numbers together with an infinite point: R ∪ {−∞} , together with the binary operations ⊕ : R max × R max → R max and ⊗ : R max × R max → R max defined by a ⊕ b = max( a, b ) and a ⊗ b = a + b . For any a ∈ D , we define ( −∞ ) ⊕ a = a ⊕ ( −∞ ) = a and ( −∞ ) ⊗ a = a ⊗ ( −∞ ) = −∞ . For simplicity , we write ε for −∞ throughout the paper . On R max , we consider the metric d ( x, y ) = | exp( x ) − exp( y ) | , with the con vention that exp( ε ) = 0 . Open and closed sets are understood with respect to the topology induced by this metric. In the remainder of the paper, we focus on the max- plus algebra R max , which we simply denote by R max . B. Semimodules, Sub-Semimodules, and Con vex Cones The Cartesian product of n copies of R max , denoted by R n max , is naturally equipped with the vector addition ⊕ : R n max × R n max → R n max defined componentwise by ( a 1 , . . . , a n ) ⊕ ( b 1 , . . . , b n ) = ( a 1 ⊕ b 1 , . . . , a n ⊕ b n ) , and the scalar multiplication · : R max × R n max → R n max defined by a · ( b 1 , . . . , b n ) = ( a ⊗ b 1 , . . . , a ⊗ b n ) . The triple ( R n max , ⊕ , · ) is called a max-plus R max -semimodule, or simply a semi- module. This space is also endo wed with the natural partial order x ≤ y ⇔ x i ≤ y i , ∀ 1 ≤ i ≤ n . The element ε n = ( ε, . . . , ε ) ∈ R n max is the minimal element. Any matrix A ∈ R n × m max defines a linear map ϕ A : R m max → R n max by ϕ A ( x ) i = L m j =1 A ij ⊗ x j , where x = ( x 1 , . . . , x m ) ∈ R m max . In the rest of the paper, we simply write ϕ A ( x ) as A ⊗ x . Similarly , for a, b ∈ R n max , their scalar product is defined by ( a | b ) = L n i =1 a i ⊗ b i . A subsemimodule is a subset S ⊆ R n max such that ∀ x, y ∈ S , ∀ µ, λ ∈ D µ · x ⊕ λ · y ∈ S . The topology on R n max is taken to be the product topology induced by the metric on R max . A con v ex cone o ver D is a subset C ⊆ R n max satisfying ∀ x, y ∈ D , ∀ λ 1 , λ 2 ∈ R , [ λ 1 ⊕ λ 2 = 0] ⇒ λ 1 · x ⊕ λ 2 · y ∈ D . For two subsets A, B ⊆ R n max , we define their max-plus sum by A ⊕ B = { a ⊕ b | a ∈ A, b ∈ B } . C. T r opical Cones, Half-Spaces and P olyhedr a A tropical half-space H is a subset of R n max of the form H = { x ∈ R n max | ( a | x ) ≤ ( b | x ) } , where a, b ∈ R n max . A tropical cone is an intersection of finitely many half-spaces, or equiv alently , a set of the form C = { x ∈ R n max | A ⊗ x ≤ B ⊗ x } (1) which we denote for simplicity as C = ⟨ A, B ⟩ . Similarly , a tropical polyhedra is a set of the form P = { x ∈ R max | A ⊗ x ⊕ c ≤ B ⊗ x ⊕ d } , (2) which we denote for simplicity as P = ⟨ ( A, c ) , ( B , d ) ⟩ . W e call such representation an outer representation, or M -form of a tropical cone (or tropical polyhedra). For simplicity , we denote ⟨ A, B ⟩ s = ⟨ A, B ⟩ ∩ ⟨ B , A ⟩ (resp ⟨ ( A, c ) , ( B , d ) ⟩ s = ⟨ ( A, c ) , ( B , d ) ⟩ ∩ ⟨ ( B , d ) , ( A, c ) ⟩ ), which are also tropical polyhedra. It has been proved by [18] that ev ery tropical cone C can be generated by a finite set such that ∃ v 1 , . . . , v N ∈ C , C = ( N M i =1 λ i · v i      λ 1 , . . . , λ N ∈ R max ) . (3) W e call { v 1 , . . . , v N } the inner representation of C , or its V - form, and denote C as Span( { v 1 , . . . , v N } ) . T ropical cones and tropical half-spaces are both subsemimodules, but the reciprocate is not true, since a subsemimodule may not be finitely generated. A bounded tropical polyhedron is a finitely generated tropical con vex cone of the following general form B = ( N M i =1 λ i · e i      N M i =1 λ i = 0 ) , where N ∈ Z > 0 and e i ∈ R n max are vectors. F or simplicity , we write B = Conv( e 1 , . . . , e N ) . It was proved in [19] that P can be decomposed as P = R ⊕ B , where R is a tropical cone and B is a bounded tropical polyhedron. The tropical cone R is called the recession cone, and it is unique for this decomposition. For a vector v ∈ R n max , we denote by v [ k : m ] the subvector consisting of the coordinates with indices ranging from k to m . For a matrix A ∈ R k × m max , we denote by A [ k 1 : k 2 ][ m 1 : m 2 ] the submatrix ( A i,j ) k 1 ≤ i ≤ k 2 ,m 1 ≤ j ≤ m 2 . For simplicity , if k 1 = 1 , k 2 = k or m 1 = 1 , m 2 = m , we denote respectiv ely A • , [ m 1 : m 2 ] and A [ k 1 : k 2 ] , • (or simplier A [ k 1 : k 2 ] ) . If k 1 = k 2 or m 1 = m 2 , we simply denote as A k 1 , [ m 1 : m 2 ] or A [ k 1 : k 2 ] ,m 1 . Next we discuss some useful properties about tropical cones and tropical polyhedra, which will be used throughout this paper . First, both tropical cones and bounded tropical polyhedras are tropical polyhedras, since they are respec- tiv ely a special case with B = { ε n } or with R = { ε n } . The following lemma characterize the relation between tropical polyhedra and Lemma 1: Let P ⊆ R n max be a tropical polyhedra, then there exists C ⊆ R n +1 max a tropical cone such that P = { x ∈ R n max | [0 , x T ] T ∈ C } . Reciprocally , all sets of this form is a tropical polyhedra. Pr oof: For simplicity , we identify the tuple ( x 1 , . . . , x n ) with the column vector [ x 1 , . . . , x n ] T , and we write [0 , x T ] T simply as (0 , x ) . Note that P is precisely the intersection of the hyperplane x 1 = 0 with the tropical cone C . Thus, by a slight ab use of notation, we write P = C ∩ R n max . Giv en a tropical polyhedra with the following form P = Span( v 1 , . . . , v N ) ⊕ Conv( e 1 , . . . , e M )) , we can construct the corresponding tropical cone C = Span(( ε, v 1 ) , . . . , ( ε, v N ) , (0 , e 1 ) , . . . , (0 , e M )) ⊆ R n +1 max then C is such that P = C ∩ R n max . Con versely , given a tropical cone C ⊆ R n +1 max generated by a set V , define U = { ( − v 1 ) · v [2: n +1] ∈ R n max | v ∈ V , v 1  = ε } , and W = { v [2: n +1] ∈ R n max | v ∈ V , v 1 = ε } . Then one can verify that C ∩ R n max = Span( W ) ⊕ Conv( U ) . I I I . P R O B L E M F O R M U L AT I ON In this section, we formulate the backward reachability problem for max-plus linear systems subject to disturbances. W e first introduce the system model and the associated backward reachability operator, and then decompose this operator into sev eral elementary set-valued operators that will be studied in the sequel. W e consider the follo wing max-plus linear system (MPLS) with disturbance over R max : x k = A ⊗ x k − 1 ⊕ B ⊗ u k ⊕ C ⊗ w k , (4) where x k ∈ R n max is the system state, u k ∈ U ⊆ R m max is the control input, and w k ∈ W ⊆ R q max is the disturbance. The matrices A ∈ R n × n max , B ∈ R n × m max , and C ∈ R n × q max are the system matrices. Giv en a target subset E ⊆ R n max , we define the (one-step) backward reachable set of E as Υ( E ) = ( x ∈ R n max      ∃ u ∈ U , ∀ w ∈ W s.t. A ⊗ x ⊕ B ⊗ u ⊕ C ⊗ w ∈ E ) . In other words, Υ( E ) consists of all states from which one can choose a control input such that, for e very admissible disturbance, the successor state remains in E after one step. T o better understand the structure of the backward reach- ability operator Υ , we next introduce three elementary set- valued operators that capture, respecti vely , the effect of the system dynamics, the universal quantification ov er distur- bances, and the existential quantification over control inputs. • In verse operator : A − 1 : 2 R n max → 2 R n max such that E 7→ { x ∈ R n max | A ⊗ x ∈ E } . • Universal in verse operator (w .r .t. the disturbance set W ): ϕ W : 2 R n max → 2 R n max such that E 7→ { x ∈ R n max | ∀ w ∈ W , x ⊕ C ⊗ w ∈ E } . • Existential in verse operator (w .r .t the control set U ): γ U : 2 R n max → 2 R n max such that E 7→ { x ∈ R n max | ∃ u ∈ U , x ⊕ B ⊗ u ∈ E } . W ith these definitions, the backward reachability operator can be decomposed as Υ = A − 1 ◦ γ U ◦ ϕ W . This decomposition separates the contributions of the dis- turbance, the control action, and the autonomous system dynamics, and will serve as the basis for the computational dev elopments in the subsequent sections. W e now state the main problem considered in this paper . Problem 1 (Backward Reachability Pr oblem): Let S be the MPLS with disturbance defined by (4). Assume that the control set U ⊆ R m max , the disturbance set W ⊆ R q max , and the target set S ⊆ R n max are tropical polyhedra. Giv en a tropical polyhedron E ⊆ R n max , compute the (one-step) backward reachable set Υ( E ) of E with respect to S . Compared with classical linear systems, the backward reachability problem considered here is substantially more in volv ed. In the max-plus framework, the operations defining the system ev olution may transform a set in a way that changes its underlying geometric structure. In particular , ev en when the initial set is a tropical polyhedron, it is not obvious a priori whether the successi ve application of in verse images, univ ersal constraints with respect to disturbances, and existential projections with respect to controls still yields a tropical polyhedron. This moti vates the main objective of this paper: to characterize conditions under which the operator Υ preserv es the tropical polyhedral structure, and to develop effective procedur es for computing Υ( E ) when E , U , and W are tropical polyhedra. The remainder of this paper addresses the solution to Problem 1 and is organized as follo ws. Section IV derives computational procedures for the operators A − 1 and γ □ , which are based on similar techniques. In contrast, the univ ersal inv erse operator ϕ □ is more challenging and is therefore treated separately . Section V analyzes its structural properties, and Section VI develops a computational method based on these results. I V . C O M P U TA T I O N F O R T H E O P E R ATO R S A − 1 A N D γ □ In the following, we explain ho w to compute the backward reachable set associated with the in verse operator A − 1 and the existential inv erse operator γ □ defined in Section III. W e prov e that the tropical polyhedra structure is preserved under both operators and derive a method to determine generating vectors for the image of a tropical polyhedron under each of these operators. W e begin with the following lemma, which is essential for the computation of these operators. Lemma 2: Let Z be a tropical cone, then the projection on its first r coordinates ( r ≤ n ): p r ( Z ) = { z ∈ R r max | ∃ v ∈ R n − r max , ( z , v ) ∈ Z } is still a tropical cone. Pr oof: Let v 1 , . . . , v k be a family of generating vectors for C . Denote by v ′ 1 , . . . , v ′ k the vectors obtained by restrict- ing v 1 , . . . , v k to their first r coordinates. Then the projection of Z onto the first r coordinates is the tropical cone in R r max generated by v ′ 1 , . . . , v ′ k . A. Computation for the ima ge of A − 1 In this section, we illustrate the procedure for computing the backward propagation set A − 1 ( Z ) , where A ∈ R n × n max is a matrix and Z ⊆ R n max is a tropical polyhedron of the form C ∩ R n max . Here, C ⊆ R n +1 max is a tropical cone generated by a matrix M ∈ R ( n +1) × q max . W ithout loss of generality , we assume that all coefficients in the first row of M are either ε or 0 . By definition, we hav e A − 1 ( Z ) = ( x ∈ R n max      ∃ u ∈ R q max , A ⊗ x = M [2: n +1] , • ⊗ u M 1 , • ⊗ u = 0 ) Note that A − 1 ( Z ) can be equi valently rewritten as A − 1 ( Z ) = ( x ∈ R n max      ∃ u ∈ R q max , z = (0 , x, u ) M A,M 1 ⊗ z = M A,M 1 ⊗ z ) where the matrices M A,M 1 and M A,M 2 are defined by M A,M 1 =  0 E 1 × n E 1 × q E n × 1 A E n × q  M A,M 2 =  ε E 1 × n M 1 , • E n × 1 E n × n M [2: n +1] , •  Observe that the vector (0 , x, u ) ranges precisely ov er the tropical polyhedron D M A,M 1 , M A,M 2 E . Consequently , we obtain A − 1 ( Z ) = n x ∈ R n max    (0 , x ) ∈ p n +1 D M A,M 1 , M A,M 2 E s o = p n +1 D M A,M 1 , M A,M 2 E s  ∩ R n max This formulation pro vides an outer description of the set A − 1 ( Z ) , and the above construction leads to the follo wing corollary . Corollary 1: Let A ∈ R n × n max be a matrix, and let Z ⊆ R n max be a tropical polyhedron. Then the set A − 1 ( Z ) is also a tropical polyhedron. B. Computation for γ □ In this section, we illustrate the computation of the backward reachable set associated with the operator γ □ . Let A ∈ R n × n max be a matrix, and let U = G ∩ R n max be a tropical polyhedron, where G is generated by a matrix R ∈ R ( n +1) × r max whose first-row coefficients are either 0 or ε . Let Z be a tropical polyhedron of the same form as defined in Section IV -A. Our goal is to compute the set γ U ( Z ) . By definition, we have γ U ( Z ) =          x ∈ R m max          ∃ y ∈ R r max , ∃ z ∈ R q max , x ⊕ A ⊗ R [2: n +1] , • ⊗ y = M [2: n +1] , • ⊗ z , R 1 , • ⊗ y = 0 , M 1 , • ⊗ z = 0          . Similarly to Section IV -A, the set γ U ( Z ) can be rewritten as γ U ( Z ) =      x ∈ R m max        ∃ ( y , z ) ∈ R r max × R q max ∃ t = (0 , x, y , z ) , M A,R,M 1 ⊗ t = M A,R,M 2 ⊗ t      . The matrices M A,R,M 1 and M A,R,M 2 are defined as M A,R,M 1 =   ε Id m A ⊗ R [2: n +1] , • E m × q ε E 1 × m R 1 , • E 1 × q ε E 1 × m E 1 × r M 1 , •   M A,R,M 2 =   ε E m × m E m × r M [2: n +1] , • 0 E 1 × m E 1 × r E 1 × q 0 E 1 × m E 1 × r E 1 × q   . Then γ U ( Z ) can be e xpressed as γ U ( Z ) = p n +1 D M A,R,M 1 , M A,R,M 2 E s  ∩ R n max . As before, this formulation provides an outer description of the set γ U ( Z ) . W e thus obtain the follo wing corollary . Corollary 2: Let A ∈ R n × n max be a matrix, and let Z , U be two tropical polyhedra. Then the set γ U ( Z ) is also a tropical polyhedron. V . C O M P U TA T I O N F O R ϕ W W e are now interested in the set propagation operator ϕ □ . Giv en a tropical polyhedron Z of the form ⟨ C , D ⟩ ∩ R n max (with C , D ∈ R q × ( n +1) max ), we aim to characterize the set ϕ W ( Z ) , where W is a tropical polyhedron in R n max . In the follo wing, we first show that this set is itself a tropical polyhedron, and then pro vide techniques to deriv e its inner representation. A. Simplification Step As shown in [19], W can be decomposed as the max-plus sum of two components: W = R ⊕ P , where P is a bounded tropical polyhedron and R is a tropical cone (its recession cone). The follo wing proposition allows us to simplify the computation of ϕ W ( Z ) . Proposition 1: Let Z and W = P ⊕ R be de- fined as abov e. A necessary condition for ϕ W ( Z ) to be nonempty is that R is contained in the tropical cone ⟨ C • , [2: n +1] , D • , [2: n +1] ⟩ . In this case, we hav e ϕ W ( Z ) = ϕ P ( Z ) . Pr oof: W e first prove that R ⊆ ⟨ C • , [2: n +1] , D • , [2: n +1] ⟩ is a necessary condition for ϕ W ( Z ) being non empty . Sup- pose by contradiction that there exists y ∈ R such that C • , [2: n +1] ⊗ y ≤ D • , [2: n +1] ⊗ y does not holds (there e xists index i , ( C i, [2: n +1] | y ) > ( D i, [2: n +1] | y ) ) and there exists x ∈ R n max s.t. ∀ u ∈ W , x ⊕ u ∈ Z , then for arbitrary y ′ ∈ P and for all λ ∈ R max , we have C ⊗ (0 , x ⊕ y ′ ⊕ λ · y ) ≤ D ⊗ (0 , x ⊕ y ′ ⊕ λ · y ) . As ( C i, [2 ,n +1] | λ · y ) > ( D i, [2: n +1] | λ · y ) for all λ > ε , fix y ′ ∈ P , then for λ large enough we will hav e ( C i | (0 , x ⊕ y ′ ⊕ λ · y )) > ( D i | (0 , x ⊕ y ′ ⊕ λ · y )) , which is a contradiction. Once we hav e verified that the recession cone R is contained in ( C i, [2: n +1] | y ) > ( D i, [2: n +1] | y ) , we can prove that ϕ W ( Z ) = ϕ P ( Z ) . Indeed, it is immediate that ϕ W ( Z ) ⊆ ϕ P ( Z ) . T o sho w that ϕ P ( Z ) ⊇ ϕ P ( Z ) , take x ∈ ϕ P ( Z ) , then for all y ′ ∈ P , we have C ⊗ (0 , x ⊕ y ′ ) ≤ D ⊗ (0 , x ⊕ y ′ ) . Let u ∈ W be of the form u = y ⊕ y ′ with y ′ ∈ P and y ∈ R , since C • , [2: n +1] ⊗ y ≤ D • , [2: n +1] ⊗ y , we ha ve C ⊗ (0 , x ⊕ y ′ ⊕ y ) ≤ D ⊗ (0 , x ⊕ y ′ ⊕ y ) , hence C ⊗ (0 , x ⊕ u ) ≤ D ⊗ (0 , x ⊕ u ) which pro ves that x ∈ ϕ W ( Z ) . Example 1: Let us consider W ′ ⊆ R 3 max the tropical cone generated by the follo wing set { [0 , 1 , 1] T , [0 , 3 , 1] T , [0 , 1 , 3] T , [ ε, 0 , 0] , [ ε, 1 , 0] } Then W = W ′ ∩ R 2 max is the tropical polyhedra of the following form: W = Span  0 0  ,  1 0  ⊕ Con v  1 1  ,  3 1  ,  1 3  where the first part is the recession cone R of W and the second part is the bounded polyhedra P . W e consider the safety set S in R 2 max defined by S = Span([1 , 0] T , [0 , 1] T ) , or equiv alently , in its M -form S =  ε ε 0 ε − 1 ε  ,  ε 1 ε ε ε 0  ∩ R 2 max which we denote as S = ⟨ C, D ⟩ ∩ R 2 max . W e can verifty that ( C 1 , ≥ 2 | [0 , 0] T ) = 0 ≤ ( D 1 , ≥ 2 | [0 , 0] T ) = 1 , and ( C 2 , ≥ 2 | [1 , 0] T ) = − 1 ≤ ( D 2 , ≥ 2 | [1 , 0] T ) = 0 , thus R ⊆ S , which implies that ϕ U ( S ) may not be empty and we ha ve ϕ U ( S ) = ϕ R ( S ) . B. Reform ϕ P ( Z ) as An Intersection of Pseudo Half-Spaces Now the problem reduces to characterizing ϕ P ( Z ) , where P is a bounded polyhedron. Let P = Conv( e 1 , . . . , e M ) , and define P ′ = Span((0 , e i ) i ) . Then we have P = P ′ ∩ R n max . The set ϕ P ( Z ) can therefore be rewritten as ϕ P ( Z ) = ( x ∈ R n +1 max      ∀ y ∈ P ′ , s.t. y 1 = x 1 C ⊗ ( x ⊕ y ) ≤ D ⊗ ( x ⊕ y ) ) ∩ R n max T o better understand its structure, we introduce the fol- lowing definition. Definition 1 (Pseudo Half-Space): A pseudo half-space is a set of the form H U c,d = ( x ∈ R n +1 max      ∀ y ∈ U , y 1 = x 1 ( c | x ⊕ y ) ≤ ( d | x ⊕ y ) , ) where c, d ∈ R n +1 max are two vectors and U is a tropical cone. W ith this definition, the set ϕ P ( Z ) has the form ϕ P ( Z ) =  T q i =1 H P ′ C i ,D i  ∩ R n max . W e now study some structural properties of this set. Lemma 3: The set H P ′ C i ,D i is a closed sub-semimodule. In particular E = T q i =1 H P ′ C i ,D i is a closed sub-semimodule. Pr oof: W e first prove that H C i, D i P ′ is a sub- semimodule. Indeed, for all λ ∈ R max \ { ε } and x ∈ H P ′ C i ,D i , for all y ∈ P ′ such that ( λ · x ) 1 = y 1 , we hav e x 1 = (( − λ ) · y ) 1 and ( − λ ) · y ∈ P ′ , thus ( C i | ( x ⊕ (( − λ ) · y ))) ≤ ( D i | ( x ⊕ (( − λ ) · y ))) , which is equiv alent to ( C i | ( λ · x ⊕ y )) ≤ ( D i | ( λ · x ⊕ y )) . F or λ = ε , we ha ve λ · x = ε n +1 , thus y 1 = ε , which implies y = ε n +1 , we then hav e ε = ( C i | ( λ · x ⊕ y )) ≤ ( D i | ( λ · x ⊕ y )) = ε . Now let x, x ′ ∈ H P ′ C i ,D i , if x 1 = x ′ 1 , then it is trivial that x ⊕ x ′ ∈ H P ′ C i ,D i . Suppose now x 1 < x ′ 1 , let y ∈ P ′ such that y 1 = ( x ⊕ x ′ ) 1 , then y 1 = x ′ 1 > ε . W e let y ′ = ( x 1 − x ′ 1 ) · y . W e verifies quickly that y ′ ≤ y and x ⊕ y ′ ∈ ⟨ C i , D i ⟩ . Since x ′ ⊕ y ∈ ⟨ C i , D i ⟩ , we have ( x ⊕ y ′ ) ⊕ ( x ′ ⊕ y ) = ( x ⊕ x ′ ) ⊕ y ∈ ⟨ C i , D i ⟩ . W e have prov ed that H P ′ C i ,D i is a subsemimodule. Further more, we can show that H P ′ C i ,D i is closed. Indeed, let { x ( n ) } n ∈ Z > 0 be a sequence in H P ′ C i ,D i con ver ging to a point x ∈ R n +1 max and let y ∈ U such that y 1 = x 1 . W e distinguish two cases: 1) If x 1  = ε then y ( n ) = ( x 1 − x ( n ) 1 ) · y is a sequence in P ′ con ver ging to y and is such that y ( n ) 1 = x ( n ) 1 for all n . Since x ( n ) ⊕ y ( n ) ∈ ⟨ C , D ⟩ and ⟨ C i , D i ⟩ is a closed set, we conclude that x ⊕ y ∈ ⟨ C i , D i ⟩ . 2) If x 1 = ε then y = ε n +1 . T ake an arbitrary y (0) ∈ U s.t. y (0) 1 = 0 . For n ∈ Z > 0 , define y ( n ) = x ( n ) 1 · y (0) , then { y ( n ) } n con ver ge to y and we hav e { x ( n ) ⊕ y ( n ) } con ver ge to x ⊕ y = x , which proves that x ∈ H P ′ C i ,D i . Another notew orthy property is that the complement of H U c,d is also stable under addition and scalar multiplication. Indeed, we have  H U c,d  c = ( x ∈ R n +1 max      ∃ y ∈ U , y 1 = x 1 ( c | x ⊕ y ) > ( d | x ⊕ y ) ) Let x, x ′ ∈  H U c,d  c . Then there exist y , y ′ ∈ U such that y 1 = x 1 , y ′ 1 = x ′ 1 and ( c | x ⊕ y ) > ( d | x ⊕ y ) , ( c | x ′ ⊕ y ′ ) > ( d | x ′ ⊕ y ′ ) . W e then ha ve ( c | ( x ⊕ x ′ ) ⊕ ( y ⊕ y ′ )) = ( c | x ⊕ y ) ⊕ ( c | x ′ ⊕ y ′ ) > ( d | x ⊕ y ) ⊕ ( d | x ′ ⊕ y ′ ) = ( d | ( x ⊕ x ′ ) ⊕ ( y ⊕ y ′ )) , and moreover ( y 1 ⊕ y 2 ) (1) = ( x 1 ⊕ x 2 ) (1) . This shows that x 1 ⊕ x 2 ∈  H U c,d  c . Now let λ ∈ R max be a scalar . W e readily verify that ( λ · y ) 1 = ( λ · x ) 1 and ( c | λ · x ⊕ λ · y ) > ( d | λ · x ⊕ λ · y ) . Therefore, λ · x ∈  H U c,d  c . C. Induction Step for Calculating Generating Set for ϕ P ( Z ) In the previous section, we established that the set E = T q i =1 H U C i ,D i possesses a fav orable mathematical structure. Building on these results, the goal of this section is to prove that E is finitely generated, i.e., that it is a tropical cone, and to develop an algorithm for computing a generating set. The following theorem plays a ke y role in the recursive procedure underlying our algorithm. Theorem 1: Let C ⊆ R n +1 max be a closed tropical cone generated by a set V of elements of R n +1 max , and let H U c,d be a pseudo tropical half-space. Then the cone C ∩ H U c,d is generated by the following set: ( V ∩ H U c,d ) ∪ ( v ⊕ ρ · w      ( v , w ) ∈ ( V ∩ H U c,d ) × ( V \ H U c,d ) ρ = max { λ | v ⊕ λ · w ∈ H U c,d } ) Pr oof: Note that ρ is well defined, in the sense that the set M = { λ | v ⊕ λ · w ∈ H U c,d } admits a maximal element. Indeed, M is nonempty since ε ∈ M . W e claim that M also admits a greatest element. Suppose, by contradiction, that this is not the case. Then ( − λ ) · v ⊕ w ∈ H U c,d holds for arbitrarily large λ . Letting λ → + ∞ , we obtain that ( − λ ) · v ⊕ w ∈ H U c,d con ver ges to w / ∈ H U c,d , which contradicts the fact that H U c,d is closed. Now , we observe that the set defined in Theorem 1 is contained in C ∩ H U c,d . Let x ∈ C ∩ H U c,d . By the tropical analogue of the Minkowski–Carath ´ eodory theorem [20], x can be expressed as a combination of at most n + 1 elements of V . That is, there exist V ′ ⊆ V ∩ H U c,d and W ′ ⊆ V \ H U c,d such that | V ′ | + | W ′ | ≤ n + 1 , and x = M v ∈ V ′ λ v · v ⊕ M w ∈ W ′ λ w · w where λ v , λ w ∈ R max \ { ε } . Denote ρ ( v ,w ) = max { λ | v ⊕ λ · w ∈ H U c,d } . W e show that for every w ∈ W ′ , there exists v ∈ V ′ such that λ v ⊗ ρ ( v ,w ) ≥ λ w . Suppose, by contradiction, that there exists w ∈ W ′ such that λ v ⊗ ρ ( v ,w ) < λ w for all v ∈ V ′ . Then λ v · v ⊕ λ w · w / ∈ H U c,d (since λ v  = ε ). Since the complement of H U c,d is stable under addition and multiplication by a nonzero scalar , it follo ws that x = M v ∈ V ′ λ v · v ⊕ λ w · w ! ⊕   M w ′ ∈ W ′ \{ w } λ w ′ · w ′   does not belong to H U c,d , which contradicts the assumption that x ∈ H U c,d . For each w ∈ W ′ , let v w be an element of V ′ such that λ v w ⊗ ρ ( v w ,w ) ≥ λ w . Since λ w  = ε , we hav e ρ ( v w ,w )  = ε . Hence, x = M v ∈ V ′ λ v · v ! ⊕ M w ∈ W ′ ( λ w − ρ ( v w ,w ) ) · ( v w ⊕ ρ ( v w ,w ) · w ) ! . T o compute a generating set for E , we proceed as follows. W e start with V = { e i } i =1 ,...,n +1 , which is a generating set for R n +1 max . W e then construct the generating set inductively: if V j is a generating set for C j = T j i =1 H U C i ,D i , then Theorem 1 allo ws us to derive a generating set for T j +1 i =1 H U C i ,D i = C j ∩ H U C j +1 ,D j +1 . Howe ver , a key difficulty is that the set described in Theorem 1 is not straightforward to compute. The next section addresses this issue. V I . C O M P U TA T I O N M E T H O D S F O R ϕ □ T o compute a generating set for E , we need to determine the set V ∩ H U c,d . Moreov er , for v ∈ V ∩ H U c,d and w ∈ V \ H U c,d , we need to compute ρ = max { λ | v ⊕ λ · w ∈ H U c,d } . W e now explain ho w to determine, for a giv en v ∈ V , whether v ∈ H U c,d . The case v 1 = ε is trivial. Suppose now that v 1  = ε , and for simplicity assume that v 1 = 0 . Then v ∈ H U c,d if and only if for all y ∈ U such that y 1 = 0 , we hav e ( c | v ⊕ y ) ≤ ( d | v ⊕ y ) . Let U be a generating set for U , and assume for simplicity that for all u ∈ U , u 1 = 0 . A necessary condition for v ∈ V ∩ H U c,d is that for all u ∈ U , ( c | v ⊕ u ) ≤ ( d | v ⊕ u ) . In fact, this condition is also sufficient. Suppose it holds, and let y = L u ∈ U λ u · u , with L u λ u = 0 . W e sho w that ( c | v ⊕ y ) ≤ ( d | v ⊕ y ) by considering two cases: Fig. 1. The overvie w of example 2. The region in skin color represents the set ⟨ c, d ⟩ ∩ R 2 max , the region in blue represents the set U ∩ R 2 max , and the region on the right-hand side of the dot line represent the set H U c,d . The two red points represent respectively v 1 and v 2 . It is easy to check that v 2 / ∈ H U c,d and v 1 ∈ H U c,d . 1) If ( c | v ) > ( d | v ) , then for all u ∈ U , since ( c | v ⊕ u ) ≤ ( d | v ⊕ u ) , we must have ( c | u ) ≤ ( d | u ) . As L u λ u = 0 , there exists u 0 ∈ U such that λ u 0 = 0 . Hence, ( c | v ⊕ y ) = ( c | v ⊕ u 0 ) ⊕   c       M u ∈ U \{ u 0 } λ u · u   ≤ ( d | v ⊕ u 0 ) ⊕   d       M u ∈ U \{ u 0 } λ u · u   = ( d | v ⊕ y ) 2) If ( c | v ) ≤ ( d | v ) , we claim that for all u ∈ U and for all λ ≤ 0 , we hav e ( c | v ⊕ λ · u ) ≤ ( d | v ⊕ λ · u ) . The case ( c | u ) ≤ ( d | u ) is immediate. If ( c | u ) > ( d | u ) , then from ( c | v ⊕ u ) ≤ ( d | v ⊕ u ) , it follows that ( c | u ) ≤ ( d | v ) . Hence, for λ ≤ 0 , we have ( c | λ · u ) ≤ ( d | v ) , which implies ( c | v ⊕ λ · u ) ≤ ( d | v ⊕ λ · u ) . Example 2: Let c = ( ε, ε, 0) , d = ( ε, 1 , ε ) , and U = Span((0 , 1 , 1) , (0 , 3 , 1) , (0 , 1 , 3)) , gi ven the vectors v 1 = (0 , 2 , 3) and v 2 = (0 , 1 , 2) , we want to decide wether v 1 , v 2 ∈ H U c,d . For the vector v 1 , we hav e v 1 ⊕ (0 , 1 , 1) = (0 , 2 , 3) , v 1 ⊕ (0 , 1 , 3) = (0 , 2 , 3) , v 2 ⊕ (0 , 3 , 1) = (0 , 3 , 3) . As we hav e ( c | (0 , 2 , 3)) = 3 = ( d | (0 , 2 , 3)) and ( c | (0 , 3 , 3)) = 3 < ( d | (0 , 3 , 3)) = 4 , we ha ve v 1 ∈ H U c,d . For the vector v 2 , we hav e v 2 ⊕ (0 , 1 , 1) = (0 , 1 , 2) , v 2 ⊕ (0 , 1 , 3) = (0 , 1 , 3) and v 2 ⊕ (0 , 3 , 1) = (0 , 3 , 2) . W e hav e ( c | (0 , 1 , 2)) = ( d | (0 , 1 , 2)) = 2 and ( c | (0 , 3 , 2)) = 2 < ( d | (0 , 3 , 2)) = 4 , but as ( c | (0 , 1 , 3)) = 3 > ( d | (0 , 1 , 3)) = 2 , we conclude that v 2 / ∈ H U c,d . W e provide a geometric representation in figure 1 for example 2 It remains to find the maximal ρ such that v ⊕ ρ · w ∈ H U c,d , for v ∈ H U c,d and w / ∈ H U c,d . Inspired by the previous method for determining whether a vector v belongs to the tropical cone H U c,d , we remark that the set H U c,d can be rewritten as \ u ∈ U  x ∈ R n +1 max | ( c | x ⊕ x 1 · u ) ≤ ( d | x ⊕ x 1 · u ) } Let us define the follo wing matrix: M u =         u 1 ⊕ 0 ε ε . . . ε u 2 0 ε . . . ε u 3 ε . . . . . . . . . . . . u n +1 ε . . . ε 0         then one verifies that the inequality ( C i | x ⊕ x 1 · u ) ≤ ( D i | x ⊕ x 1 · u ) can be rewritten as C T i ⊗ M u ⊗ x ≤ D T i ⊗ M u ⊗ x , or simply , ( M T u ⊗ C i | x ) ≤ ( M T u ⊗ D i | x ) , which defines exactly a tropical half-space. Denote H u c,d the tropical half-space defined by this equation, then we hav e H U c,d = T u ∈ U H u c,d and we have the following equality ρ v ,w max { ρ | v ⊕ ρ · w ∈ H U c,d } = min u ∈ U  max { λ | v ⊕ λ · w ∈ H u c,d }  It is worth noting that for u ∈ U such that w ∈ H u c,d , we shall have max { λ | v ⊕ λ · w ∈ H u c,d } = + ∞ . But the right- hand side of the equation is always a finite value since there is at least a u ∈ U such that w / ∈ H u c,d , for such a u , we hav e ( M T u ⊗ c | w ) > ( M T u ⊗ d | w ) ≥ ε , and we can derive that the largest λ such that v ⊕ λ · w ∈ H u c,d is giv en by λ u = ( M T u ⊗ d | v ) − ( M T u ⊗ c | w ) = ( d | v ⊕ v 1 · u ) − ( c | w ⊕ w 1 · u ) and ρ v ,w = min u ∈ U, w / ∈ H u c,d (( d | v ⊕ v 1 · u ) − ( c | w ⊕ w 1 · u )) Example 3: Consider the case c = ( ε, ε, 0) , d = ( ε, 1 , ε ) , U = { (0 , 1 , 1) , (0 , 1 , 3) , (0 , 3 , 1) } . v = (0 , 3 , 1) and w = (0 , 1 , 2) . It is easy to verify that v ∈ H U c,d while w / ∈ H U c,d . For u (1) = (0 , 1 , 1) , u (3) = (0 , 3 , 1) , we have w ∈ H u (1) c,d and w ∈ H u (3) c,d . For u (2) = (0 , 1 , 3) , we hav e ( c | w ⊕ w 1 · u (2) ) = 3 > ( d | w ⊕ w 1 · u (2) ) = 2 , thus w / ∈ H u (2) c,d , we thus have λ u (2) = ( d | v ⊕ v 1 · u (2) ) − ( c | w ⊕ w 1 · u (2) ) = 4 − 3 = 1 . W e thus hav e ρ v ,w = 1 . Indeed v ⊕ 1 · w = (1 , 3 , 3) belongs to H U c,d and its on the boundary of H U c,d . W ith these useful tools above, we are able to compute the generating set for E = T N i =1 H U C i ,D i with the following steps: 1. W e start with the set V = { e i } i , which is a generating set for R n max . 2. For i ranging from 1 to N , update V by the generating set for Span( V ) ∩ H C i ,D i . The resulting set V is then a generating set for E . Remark 1 ( Extremal P oints Filtering Pr ocedure): In the work of [16], a procedure of extremal points filtering is ex ecuted to enhance the performance of the algorithm. For a point x belonging to a tropical cone ⟨ C , D ⟩ , one can check wether x is an extremal point by building up the associated directed hypergraph and determine if it has a maximal strongly connected component, for details we in vite the reader to refer to [16]. Inspired by their approach, we will use the similar method to fasten the computation of extremal points for the tropical cone E defined in section IV. T o reduce the complexity of the algorithm described in the end of section IV for computation of the generating set E = T N i =1 H U C i ,D i , we shall insert an extremal points filtering procedure after each iteration: At iteration step k , the set V is currently the generating set for the cone E k = T k i =1 H i , we delete points in V that are not extremal in E k (which still keeps V a generating set), in this way we can reduce the complexity to a large extent. As it has been prov ed in the previous section that each H U C i ,D i is a finite intersection of tropical half-spaces H U C i ,D i , E can thus be rewritten as the intersection of finitely many tropical half- spaces and we can apply exactly the same method described in [16] to select the extremal points in the generating set of E : for x ∈ E = T j H j , we build the directed hypergraph G ( x, E ) associated to x and determine whether G ( x, E ) has a maximal strongly connected component (SCC). V I I . C A S E S T U D Y In the following, we use an example to illustrate how we effectuate the backw ard propagation for the following system S " x ( k ) 1 x ( k ) 2 # =  2 3 5 1  ⊗ " x ( k − 1) 1 x ( k − 1) 2 # ⊕  ε 0  ⊗  u k  ⊕ " w ( k ) 1 w ( k ) 2 # where [ w ( k ) 1 , w ( k ) 2 ] T ranges in the tropical polyhedra W = ( λ 1 ·  1 1  ⊕ λ 2 ·  3 1  ⊕ λ 3 ·  1 3      3 M i =1 λ i = 0 ) and u ranges in the polyhedra U = R max . The safety set is the tropical cone in R 2 max defined by S = Span([1 , 0] T , [0 , 1] T ) , or equiv alently , in its M -form S =  ε ε 0 ε − 1 ε  ,  ε 1 ε ε ε 0  ∩ R 2 max which we denote as S = ⟨ C , D ⟩ ∩ R 2 max Giv en S , our goal is to find the backward reach set of S w .r .t the system S . W e proceed step by step: we first calculate the set ϕ W ( S ) , then we calculate γ U ( ϕ W ( S )) , and finally the backward reach set is giv en by A − 1 ( γ ( ϕ W ( S ))) . T o calculate ϕ W ( S ) , we follow the steps in described in section VI. Let W ′ be the tropical cone in R 3 max gen- erated by U = { [0 , 1 , 1] T , [0 , 3 , 1] T , [0 , 1 , 3] T } , then we hav e S = S ′ ∩ R 2 max . W e first initialize the set V = { [0 , ε, ε ] T , [ ε, 0 , ε ] T , [ ε, ε, 0] T } , which is the canonique base for R 3 max . W e then decide the generators for the tropical cone H C 1 ,D 1 W , where C 1 = [ ε, ε, 0] , D 1 = [ ε, 1 , ε ] , which in volv es calculating the set V in = V ∩ H C 1 ,D 1 W and V out = V \ V in and combine the vectors of V in and V out to get the generating set for H C 1 ,D 1 W . As [0 , ε, ε ] T ⊕ [0 , 1 , 3] T / ∈ ⟨ C 1 , D 1 ⟩ , we kno w that [0 , ε, ε ] ∈ V out , and it is easy to verify that [ ε, 0 , ε ] T ∈ V in and [ ε, ε, 0] T ∈ V out . In conclusion, we hav e V in = { [ ε, 0 , ε ] T } = { v 1 } , V out = { [0 , ε, ε ] T , [ ε, ε, 0] T } = { w 1 , w 2 } . The generating set for H C 1 ,D 2 W can be expressed as { v 1 , v 1 ⊕ ρ 1 · w 1 , v 1 ⊕ ρ 2 · w 2 } , where ρ i = max { λ | v 1 ⊕ λ · w i ∈ H C 1 ,D 1 W } . For v 1 and w 1 , as v (1) 1 = ε and w (1) 1  = ε , we hav e ρ 1 = ( D 1 | v 1 ) − ( C 1 | w 1 ⊕ L u ∈ U u ) = ([ ε, 1 , ε ] T | [ ε, 0 , ε ] T ) − ([ ε, ε, 0] T | [0 , 3 , 3] T ) = 1 − 3 = − 2 , thus v 1 ⊕ ρ 1 · w 1 = [ − 2 , 0 , ε ] T . For v 1 and w 2 , as v (1) 1 = ε and w (1) 2 = ε , we have ρ 2 = ( D 1 | v 1 ) − ( C 1 | w 2 ) = 1 − 0 = 1 , thus v 1 ⊕ ρ 2 · w 2 = [ ε, 0 , 1] T . W e find the generating Fig. 2. The overvie w of the three regions: S , W and ϕ W ( S ) . The skin color region represents S , W corresponds to the blue region, and ϕ W ( S ) is surrounded by the line of dots. set for H C 1 ,D 1 W is V 1 = { [ ε, 0 , ε ] T , [0 , 2 , ε ] T , [ ε, 0 , 1] T } . Now we aim to calculate the generating set for H C 1 ,D 1 W ∩ H C 2 ,D 2 W . Again, we need to decide the set V ′ in = V 1 ∩ H C 2 ,D 2 W and the set V ′ out = V 1 \ V ′ in . After calculation, we get V in = { [ ε, 0 , 1] T } , which we denote as V 1 = { v ′ 1 } , and we ha ve V out = { [ ε, 0 , ε ] T , [0 , 2 , ε ] T } , denoted as V out = { w ′ 1 , w ′ 2 } . For v ′ 1 and w ′ 1 , we hav e ρ ′ 1 = ( D 2 | v ′ 1 ) − ( C 2 | w ′ 1 ) = ([ ε, ε, 0] T | [ ε, 0 , 1] T ) − ([ ε, − 1 , ε ] T | [ ε, 0 , ε ] T ) = 1 − ( − 1) = 2 . For v ′ 1 and w ′ 2 , we hav e ρ ′ 2 = ( D 2 | v ′ 1 ) − ( C 2 | w ′ 2 ⊕ L u ∈ U u ) = ([ ε, ε, 0] T | [ ε, 0 , 1] T ) − ([ ε, − 1 , ε ] T | [0 , 2 , ε ] T ⊕ [1 , 3 , 3] T ) = − 1 . A generating set for H C 1 ,D 1 W ∩ H C 2 ,D 2 W is { v ′ 1 , v ′ 1 ⊕ ρ ′ 1 · w ′ 1 , v ′ 1 ⊕ ρ ′ 2 · w ′ 2 } = { [ ε, 0 , 1] T , [ ε, 2 , 1] T , [ − 1 , 1 , 1] T } , which we normalize to be { [ ε, 0 , 1] T , [ ε, 2 , 1] T , [0 , 2 , 2] T } . A geometric representation for the example is represented in figure 2. By following the procedure described in section IV, we get finally the backward reachable set Υ( S ) = A − 1 ◦ γ U ◦ ϕ W ( S ) = A − 1 ◦ γ U   Span     ε 0 1   ,   ε 2 1   ,   0 2 2     ∩ R 2 max   = A − 1   Span     ε 0 ε   ,   ε 0 1     0 2 2       = Con v  0 ε  ,  − 2 1  ,  1 3  ,  − 2 3  V I I I . C O N C L U S I O N In this paper , we inv estigated the problem of backward reachability analysis for uncertain max-plus linear sys- tems. 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