MULTISTAGE PRODUCTION CONTROL ; AN LMI APPROACH. Jean-Claude Hennet

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1 MULISAGE PRUCIN CNRL ; AN LMI APPRACH Jean-Claude Hennet LAAS-CNRS, 7, Ave du Colonel Roche, 377 oulouse Cédex, FRANCE hennet@laasfr João Manoel Gomes da Silva Jr UFRGS-epto de Eng Eletrica, Av svaldo Aranha Porto Alegre,RS, BRAZIL jmgomes@ieeufrgsbr Abstract: he objective of this study is to construct a closed-loop ion policy based on an input-output description of the manufacturing system A -invariance approach is developed for ellipsoidal domains he aim is to combine in the same control law the integration of ion capacity constraints and the responsiveness to demand and supply changes Constraints satisfaction is achieved through inclusion of a -invariant ellipsoid in the domain of constraints he proposed control law is of the feedback type It is obtained by solving a sequence of LMI problems Keywords: Manufacturing System, Production Planning, Material Requirement Planning, Positive Invariance, Linear Matrix Inequalities INRUCIN Production planning problems are generally formulated as optimal control problems However, due to the large number of variables and constraints, the solution to such problems is generally investigated within the class of open-loop policies he main drawback in this approach is that the planned ion levels highly depend on the accuracy of predictions on future demand and ion data As these data are generally subject to unpredictable variations, the optimal plan often has to be revised In theory, replanning and receding (or rolling) horizon procedures are the most current techniques proposed for tackling this problem But such updating techniques call in question the validity and even the meaning of the planning problems In addition, they are very heavy to implement and may cause delays and sudden changes in the ion policy Safety stocks and hedging policies may provide the system with some degree of responsiveness in the case of occasional supply or demand changes (Higgins et al, 996), but they are unable to permanently respond to demand changes, ion disturbances and machine failures herefore, the difficulty to accurately predict future ion and demand considerably limits the validity of open-loop ion plans In practice, some system feedback always has to be introduced to provide the ion schedule with responsiveness his often leads to local ion adjustments, in particular the net change approach in MRP (see eg (Fogarty and Hoffmann, 983)), with no global view of the ion schedule he MRP technique itself is an attempt to implement a feedback on stock levels, but it is only partly reactive, because it combines such closedloop measurements with open-loop data on forecasted demands and requirements he Kanban technique (Kimura and erada, 98) can be seen as a feedback control technique, based both on ion capacity, real stocks and real demands measurements It is reactive and robust, but limited to manufacturing systems with a simple structure and a regular demand pattern Rather than a reactive planning technique, it can be seen as an attempt to suppress the planning stage, sometimes with the drawback of shortsightedness A natural consequence of this analysis is to try to apply modern feedback control techniques to ion planning problems he input-output approach to describe ion systems provides a simple and concise model of a ion system Such a model can be used in the analysis phase, in the evaluation phase, but also for synthesis of

2 a ion control policy he extended state approach described in this paper allows for a reformulation of the stochastic constrained control problem as an invariance problem for a well-chosen region of the state space he shape of the invariant region is chosen ellipsoidal It is associated with a classical quadratic Lyapunov function which can be determined, as well as the control gain matrix, by solving a sequence of LMI (Linear Matrix Inequality) problems Finally, the method is illustrated on a s example Notations By convention, inequalities between vectors and inequalities between matrices are componentwise he absolute value v of a vector v is defined as the vector of the absolute value of its components A matrix M having all its components nonnegative is noted M A matrix P which is symmetric and semi-definite non-negative, is noted P Matrix I denotes the identity matrix of appropriate dimension; matrix denotes the zero matrix of appropriate dimension; vector is the unity vector of appropriate dimension, defined by = he advance operator, q, applies to vector sequences (v k ) under the form: qv k = v k+ he delay operator is the inverse operator, denoted q, such that: q v k = v k 2 HE PRUCIN SYSEM MEL he considered ion structures are of the assembly type An example of such a structure is represented on Fig Such a structure relates to s and activities (or operations) rather than to physical resources If different ion activities use the same machine, they are represented as separate blocks, one for each activity he resource sharing process is not represented in the graph, but in the model, under the form of resource capacity constraints In the example, activities A and B are distinguished, even if they are performed on the same machine Under this convention, there is a one-to-one correspondence between s and activities: each ion activity has several input s and one output 2 he MIM model he evolution of ion and stocks is described by a discrete-time model o each i, i =,, N is associated at period k : an external demand d ik, such that d ik = if i is not a final, a ion level u ik, generating the output u ik at period k + θ i ; by definition, θ i is the ion delay for i, the external output of i at period k, z ik, the stock level at the end of period k, s ik Supply in raw parts r 4 Activity A A elivery of 3 34 Assembly Activity Assembly Activity C C 23 Supply in raw parts r 2 Activity B B Assembly Activity E E Fig An assembly ion system elivery of E he ion of one unit of i requires the combination of components j =,, N in quantities π ji, for j =,, N he ion process is thus described by : the input matrix Π = (π il ), the output matrix,which both represents the units of output produced and their lead times: S = diag(q θi ) he net ion matrix is : (q ) = S Π = diag(q θi ) Π () In the sequel, matrix (q ) is decomposed as follows: (q ) = + q + + q θ θ where = Π, ( j ) = diag(t j i ) with tj i = if and only if θ i = j, t j i = in any other case, j =,, θ he system delay, denoted θ, is defined as follows: θ = N max i= θ i (2) In vector form, the stock evolution equation is : s k = s k + (q )u k z k (3) In equation (3), all the vectors should be nonnegative, from their physical meaning he primary objective of the plan is to satisfy the demand, and to satisfy it on time if possible (z k = d k ) However,

3 the possibility of backorders has to be introduced if demand satisfaction is not always feasible he stock evolution equation is modified as follows: ( q )y k = (q )u k d k (4) he vector of stock levels s k can then be deduced from vector y k by : { sik = y ik if y ik s ik = if y ik () with a backorder of y ik Assuming the demand stationary, the external demand vector can be decomposed as follows: with mean values d k = d, δk = d k = d + δ k (6) A realistic assumption is that the disturbance vector δ k has a bounded norm and may take any value within the bounded domain defined by: δ k 2 2 = δ k δ k ζ (7) Using the decomposition (6) of the demand vector, the demand tracking problem can be transformed into a regulation problem by introducing the translated control vector: v k = u k u (8) Replacing d k by d+δ k and u k by v k +u in equation (4) yields : ( q )y k = (q )v k δ k (9) u is the steady-state nominal control vector associated with the stationary mean demand vector d: u = (I Π) d () Assuming that the ion system can be represented by an assembly graph, matrix Π can be ordered as an upper-triangular non-negative matrix with zeros on its main diagonal hen, clearly, matrix (I Π) is well-defined and non-negative, and since vector d is also nonnegative, vector u is welldefined by relation () and nonnegative (Hennet and Barthes, 998) In the case of primary and intermediate s, there is no possibility of backorder For these s, the nonnegativity constraint on stocks is formulated as follows : y k k N (2) he possibility of having backorders is normally restricted to final s It pushes back the nonnegativity constraints on stocks, for the concerned components y ik of vector y k However, to keep the same representation (2) for all the s, it suffices to replace, for final s with possible backorders, the constant s i by a large integer number, say M 222 Capacity constraints Production resources may be manufacturing units, machines, work teams, raw materials, pallets, Aggregated ion capacity constraints are associated with the R resources, and similarly, stock capacity constraints are associated with the P storage zones Variable u ik represents the ion release order for i at time k he ion is actually processed at periods k,, k + θ i, and delivered at period k + θ i Assuming a constant use of resources along the ion cycle, resource capacity constraints take the following form, for r =,, R, k : N θ i N θ i m r i v i,k l Mk r m r i u i (3) i= l= i= l= m r i is the amount of ressource r currently needed to produce one unit of i Mk r is the capacity of resource r at period k By convention, v k = for k = θ,, Storage capacity constraints take the form, for p =,, P, k =, : : N n p i s ik N p k (4) i= If backorders are not allowed, constraints (4) can be re-written under the form : N N n p i y ik N p k n p i s i () i= i= 22 he constraints of the problem 22 Nonnegativity constraints From the change of variables (8), nonnegativity of the ion vector u k can be achieved by imposing the constraint : v k u () 3 HE PSIIVE INVARIANCE APPRACH 3 he extended state model It is proposed to investigate the class of state feedback control laws, with respect to the following state vector

4 x k = y k v k v k θ Rn with n = N(θ + ) (6) he control objectives are to combine the demand tracking requirement with output and control constraints he state feedback control law is written: v k = Fx k with F = F G G θ (7) he dynamic closed-loop state equation associated to a feedback (7) is as follows : x k+ = Ax k + BFx k + δ k (8) with I θ A = I, B = I I, = I System (8) is subject to the set of linear constraints (), (2), (3) and (), written in the state form x k RQ, ρ = {x; Qx ρ} (9) with Q = Q ρ, ρ =, Q 2 ρ 2 dim(q) = q n, dim(ρ) = q and I N M M θ I Q = M, Q 2 =, M I ν s µ u ρ =, ρ 2 = µ u 32 he invariance approach he positive invariance approach is able to provide local solutions to constrained control problems by imposing the positive invariance of a domain Ω in the state space satisfying the following conditions: () he zero state lies in the interior of Ω, (2) Ω is positively invariant with respect to the controlled system, meaning that any state trajectory starting in Ω remains in Ω, for any disturbance vector δ k satisfying the boundedness condition (6) In this case, the set Ω is said to be -invariant with respect to system (8) (3) Ω RQ, ρ, where RQ, ρ describes the polyhedron of constraints A set Ω which satisfies these three conditions is a set of admissible initial states for the constrained closed-loop system (8) Among the three conditions of section 4, the first one and the third one are directly satisfied by the choice : Ω = RQ, ρ However, in general, it is not possible to satisfy the second condition for any given polyhedron of constraints If the system is stabilizable, the constrained control problem can always be solved locally, at least for the nondisturbed system, by constructing an ellipsoid Ω = {x x P x ω} which satisfies the three conditions of section 4 he existence of a positive definite symmetric matrix P R n n and a positive scalar ω is guaranteed by the celebrated Kalman heorem stating the existence of a quadratic Lyapunov function for any stable linear system Furthermore, if the system is asymptotically stabilizable, the contractivity of the ellipsoid with respect to the closed-loop system trajectory guarantees the possibility to absorb bounded disturbances as long as the components of the bound vector are small enough Lyapunov closed-loop stability condition : (A + BF) P (A + BF) P, P (2) Using the Schur complement result (see eg (Boyd et al, 99)), this matrix inequality can be rewritten as an LMI under the form : W W (A + BF), (A + BF)W W with W = P (2) Invariance of the ellipsoid with respect to the disturbed controlled system is characterized by the following implication : { δ k δ k ζ x k P x k ω = x k+p x k+ ω (22) he right part of this implication can be re-written in the form : x k δk (A + BF) xk P (A+BF) ω δ k Using the S-procedure (Boyd et al, 99), a sufficient condition for implication (22) to be true is the existence of two scalars α and β such that, for all pair (x, δ) : x δ (A + BF) P (A + BF) x δ ω α(x P x ω) β(δ δ ζ) his condition is equivalent to the following set of two conditions, (23), (24): and ω αω βζ (23) αp βi

5 (A + BF) P (A + BF) (24) Classically, the gain matrix is factorized as follows: F = Y W Under this change of variables, the matrix inequality (24) is re-written in the form : αw βi (AW + BY ) W (AW + BY ) () hen, using the Schur complement result, the matrix inequality () is equivalent to : αw (AW + BY ) βi (26) (AW + BY ) W Hence, the existence of W, Y, α and β verifying (26) and (23) guarantees the satisfaction -positive invariance of Ω wrt (8) Besides, the satisfaction of constraints (9) is guaranteed for each initial state in the invariant ellipsoid Ω if this ellipsoid is included in the polyhedron RQ, ρ For ρ, this inclusion condition is equivalent to the set of inequalities : Q i W Q i ρ 2 i /ω for i =,, q (27) If we consider (W, Y, α, β, ω) as variables, matrix inequalities (26), (23) and (27) are nonlinear However, it is easy to see that, without loss of generality, we can assume ω = In this case, for a given α, the inequalities (26) and (27) become LMIs Furthermore, considering ω = and η = ζ >, inequality (23) can be rewritten as: η αη β (28) Considering now that the variations of the final demand vector are not known, the idea is to compute a matrix F in order to maximize the bound ζ on the disturbance vector norm Hence, for ω = and a given α, the following convex optimization problem can be formulated: min η W,Y,β,η subject to W = W η αη β LMIs (26) and (27) (29) With respect to the conditions stated above and the given α, the optimal solution η of this problem corresponds to the maximal bound on the disturbance ζ = /η for which is possible to compute a matrix F that guarantees the non violation of the constraints for any disturbance vector belonging to Note that we can vary α iteratively in order to obtain matrices F guaranteeing the solution of the problem for higher bounds on the disturbance In the case of ζ is given, other optimization problems can be formulated in order to compute the matrix F that guarantees both the -invariance of Ω with respect to the closed-loop system and other control objectives It should be pointed out that the main advantage of the LMI approach is its possibility to include other convex conditions in the control design problem Moreover, efficient numerical algorithms and software packages are available to solve convex optimization problems with LMI constraints (Boyd et al, 99) 4 EXAMPLE Consider the ion system of Fig he gozinto coefficient and ion lead-times are represented on the timed-petri net of Fig2 θ A = θ =2 2 A 2 C 3 θ C = B 2 2 Fig 2 he assembly process θ B =2 E θ E = he dynamics of the controlled ion system are described by equation (9), with : q 2 2 q 2 3 (q ) = q 2 2 q 2 q he extended state model is then written as in section 3, with θ = 2, = 2 2, =, 2 =,

6 N = M = I, M 2 = 2 he extended state vector has dimension n = he vector of average demands per period is given by: d = 2 hen, using (), the vector of nominal ions per period is : u = he constraints are defined as in section 3, with s = 4 4 2, µ = ν = For this data and considering α = 8, the maximal bound on ζ obtained from the optimization problem (29) is equal to In this case the matrix F guaranteing the invariance of Ω and the respect of constraints is given by: F = Fig4a Production A Figa Production B Fig6a Production C 3 2 Fig7a Production Figbb Stock A Figb Stock B 2 Fig6b Stock C 2 Fig7b Stock 2 he optimization problem was solved using the LMI Control oolbox (Gahinet et al, 99) of MA- LAB Uniformly distributed random sample demand trajectories have been generated for s and E, around their nominal values under the deviation constraint (7), with ζ = hey are represented on Fig3a and 3b over a time-horizon of 4 periods 3 2 Fig3a emand for 3 2 Fig3b emand for E Under the closed-loop ion policy (7) defined by the gain matrix F, a positively invariant domain Ω = {x x P x } is constructed his domain of admissible initial states is included in the set of storage and capacity constraints he following figures describe the ion amounts and the stock evolutions of the s during the 4 time periods, starting from an arbitrary state in Ω : x = 4 4 Fig8a Production E REFERENCES Fig8b Stock E Boyd, S, L El Ghaoui, E Feron and V Balakrishnan (99) Linear Matrix Inequalities in System and Control heory SIAM Studies in Applied Mathematics Fogarty, W and R Hoffmann (983) Production and inventory management South- Western Publishing Company Gahinet, P, A Nemirovski, A Laub, and M Chilali (99) LMI Control oolbox User s Guide he Math Works Inc Hennet, JC and I Barthès (998) Closed-loop planning of multi-level ion under resource constraints In: Proceedings of the IFAC Symposium INCM 98, Nancy, (France) Higgins, P, P le Roy and L ierney (996) Manufacturing Planning and Control Chapman et Hall Kimura, and H erada (98) esign and analysis of pull system, a method of multistage ion control Intl J Production Research, vol 9, No3, pp 24-3