Randomly switching max-plus linear systems and equivalent classes of discrete event systems

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1 Delft University of Technology Delft Center for Systems and Control Technical report 8-8 Randomly switching max-plus linear systems and equivalent classes of discrete event systems T. van den Boom and B. De Schutter If you want to cite this report, please use the following reference instead: T. van den Boom and B. De Schutter, Randomly switching max-plus linear systems and equivalent classes of discrete event systems, Proceedings of the 9th International Workshop on Discrete Event Systems (WODES 8, Göteborg, Sweden, pp , May 28. Delft Center for Systems and Control Delft University of Technology Mekelweg 2, 2628 CD Delft The Netherlands phone: (secretary fax: URL: This report can also be downloaded viahttp://pub.deschutter.info/abs/8_8.html

2 Randomly switching max-plus linear systems and equivalent classes of discrete event systems Ton van den Boom and Bart De Schutter Abstract In switching max-plus-linear discrete event systems we can switch between different modes of operation. In each mode the discrete event system is described by a maxplus-linear state space model with different system matrices for each mode. In randomly switching max-plus-linear systems, the switching between the modes can be both deterministic and stochastic. The switching changes the structure of the system, which allows the system to break synchronization and to change the order of events. In this paper two equivalent descriptions for randomly switching max-plus-linear systems will be discussed. Furthermore, we will show that a randomly switching maxplus-linear system can be written as a piecewise affine system or a max-min-plus-scaling system. The last transformation can be established under (rather mild additional assumptions on the boundedness of state and input. I. INTRODUCTION The class of discrete event systems (DES essentially consists of man-made systems that contain a finite number of resources (such as machines, communications channels, or processors that are shared by several users (such as product types, information packets, or jobs all of which contribute to the achievement of some common goal (the assembly of products, the end-to-end transmission of a set of information packets, or a parallel computation. In this paper we consider switching max-plus-linear (SMPL systems, discrete event systems that can switch between different modes of operation, in which the mode switching depends on a stochastic sequence or on the input and previous state. In each mode the system is described by a max-plus-linear state equation and a max-plus-linear output equation, with different system matrices for each mode. In 8 we have discussed SMPL systems with deterministic switching, and in 9, we have discussed SMPL systems with random switching. The class of SMPL systems contains discrete event systems with synchronization but no concurrency, in which the order of synchronization of the event steps may vary randomly, or is determined by input signals or the previous state. This work was partially funded by the Dutch Technology Foundation STW project Multi-agent control of large-scale hybrid systems (DWV.688, by the European 6th Framework Network of Excellence HYbrid CONtrol: Taming Heterogeneity and Complexity of Networked Embedded Systems (HYCON (FP6-IST-5368, by the Delft Research Center Next Generation Infrastructure, and by the Transport Research Cnter Delft. Ton van den Boom and Bart De Schutter are with the Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands a.j.j.vandenboom@tudelft.nl, b@deschutter.info Bart De Schutter is also with the Maritime & Transport Technology Group, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Typical examples of SMPL systems are flexible manufacturing systems, telecommunication networks, logistic networks, and signal controlled urban traffic networks. Mode switching depending on input signals allows us to model a change in the structure of the system, such as breaking a synchronization or changing the order of events. Mode switching depending on the state may be due to concurrency between various events (see 8. Random mode switching between may be due to e.g. (randomly changing production recipes, varying customer demands or traffic demands, or failures in production units, transmission lines, or traffic links. In this paper we review the type RSMPL system description introduced in 9, and we introduce a type 2 RSMPL system description. We show that these two RSMPL systems descriptions are in fact equivalent. Furthermore, we show that a type 2 RSMPL system can be rewritten as a piecewise affine (PWA system or a max-min-plus-scaling (MMPS system. Using the results of 5 we also implicitly prove that a RSMPL system can be rewritten as an (extended linear complementarity (ELC/LC system or a mixed logic dynamical (MLD system, which are well-known system descriptions used in the field of hybrid systems. Using these results we can transfer properties of and methods for PWA systems and MMPS systems to RSMPL systems. II. SWITCHING MAX-PLUS LINEAR SYSTEMS Max-plus algebra We start this section by giving the some basic definitions in the max-plus algebra, 4, 6. Define ε = and R ε =R {ε}. The max-plus-algebraic addition ( and multiplication ( are defined as follows: x y=max(x,y, x y=x+y for numbers x,y R ε and A B i j = a i j b i j = max(a i j,b i j n A C i j = a ik c k j = max (a ik+ c k j k=,...,n k= for matrices A,B R m n ε and C R n p ε. Randomly switching Max-Plus-Linear systems Switching Max-Plus-Linear (SMPL systems are discrete event systems that can switch between different modes of operation 8. In each mode l {,...,L}, the system is

3 described by a max-plus-linear state equation and a maxplus-linear output equation: x(k=a (l(k x(k B (l(k u(k ( y(k= C (l(k x(k (2 in which the matrices A (l R n x n x ε, B (l R n x n u ε, C (l are the system matrices for the l-th mode.the index R n y n x ε k is called the event counter. For discrete event systems the state x(k typically contains the time instants at which the internal events occur for the kth time, the input u(k contains the time instants at which the input events occur for the kth time, and the output y(k contains the time instants at which the output events occur for the kth time. In 8 we have considered deterministic switching, that was a function of the previous state or an input signal. In 9 we have introduced random switching, i.e. the mode switching depended on a stochastic sequence. In we have combined both switching types. For the SMPL system (-(2, the mode switching variable l(k then depends on both stochastic variables as well as deterministic variables (state and inputs. The switching times are determined by a switching mechanism. The mode switching variable l(k is a stochastic process, and the probability of switching depends on the previous mode l(k, the previous state x(k, the input variable u(k, and an (additional control vector v(k R n v. Definition (9, : A type Randomly Switching Max-Plus-Linear (RSMPL system is defined as follows: Consider system (-(2 with L possible modes. The probability of switching to model(k givenl(k, x(k, u(k, v(k is denoted by P(l(k l(k,x(k,u(k,v(k. For any given l(k {,...,L}, the probability P is piecewise affine on polyhedral sets in the variables l(k, x(k, u(k, v(k. Define w(k = l(k x T (k u T (k v T (k T, then for any m {,...,L} there exist matrices S i,m, vectors α i,m, s i,m and scalars β i,m such that the probability P can be written as P(l(k w(k=α T i,l(k w(k+β i,l(k, if w(k Γ i,l(k for i=,...,n l(k where ;Γ i,l(k = { w(k S i,l(k w(k s i,l(k }, and the sets Γ i,l(k are such that n m Γ i,m =R n w and int(γ i,m int(γ j,m = / for i j. i= where int( denotes the interior. Definition 2: A type 2 Randomly Switching Max-Plus- Linear (RSMPL system is defined as follows: Consider system (-(2 with L possible modes. The mode l(k=m if z(k= l(k x T (k u T (k v T (k d(k T Ωm Note that P is a probability, so obviously for any w(k we find P(m w(k, m=,...,l and L m= P(m w(k=. where d(k, is a uniformly distributed scalar signal, and where Ω m = n m j= Ω m, j in which Ω m, j are closed convex polyhedra (i.e. given by a finite number of linear inequalities with non-overlapping interiors in the variable z(k: Ω m, j ={z(k R m, j z(k r m, j }, for j=,...,n m In the next section we will show that the two types of RSMPL systems are equivalent and therefore model the same class of discrete event systems. A type RSMPL model is easy for modeling where we consider the probabilities of switching from one mode to another. The model gives a lot of physical insight into the system and is usually more intuitive for the user. A type 2 RSMPL model is signal based and the properties of probabilities of switching are translated into the properties of a stochastic signal. The fact that type 2 RSMPL models are signal based makes that this type of RSMPL system can easily be translated into another model in one of the well-known classes of hybrid systems (see Section IV. III. EQUIVALENCE IN CLASSES OF RSMPL SYSTEMS Proposition 3: Every type RSMPL system can be written as a type 2 RSMPL system. Proof: Consider a type RSMPL system, let w(k be fixed and let (i,i 2,...,i L be such that w(k Γ i j, j for j =,...,L. Now define the function { m j= η(m,w(k= αt i j, j w(k+β i j, j for m=,...,l for m= Note that for varying w(k the function η(m, w(k is a sum of piecewise affine functions in the variable w(k, and therefore is piecewise affine in w(k itself. Now define a system (-(2 and a uniformly distributed scalar signal d(k,, and let l(k=m if η(m,w(k d(k η(m,w(k This means that the probability that l(k=m is equal to: ( P(m w(k=p η(m,w(k d(k η(m,w(k = η(m,w(k η(m,w(k = α T i m,m w(k+β im,m. like in a type RSMPL system. So we obtain d(k+η(m,w(k l(k=m if d(k η(m,w(k together with the constraints S i j, jw(k s i j, j for j=,...,l. Define z(k= w T (k d(k T and the function d(k+η(m,w(k d(k η(m,w(k S i,w(k s i, ξ(m,z(k=. S il,lw(k s il,l,

4 Define the set Φ = {φ,...,φ nt } = {φ t = (i,i 2,...,i L S i j, jw(k s i j, j for j=,...,l} of all possible permutations, such that S i j, jw(k s i j, j for j =,...,L. Now there exist matrices R m,t and scalars r m,t such that the function ξ(m,z(k for the corresponding φ t can be written as ξ(m,z(k=r m,t z(k r m,t This means that l(k=m if R m,t z(k r m,t, t =,...,n t, which is equal to a type 2 RSMPL system. Proposition 4: Every type 2 RSMPL system can be written as a type RSMPL system. Proof: Consider a type 2 RSMPL system, so Proposition 5: A type 2 RSMPL system can always be rewritten in the compact form: x(k=ā (κ(k x(k B (κ(k u(k (4 y(k= C (κ(k x(k (5 The mode κ(k=m if z(k= κ(k x T (k u T (k v T (k d(k T Ω m where d(k, is a uniformly distributed scalar signal, and where Ω m are polyhedra (i.e. given by a finite number of linear inequalities with non-overlapping interior in the variable z(k: Ω m ={z(k R m z(k r m } l(k=m if R m,t z(k r m,t Define R m,t, and R m,t,2 such that R m,t z(k=r m,t, w(k+r m,t,2 d(k r m,t (3 Note that d(k, is a scalar. Let d m,t,max (w(k be the maximum value d such that (3 is satisfied, and let d m,t,min (w(k be the minimum value d such that (3 is satisfied. If for some w(k there exists no d(k, such that (3 is satisfied, we define d m,t,max (w(k = d m,t,min (w(k =. Finding d m,t,max (w(k and d m,t,min (w(k can be done using a linear programming algorithm, which means that d m,t,max (w(k and d m,t,min (w(k are piecewise affine in w(k (3. So there exist vectors p m,t,i,max and p m,t,i,min and scalars q m,t,i,max and q m,t,i,min, such that d m,t,max (w(k= p T m,t,i,max w(k+q m,t,i,max d m,t,min (w(k= p T m,t,i,min w(k+q m,t,i,min if S i,m,t w(k s i,m,t for i=,...,m m, t =,...,n t. The probability that l(k=m, given w(k can be written as P(m w(k=p(d m,t,min d(k d m,t,max = d m,t,max d m,t,min =(p T m,t,i,max p T m,t,i,min w(k +(q m,t,i,max q m,t,i,min Proof: Consider the type 2 RSMPL system of definition 2, and introduce a new numbering l κ = j+ n i, for l=,...,l and j=,...,n l then we find and i= Ω κ = Ω l, j Āκ B κ C κ = A (l(k B (l(k C (l(k Definition 6: An RSMPL system is bounded if for any bounded (x(k, u(k we have that x(k and y(k are bounded for all l(k {,...,L} and d(k,. Corollary 7: For a bounded RSMPL system the matrix M (l = A (l B (l ε ε ε C (l is row-finite for all l =,...,L (i.e. for all l every row of the matrix M (l has at least one finite entry. IV. RSMPL SYSTEMS AND EQUIVALENT SYSTEM DESCRIPTIONS for S i,m,t w(k s i,m,t for i =,...,M m, t =,...,n t, which is a type RSMPL system. The following definition is an extension of 7: Definition 8: Piecewise Affine (PWA systems are described by x(k x(k=a i x(k +B i u(k+ f i for u(k Ω MMPS y(k=c i x(k+d i u(k+g i, (6 i d(k TYPE 2 TYPE RSMPL for i =,...,N where Ω,...,Ω N are closed polyhedra RSMPL (i.e. given by a finite number of linear inequalities with non-overlapping interiors in the variables x(k, u(k, PWA and d(k. The signal d(k, is a uniformly distributed scalar signal.

5 Proposition 9: Every bounded RSMPL system of type 2 can be written as a piecewise affine (PWA system. Proof: Consider the bounded RSMPL system of type 2 with state and output equations ( and (2 where l(k=i if z(k satisfies R i z(k r i. Define x(k x(k ω(k=, η(k= u(k (7 y(k x(k (8 M (l A (l B = (l ε ε ε C (l then ( and (2 can be written as ω(k=m (l (k η(k for R l z(k r l The RSMPL system is well-defined, which means that if η is bounded, then ω will bounded. Define the set Φ l of all elements φ t = (l t, p,t, p 2,t,..., p n,t, t =,...,n t, where l t {,...,L} and p i,t {,...,n} such that there exists an η(k satisfying ω j (k=max p (m(l t j,p +η p(k=m (l t j,p j,t +η p j,t (k, j=,...,n or equivalently j,p j,t + η p j,t (k m (l t j,i + η i (k for i, j=,...,n m (l t Now by collecting all entries for i, j=,...,n we obtain that ω(k=h (t η(k+h (t (9 if E (t η(k e (t and R lt z(k r lt where, for i, j=,...,n, and for t =,...,n t, we have { H (t for j= p i,t i, j =, h (t i = m (l i,p otherwise i,t for l = j and j p i,t E (t i(n + j,l = for l = p i,t and j p i,t otherwise e (t i(n + j,l = m (l t i,p i,t m (l t i, j It is clear that from equation (9 we can derive the matrices A i, B i, C i, D i, and vectors f i and g i. Note that the polyhedra are described by the inequalities E (t,l η(k e (t,l and R l z(k r l. The number of polyhedra is less than or equal to N = Ln n. Definition : An MMPS expression f of the variables x,..., x n is defined by the grammar 2 f := x i α max( f k, f l min( f k, f l f k + f l β f k ( with i {,2,...,n}, α, β R, and where f k, f l are again MMPS expressions. 2 The symbol stands for OR and the definition is recursive. Definition : Consider systems that can be described by x(k+ = f x (x(k,u(k,d(k ( y(k= f y (x(k,u(k,d(k, (2 where f x, f y are MMPS expressions in terms of the components of x(k, u(k, and the auxiliary variables d(k, which are all real-valued. Such systems will be called MMPS systems. If in addition, we have a condition of the form f c (x(k,u(k,d(k c(k, with f c an MMPS expression, we speak about constrained MMPS systems. Proposition 2: Every bounded RSMPL system of type 2 can be written as a constrained max-min-plus-scaling (MMPS system provided that the variables x and u are bounded. Note that this Proposition is a direct consequence of the equivalence between the class of PWA systems and constrained MMPS system (see 5. However we will provide a direct proof here which transfers RSMPL systems directly into MMPS systems. Proof: Consider a bounded RSMPL system of type 2 with state and output equations ( and (2 where l(k=i if z(k satisfies R i z(k r i. Define ω, η, and M (l as (7 and (8. Then so ω(k=m (l(k (k η(k for R l(k z(k r l(k ω i (k=max m (l(k j i, j + η j (k for R l(k z(k r l(k where M (l i, j = m (l i, j. Introduce the binary variables δ i {,}, for all i=,...,l such that δ i = R i z(k r i (3 Now assume that x and u are bounded. Then with l {,...,L} and d, we find that there exist bounded sets E and Z such that η E and z Z. Let ρ i = max z Z R i z(k r i then according to 2, using the fact that the interiors of the sets { R i z(k r i } are non-overlapping, we can rewrite (3 as R i z(k r i ρi δ i (k (4 L i= δ i (k= (5 Note that constraints (4 and (5 are linear constraints. Define the lower bound σ min = min l min min i max η E j (m (l i, j + η j(k

6 (note that σmin is bounded because both m(l i, j and η j (k are bounded, then the system is described by the following equations: ω i (k=max (m l i, j+ η j (k+( δ l (kσ min l, j Note that this is an MMPS function in the variables η and δ i. Together with the linear constraints (4 and (5 (which are also MMPS constraints we have proven that a type 2 RSMPL system, with bounded x and u, can be rewritten as a constrained MMPS system. V. EXAMPLE Consider a type RSMPL system x(k=max x(k +,u(k+.2 ( x(k=max x(k.,u(k+. with y(k=x(k P(,x(k = P(2,x(k = P( 2,x(k =sat( x(k for for P(2 2,x(k =sat(+x(k for α < where sat(α = α for α. for α > l(k= l(k=2 Define the vector w(k = l(k x(k u(k T. The probability P(l(k w(k can be written as a piecewise function: P(l(k w(k=α T i,l(k w(k+β i,l(k, if S i,l(k w(k s i,l(k, i=,...,4 where i= S,l(k = s,l(k =.5 α T, = β, = α,2 T = β,2 =.5 i=2 S 2,l(k = s 2,l(k = α T 2, = β 2, = α2,2 T = β 2,2 = i=3 S 3,l(k = s 3,l(k =.5 α3, T = β 3, = α3,2 T = β 3,2 =.5 i=4 S 4,l(k = s 4,l(k = α T 4, = β 4, = α T 4,2 = β 4,2 = This is a type RSMPL system. Now we will rewrite this system as a type 2 RSMPL system. First define η(,w(k= η(,w(k=α T i, w(k+β i, for S i, w(k s i,, i=,...,4 η(2,w(k= Now define a uniformly distributed scalar signal d(k,, define z(k= w(k d(k T, and the functions ξ(,z(k= ξ(2,z(k= d(k+ d(k α T i, w(k β i, S i, w(k s i, d(k+α T i, w(k+β i, d(k S i, w(k s i, for i=,...,4. Note that the first entry of ξ(,z(k is always smaller than zero, and the same holds for the second entry of ξ(2,z(k, so we can remove these entries. Now we can compute the matrices for i=,...,4: α T R,i = i, βi, r S,i = i, s i, α T R 2,i = i, βi,2 r S 2,i = i,2 s i,2 which results in R, = R 2, = R,2 = R 2,2 = R,3 = R 2,3 = R,4 = R 2,4 = r,2 = r 2,2 = r,3 = r 2,3 = r,4 = r 2,4 =,.5 r, =.5 r 2, =

7 To rewrite this type 2 RSMPL system in a compact form we have to introduce 8 modes, where we have x(k=max x(k +,u(k+.2 for κ(k=,2,3,4 x(k=max x(k.,u(k+. for κ(k=5,6,7,8 y(k=x(k (6 We redefine R κ = R (i 4+ j = R i, j and we define r =, r = 4.5, r 3 = 4.5, r 4 = 4.5, r 5 =, r = 4.5, r 7 = 4.5 and we obtain that the now mode κ(k=m if, r 8 = 4.5 z(k= κ(k x T (k u T (k v T (k d(k T Ω m where Ω m ={z(k R m z(k r m } Now it is straightforward to derive the PWA system: Note that x(k=x(k + x(k=u(k+.2 if κ(k<4.5, x(k + u(k+.2 if κ(k<4.5, x(k +<u(k+.2 x(k=x(k. if κ(k>4.5, x(k. u(k+. x(k=u(k+., if κ(k>4.5, x(k.<u(k+. This translates to the following PWA system: for κ =,...,4 : x(k=x(k + if R κ z(k r κ, x(k +.8 u(k x(k=u(k+.2 if R κ z(k r κ, x(k +.8 u(k for κ = 5,...,8 : x(k=x(k + if R κ z(k r κ, x(k.2 u(k x(k=u(k+. if R κ z(k r κ, x(k.2 u(k We conclude that this system is a PWA system with 6 polyhedral regions. Finally we rewrite the system as a constrained MMPS system. Assume the input is in the bounded set u min u(k u max and that the initial state x( satisfies x min x( x max. It is easy to derive that in that case x(k, k will be bounded: max x min.,u min +. x(k max,x max +,u max +.2. We compute and ρ i = max z Z σ min = min l R i z(k r i = min( 3.5,x min. min min i max η E j (m (l i, j + η j(k=u min +. Introduce the binary variables δ i {,}, for all i=,...,8. Now we obtain the following model: ( x(k=max max(x(k +,u(k+.2+ ( δ δ 2 δ 3 δ 4 σmin, ( max(x(k.,u(k+.+ ( δ 5 δ 6 δ 7 δ 8 σmin for δ i (k, and subject to R i z(k r i + ρ i ( δ i (k for all i=,...,8 8 i= δ i (k= This is a constrained MMPS system. VI. DISCUSSION In randomly switching max-plus-linear (RSMPL discrete event systems we can switch between different modes of operation. The switching between the modes can be both deterministic and stochastic, and in each mode the discrete event system is described by a max-plus-linear state space model with different system matrices for each mode. In this paper we have revisited the type randomly switching max-plus-linear (RSMPL systems, introduced a new type 2 RSMPL system, and showed that these two classes of RSMPL systems are equivalent. Furthermore we have proven that every bounded RSMPL system of type or 2 can be written as a piecewise affine (PWA system or a constrained max-min-plus-scaling (MMPS system (provided that the variables x and u are bounded. REFERENCES F. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat. Synchronization and Linearity. John Wiley & Sons, New York, A. Bemporad and M. Morari. Control of integrated logic, dynamics, and constraints. Automatica, 35(3:47 427, F. Borelli. Constrained Optimal Control of Linear and Hybrid Systems. Springer-Verlag, Berling, Germany, R.A. Cuninghame-Green. Minimax Algebra, volume 66 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, W. Heemels, B. De Schutter, and A. Bemporad. Equivalence of hybrid dynamical models. Automatica, 37(7:85 9, July 2. 6 B. Heidergott, G.J. Olsder, and J. van der Woude. Max Plus at Work. Princeton University Press, Princeton, E. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE Transactions in Automatic Control, 26(2: , T.J.J. van den Boom and B. De Schutter. Modelling and control of discrete event systems using switching max-plus-linear systems. Control Engineering Practice, 4(:99 2, October T.J.J. van den Boom and B. De Schutter. Stabilizing model predictive controllers for randomly switching max-plus-linear systems. In Proceedings of the ECC 27, Kos, Greece, July 27. T.J.J. van den Boom and B. De Schutter. Model predictive control for switching max-plus-linear systems with random and deterministic switching. In IFAC World Congress 28, Seoul, South Korea, July 28.