EMPIRICAL COMPLEXITY ANALYSIS OF A MILP-APPROACH FOR OPTIMIZATION OF HYBRID SYSTEMS

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1 EMPIRICAL COMPLEXITY ANALYSIS OF A MILP-APPROACH FOR OPTIMIZATION OF HYBRID SYSTEMS Jochen Till, Sebastian Engell, Sebastian Panek, and Olaf Stursberg Process Control Lab (CT-AST), University of Dortund, Dortund, Gerany. E-ail: {j.till s.engell s.panek o.stursberg}@ct.uni-dortund.de Abstract: Many techniques for designing controllers for hybrid systes suffer inherently fro the coplexity of coputation, such that the applicability is liited to relatively sall probles. It is often not obvious, however, which particular part of a proble forulation has a doinant ipact on the increase of coplexity with the proble size. This paper describes a thorough epirical investigation of the sources of coplexity for an approach to optial control of hybrid systes. This approach transfors the control task into a ixed-integer prograing proble that is solved by Branch&Bound techniques. The influence of various paraeters on the coputational costs are investigated for a scalable technical exaple. Keywords: Coplexity, Epirical Analysis, Hybrid Systes, Mixed Integer Prograing, Optial Control. 1. INTRODUCTION In coparison to designing controllers for purely discrete or purely continuous systes, the design of controllers for hybrid systes is a particularly challenging task: even in phases of the evolution of a hybrid syste in which the structure of the dynaic equations reains constant, the evolution can be governed by continuous and discrete inputs. Design techniques have to account for the interaction of both degrees of freedo, with possibly opposing effects. If, in addition, the odel structure changes discretely at the occurrence of specific internal or externally triggered events, the design has to consider further discontinuities. Nevertheless, various algorithic approaches to controller design for different types of hybrid systes have eerged in recent years, notably those for algorithic synthesis and verification as can be found, e.g., in the proceedings of the HSCCworkshops (Tolin and Greenstreet, 2002). A coon property of at least those techniques that consider nonlinear dynaics in the continuous part of hybrid systes is that the coputational costs increase drastically with the proble size (see, e.g., (Silva et al., 2001)). A first step in aking existing techniques applicable to real-world probles is to investigate which degrees of freedo or which particular part of a proble forulation are doinating sources of coplexity. The intention of this contribution is to identify those effects that prohibit the application to large systes. The investigation is carried out exeplarily for an approach to copute optial control strategies for hybrid systes with nonlinear continuous dynaics and continuous as well as discrete inputs. The ethod is based on a transforation into a ixed-integer linear prograing (MILP) proble that is solved

2 by a Branch&Bound algorith (Stursberg and Panek, 2002; Stursberg et al., 2002). After sketching the approach in Sec. 2, we eploy two different ethods to assess the coplexity, as described in (Nehauser et al., 1994): In Sec. 3, a worstcase analysis provides guaranteed lower and upper bounds for the coplexity as functions of key paraeters of the proble forulation. Since these bounds are often overly conservative, additional inforation is obtained fro an epirical analysis (Sec. 4). In this step, the coputation tie is easured for different variations of the proble forulation, using a scalable exaple. Of course, the results presented are specific for the considered approach, but we nevertheless think that they point to typical probles arising in algoriths for hybrid systes and are thus helpful in assessing the coplexity of other approaches as well. 2. AN OPTIMAL CONTROL PROBLEM FOR HYBRID SYSTEMS We consider the task of deterining a control strategy to drive a given hybrid syste fro an initial state x 0 to a target region R tar with inial cost. Such probles typically arise in start-up procedures of processing systes where the ain goal is to reach a new desired operating region as quickly as possible while state and input constraints have to be et. The solution approach described in (Stursberg and Panek, 2002; Stursberg et al., 2002) consists of three steps: The starting point is to odel the given syste as a hybrid autoaton with possibly nonlinear continuous-tie dynaics and continuous and discrete control inputs. The state space X R nx of diension n x is partitioned by a set of n E hyperplanes such that ñ R polyhedral regions R r result: r R r = X. For reach region R r the dynaics is defined by a function ẋ = f r (x, u, v), where x is the current state, and u and v are continuous and discrete inputs. The nuber of all possible discrete input values is denoted by n V. The dynaics switches autonoously fro f r to f r if the state trajectory x(t) crosses the hyperplane that separates the regions R r and R r. The second step coprises the following: (1) Tie is discretized by transforing the ODEs fro step one into a set of difference equations defined for a set T of n T discrete points of tie. (2) All functions f r are replaced by linear approxiations for appropriate linearization points of x L r (in each region R r ) and u L. In order to iprove this approxiation, it can be necessary to introduce additional hyperplanes that lead to a finer partition of the state space, and hence to a larger nuber n R ñ R of regions. The result is a set of linear discrete tie functions x k+1 = f r,v (x k, u k ) = A r,v x k + B r,v u k + L r,v for each region R r and each discrete input value v. The tie index is k. In the third step, the optiization proble is forulated using the dynaic odel as constraints: The linear dynaics define restrictions on the continuous variables x k and u k, i.e., they relate the states x k, x k+1 at successive points of tie. Since only one linearized dynaics f r,v (x k, u k ) can be valid for each t k T, all other dynaic equations ust be disabled. For this purpose, a set of binary variables b k,r,v is defined, each of which enables or disables the corresponding function f r,v (x k, u k ) at k. The choice of the dynaics for a specific region R r and discrete input v k is forulated by: x k+1 = r,v b k,r,v (A r,v x k + B r,v u k + L r,v ). A transition between different dynaics hence corresponds to a change of values for exactly two binary variables. Since the objective is to forulate a linear optiization proble, the products of variables are linearized by introducing continuous auxiliary variables x b k,r,v := b k,r,v x k, u b k,r,v := b k,r,v u k and special sets of inequalities. This construction was proposed in (Stursberg and Panek, 2002) as disjunctive forulation. It is appropriate to define the relationship between continuous and binary variables without requiring large nubers of binary auxiliary variables. The forulation of the dynaics results in: x k+1 = r,v A r,vx b k,r,v + B r,v u b k,r,v + b k,r,vl r,v, and it ust hold that r,v b k,r,v = 1 for all k T. The set of constraints additionally contains inequalities that specify the regions R r and the check wether x k R r. For regions specified by C r x d r, the latter is written as b k,r (C r x d r ) 0, r b k,r = 1 with additional binary variables b k,r := v b k,r,v. Also here, the product of variables can be replaced by the disjunctive forulation. As described in (Stursberg et al., 2002), different possibilities exist to specify appropriate objective functions in the given context. We consider here a version that forulates the distance to a given rectangular target region R tar X in each step k: J = k x k z 1, z R tar. Since used in a iniization in u,v J, those trajectories of continuous (u k ) and discrete inputs (v k ) are selected, which drive the syste as close as possible to R tar in each step. The objective function together with the listed constraints represents a ixed integer linear prograing proble (MILP) that can be solved by standard techniques such as Branch&Bound algoriths.

3 3. THEORETICAL COMPLEXITY ANALYSIS For the optial control proble stated in Sec. 2, the objective of the coplexity studies is to investigate the coputational costs as a function of of the proble size. To deterine the latter, the following three easures are eployed: M 1 the size of the original proble easured by the values of the characteristic paraeters n x, n R, n v, and n T ; M 2 the size of the prograing proble specified by the overall nuber of variables n, the nuber of binary variables d (contained in n), and the overall nuber of constraints ; M 3 the coputational cost to solve the MILP easured as CPU-tie. This section describes the dependencies between the three easures prior to epirical studies. 3.1 Relation between M 1 and M 2 The nuber of variables n and the nuber of equations grow linearly with n x, n v, and n R. While d is not affected by n x, it depends according to log 2 (n R ) and log 2 (n v ) on the nuber of regions and discrete input cobinations due to the binary encoding of the discrete variables n R and n v. The nuber of regions itself is bounded by (n E +1) n R (2 n E ). In addition, all paraeters related to the easure M 2 grow linearly with the considered nuber n T of tie points. The coplexity of M 2 depending on M 1 is O(n T n x n v n R ) for and O(n T log 2 (n v + n R )) for d. is a sall ultiple of, but for large-scale probles it is reported that a ore realistic estiation is O() for the nuber of iterations and O( n) for the nuber of arithetic operations in each iteration. In (Hocks, 1995) it is stated that the coplexity of the siplex algorith grows fro linear to polynoinal with increasing proble size (Table 1), whereas barrier ethods proise lower coplexity for large scale probles. Table 1. Coplexity of the siplex algorith according to Hocks Scale Diension Coplexity sall < 100 O() ediu 100 < O( 2 ) large O( ) worst case O(2 ) The classical approach for solving an IP (with d binary variables) is the tree search by a Branch&Bound algorith with linear prograing relaxation, which generally has an exponential coplexity (NP-hard) (Floudas and Pardalos, 2001). In the best case, the coplexity of the Branch&Bound algorith is linear in the nuber of binary variables d, and in the worst case the full tree has to be searched. Heuristics ay be used to generate a first feasible solution and upper bounds, which reduce the size of the relaxed proble and also the tree size. An exact relation between the nuber of binary variables and the average coplexity of practical probles is still unknown, but the average coplexity varies fro fro O(d) over O(d 2 ) to O(2 d ) (Zhang, 1999). 3.2 Relation between M 2 and M 3 This section ais at assessing the relation between (n, d, ) and the coputation tie by reviewing published literature on the coplexity of MILP probles. By Branch&Bound algoriths, MILPs are decoposed into an integer progra (IP) and a series of subordinated linear progras (LP). The size of an LP is defined by its n variables and constraints. A standard solver for LPs is the Siplex Algorith, which searches along the vertices of the feasible region in direction of a decreasing value of the objective function. In the worst case, all vertices have to be investigated what leads to ψ = ( n ) iterations. The relation between the proble size of an LP and the average coplexity of the solution is still an open issue, but it has been found experientally that the effort is often considerably saller than in the worst-case: (Edgar and Hielblau, 1988) ention that the nuber of iterations usually depends uch ore on than on n, and is often between and 3. This range is confired in (Nehauser et al., 1994) for probles in which n 4. EMPIRICAL ANALYSIS Since especially the relation between M 2 and M 3 cannot be established precisely fro theoretical considerations, we now epirically analyze the perforance of the approach. We use a scalable ulti-tank syste to investigate directly the relation between M 1 and M 3. Scaling eans in this context that one of the paraeters dterining M 1 is varied (while the others are kept constant) and the change of M 3 is observed. The scalable syste under consideration is shown in Fig. 1: It consists of a series of tanks connected by pipes. The feed into the first tank is located above this tank, while all other feed tubes are located at the ediu height (H) of the subsequent tank. The feed and the outflow pipes are equipped with valves to adjust the flow rates. Autonoous changes of the dynaics occur, if the liquid level h in a tank exceeds the height of the feed tube. A corresponding hyperplane and two distinct dynaics account for this in the hybrid odel. The fact, that a reversed flow fro a tank into a preceding one is assued to be not possible,

4 target R tar forbidden regions V0 h2 H2 h1 h2 V1 h3 H2 V2 h1 H3 H3 h3 switching of the dynaics n T (a) tie steps n T n x (b) states n x Fig. 1. Reference syste with three tanks; partitioning of X is odeled by a forbidden region in the state space (see (Stursberg and Engell, 2002) for details on specifying forbidden regions). The syste as shown in Fig. 1 acts as a reference odel, i.e., it is the basis for investigating variations of one of the paraeters n x, n R, n v, n T. The reference odel is defined by three tanks (n x = 3) with only continuous inputs, thus only one discrete input cobination is considered (n v = 1). The continuous valves V represent bounded continuous inputs. Modelling of the syste leads to n E = 4 separating hyperplanes which define n R = 7 regions. The optiization goal is to iniize the objective function as specified in Sec. 2 for the transition fro the initial state x 0 = (0.1, 0.1, 0.1) T to the target region R tar = [0.7, 0.8] [0.7, 0.8] [0.7, 0.8] within n T = 10 tie steps. n R (c) regions n R (d) input cob. n v Fig. 2. Analysis of coputing tie: scaled tie c vs. proble size M 1 feeding pipes located at the top (as for the first tank). The results for n x between 3 and 8 are shown in Fig. 2(b) Regions: In order to study the ipact of n R, hyperplanes are successively reoved fro the odel (by setting the height of the inflow pipes to a axiu value of the level), or added respectively. For exaple, by adding one additional separating hyperplane to the first tank of the references odel (i.e., n E = 5), the nuber of regions is increased to n R = 18. The effect on the CPU-tie is shown in Fig. 2(c). n v 4.1 Analysis of the Coputation Tie The MILPs are solved by using the package GAMS/CPLEX 7.5 which is based on a Branch& Bound algorith with cuts and heuristics, and a siplex algorith to solve the LP subprobles. Upper bounds for the nuber of iterations and the axiu resource usage are set to 10 6, and 10 5 s( 27.7h) respectively. The relative gap to optiality is set to ɛ All coputations were perfored on a Workstation Sun Ultra-Sparc with 300 MHz. In all subsequent figures, we use a scaled CPU-tie c = t/t ref where t ref = 19.74s is the coputing tie for the reference syste Tie steps: To observe the effect of altering the nuber of tie steps, the tie horizon is kept constant as t nt = 50s while n T is varied between 5 and 11. As shown in Figure 2(a), the CPU-tie grows exponentially Cont. State Space: The diension n x of X is increased by adding additional tanks, with Discrete Input Cobinations: Finally, the effects of a changed nuber of input cobinations is investigated by allowing different settings for the two valves V1 and V2. Figure 2(d) suarizes the results for n v between 1 and Suary of the Experiental Results As shown in Figures 2(a)-2(d), the following dependencies of M 3 fro M1 were observed: O(2 n T ), O(n 2 x ), O(n 2 R ), and O(n 2 v ). The ost critical effect is the exponential growth by an increase of n T. If n x, n R, and n v are increased (independently), the coplexity grows only polynoially. Fro about one hundred optiization studies, it was found that probles with 5 to 10 continuous state variables and a relatively sall nuber of tie steps (n T 5) are solvable for up to 8 discrete input cobinations and less than 10 regions within the given resource bounds.

5 T IT T IT I LP (a) Iterations per LP I LP LP size vs. (b) Cop. tie per LP iteration T IT (s) vs. LP size (c) B&B nodes N IP vs. IP size d Fig. 3. Iterations, coputing tie and B&B nodes vs. MILP size M 2 for series A T IT N IP I LP d (a) Iterations per LP I LP LP size vs. (b) Cop. tie per LP iteration T IT (s) vs. LP size (c) B&B nodes N IP vs. IP size d Fig. 4. Iterations, coputing tie and B&B nodes vs. MILP size M 2 for series B 4.3 Analysis of the Algorith To get further insight into the dependencies, the coplexity of the solution algorith is studied by observing the nuber of nodes N IP explored by the Branch&Bound algorith, and the average tie required to solve an LP. The tie required for one iteration during the LP solution varies due to the nuber of necessary algebraic operations. Therefore, the coputing tie of the LP solution is coputed by the average nuber of iterations I LP per LP ties the average CPU-tie per iteration T IT. This analysis includes the configurations of Sec. 4.1 (series A) and a larger set of about 70 experients with changes of ore than one syste paraeter at a tie (series B). We assue that the LP odel size is characterized by and the IP size by d. In the following figures, the paraeters N IP, I LP, and T IT are plotted against or d, respectively. The nuber of iterations per LP I LP is alost linear in for series A (Fig. 3(a)) with a few exceptions. For series B, Fig. 4(a) shows that especially for ore than 5000 equations the spread of points is rather large. The coputation ties per LP iteration T IT of series A are shown in Fig. 3(b) and of series B in Fig. 4(b). In both figures T IT appears linear in. It is ore difficult to estiate the dependency between N IP and d: Series A in (Fig. 3(c)) shows siilar results as series B (Fig. 4(c)). Both figures do not show a linear or quadratic relation clearly. 4.4 Interpretation of the Results The ai of the epirical analysis is to define the gap between the worst-case theoretical coplexity and the estiated coplexity of practical probles. The expected coputing tie of a MILP instance can be related to and d by: T MILP = K I K T K N d. The paraeters K are obtained fro the epirical series by: K I = I LP /, K T = T IT /, and K N = N IP /d. Now the epirical upper and lower bound of the total coputing tie can be estiated by taking the iniu and the axiu value of each K (Tab. 2) and relate and d to the respective values of the MILP instance. Figure 5 shows the coputing tie of the probles of series B and the epirical upper and lower bounds. We assue a constant value of the coputing tie per iteration by LP size K T for that. Copared to the epirical results, the theoretical worst case coputing tie can be ore than ties higher. If we copare the results of Table 2 with the considerations of Section 3, we can conclude that

6 Table 2. Epirical results of K Coefficient in ean ax K I (B) K T (B) (s) K N (B) K I K T K N (s) Series B Ep. axiu Ep. iniu to be to introduce the notion of odularity. If the proble can be split into several odules (each with sall values for n x, n R, and n v ), the solution of the optial control probles should be efficiently possible for each of the. The challenging task is to construct a solution for the coplete syste fro the coposition of the single solutions. 1 E+05 Total cop. tie (s) 1 E+04 1 E+03 1 E+02 1 E+01 1 E+00 1 E-01 1 E+06 1 E+07 1 E+08 1 E+09 1 E+10 MILP proble size: 2 d Fig. 5. Worst case and epirical bounds the series of MILP probles was solved very efficiently with respect to their proble size. To obtain a ore general result, we roughly estiate the coplexity fro Figures 3(a) - 4(c). We find I LP to be between O() and O( 2 ). For T IT it is O() and for N IP we assess a range between O(d) and O(d 2 ). As a result we can state the coputational coplexity M 3 of practical probles depending on M 2 to be O( 2 d) to O( 3 d 2 ). If we cobine this with the results of Section 3.1, we obtain the relation between M 1 and M 3 to be bounded by O((n T n x n v n R ) 2 n T log 2 (n v + n R )) and O((n T n x n v n R ) 3 (n T log 2 (n v + n R )) 2 ). 5. CONCLUSIONS The investigation shows (not surprisingly) that the tie horizon (n T ) has the doinant influence on the solution perforance: The nuber of equations and variables grows linearly with n T what leads to an exponential growth of the nuber of degrees of freedo. Two odifications to weaken this effect have been introduced: An obvious idea ist the use of a MPC-strategy that solves the optiization proble on short and oving horizons (Stursberg et al., 2002). A further reduction is obtained by using variable tie steps what allows to cover larger periods of tie with constant inputs (Stursberg and Engell, 2002). But even with these extensions, large scale probles reain unsolvable by the considered approach. This calls for reducing the size of the proble forulation for each point of tie. Considering the dependency of the coplexity on n x, n R, and n v, their ipact sees to be siilar, i.e., there is no clear candidate to be tackled first. Hence, a reasonable strategy for enhancing the applicability of this or siilar approaches sees REFERENCES Edgar, T. F. and D. M. Hielblau (1988). Optiization of cheical processes. McGraw- Hill. Floudas, C.A. and Pardalos, P.M., Eds.) (2001). Encyclopedia of Optiization. Kluwer Acadeic Publishers. Hocks, M. (1995). Innere-Punkt-Methoden und autoatische Ergebnisverifikation in der Linearen Optiierung (in Geran). PhD thesis. Universitaet Karlsruhe (TH). Nehauser, G.L., A.H.G. Rinnooy Kan and M.J. Todd (1994). Optiization. North-Holland. Silva, B.I., O. Stursberg, B.H. Krogh and S. Engell (2001). An assessent of the current status of algorithic approaches to the verification of hybrid systes. In: Proc. IEEE Conf. on Decision and Control. pp Stursberg, O. and S. Engell (2002). Optial control of switched continuous systes using ixed-integer prograing. In: Proc. 15 th IFAC World Congress on Autoatic Control. Paper: Th A06-4. Stursberg, O. and S. Panek (2002). Control of switched hybrid systes based on disjunctive forulations. In: Hybrid Systes: Coputation and Control. Vol of LNCS. Springer. pp Stursberg, O., S. Panek, J. Till and S. Engell (2002). Generation of optial control policies for systes with switched hybrid dynaics. In: Modelling, Analysis, and Design of Hybrid Systes. Vol. 279 of LNCIS. Springer. pp Tolin, C.J. and M.G. Greenstreet (2002). Hybrid Systes: Coputation and Control. Vol of LNCS. Springer. Zhang, Weixiong (1999). State-Space Search: Algoriths, Coplexity, Extensions, and Applications. Springer.