Proceedings of the 44th IEEE Conference on Decision and Control, and the Eropean Control Conference 2005 Seville, Spain, December 12-15, 2005 TC08.5 Nonlinear parametric optimization sing cylindrical algebraic decomposition Ioannis A. Fotio*, Pablo A. Parrilo and Manfred Morari* Abstract In this paper, a new method is presented for optimization of parametric families of polynomial fnctions sbject to polynomial constraints. The method is based on cylindrical algebraic decomposition (CAD). Given the polynomial objective and constraints, the method constrcts the corresponding CAD offline, extracting in advance all the relevant strctral information. Then, given the parameter vale, an online procedre ses the precompted information to efficiently evalate the optimal soltion of the original optimization problem. The method is very general and can be applied to a broad range of problems. I. INTRODUCTION Model predictive control is a very active area of research with broad indstrial applications [1]. It is among the few control methodologies that provides a systematic way to perform nonlinear control synthesis nder state and inpt constraints. This ability of dealing with constraints is one of the main reasons for the practical sccess of model predictive control (MPC) [2]. MPC ses on-line optimization to obtain the soltion of an optimal control problem in real time. This method has been proven most effective for applications. Typically, the optimal control problem can be formlated as a discretetime mathematical program, whose soltion yields a seqence of control moves. Ot of these control moves only the first is applied, according to the receding horizon control (RHC) scheme. The specific form of the corresponding mathematical program can be a linear program (LP), a qadratic program (QP) or a general nonlinear program (NLP). Technology and cost factors, however, make the direct implementation of receding horizon control difficlt, or in some cases impossible. In the standard linear MPC case, the corresponding optimization problem is a convex QP, with linear constraints. In this case, an alternative approach to the soltion of the optimal control problem is to compte the control law off-line, by solving the corresponding program parametrically [3]. That is, we compte the explicit formla giving the soltion of the mathematical program (control inpts) as a fnction of the problem parameters (measred state). The soltion is then efficiently implemented on-line as a lookp table. In the general case of nonlinear systems and constraints, the MPC formlation gives rise to a parametric optimization Atomatic Control Laboratory, Swiss Federal Institte of Technology, CH-8092 Zrich, Switzerland. Laboratory for Information and Decision Systems, Massachsetts Institte of Technology, Cambridge, MA 02139-4307, USA. Corresponding Athor: Email: fotio@control.ee.ethz.ch 0-7803-9568-9/05/$20.00 2005 IEEE 3735 problem, where as before the soltion can be expressed as an explicit fnction of the parameters. However, in contrast to the linear MPC case, no simple expression of the optimal soltion is possible, as it necessarily involves implicit algebraic fnctions. Nevertheless, while a closed form expression is not possible, a parametrization of the optimal soltion is still possible by combining a precomptation stage sing algebraic techniqes and the on-line soltion of nivariate polynomial eqations. In this paper, we se cylindrical algebraic decomposition (CAD) to perform nonlinear parametric optimization of polynomial fnctions sbject to polynomial constraints. The proposed method encompasses linear and qadratic parametric optimization as special cases. II. PARAMETRIC OPTIMIZATION AND CAD A nonlinear parametric optimization problem generally assmes the form min J(,x) s.t. g(,x) 0 (1) where J(,x) is a polynomial fnction in and x, R n is the decision variable vector, x R m is the parameter vector and g(, x) is a vector polynomial fnction. The ineqality is meant in the sal componentwise fashion. By parametric optimization, we mean minimizing the fnction J(, x) with respect to for any given parameter x in the region of interest. Therefore, the nonlinear parametric optimization problem this paper addresses is finding a comptational procedre for evalating the maps (x) : R m R n where J (x) : R m R = argmin J(, x) J = min J(,x). To keep or discssion simple, we restrict or attention to those parameters x for which problem (1) has a niqe minimizer. That way x is a fnction and not a pointto-set map. We also assme that the feasible set is compact. Nevertheless, the algorithm presented in this paper is easily extended to the cases where the minimizer is not niqe or the feasible set is not compact. Before presenting or approach we first have to describe some basic concepts. (2)
A. Cylindrical algebraic decomposition Given a set P R[,...,x n ] of mltivariate polynomials in n variables, a CAD is a special partition of R n into components, called cells, over which all the polynomials have constant signs. The algorithm for compting a CAD also provides a point in each cell, called sample point, which can be sed to determine the sign of the polynomials in the cell [4]. To perform optimization, a CAD is associated with a Boolean formla. This Boolean formla can either be qantified or qantifier free. By a qantifier-free Boolean formla we mean a formla consisting of polynomial eqations ( f i (x) =0) and ineqalities ( f j (x) 0) combined sing the Boolean operators (and), (or), and (implies). In general, a formla is an expression in the variables x =(,...,x q ) of the following type: Q 1...Q s x s F ( f 1 (x),..., f r (x)) (3) where Q i is one of the qantifiers (for all) and (there exists). Frthermore, F ( f 1 (x),..., f r (x)) isassmedtobea qantifier-free Boolean formla. B. Constrction of CAD Obtaining the CAD involves compting with discriminants and resltants. This procedre is called projection phase and has as many steps as the nmber of variables x i in F ( f 1 (x),..., f r (x)) mins one. The main idea is, given the set of polynomials { f i (x)}, to obtain in each step k = 1,...,q 1 a new set of polynomials P k ( f i (x)) eliminating one variable at a time. That is, the new set of polynomials will depend only on q 1 variables {,,...,x q 1 }. Along with the new set of polynomials, the CAD constrction algorithm provides s with a special set of polynomials attached to each projection level, called the projection level factors denoted by {Li d} i=1..t d. The set of the real roots of these polynomials contains critical information abot the CAD, defining the bondaries of its cells. These roots can be isolated points in R n, crves, srfaces or hypersrfaces, depending on the dimension of the projection space. C. Posing the problem Sppose we have to solve problem (1). We associate with problem (1) the following boolean expression (g(,x) 0) ( J(,x) 0) (4) We then compte the CAD defined by the polynomial expressions in (4). For this, we se QEPCAD [5]. The signs of the polynomials appearing in (4) as well as of P k ( f i (x)) reslting from the projection steps are determined in each cell. These signs, in trn, determine wether (4) is tre or false in each particlar cell. For or prposes, it is enogh that QEPCAD determines the trth or falsehood of flldimensional cells only. The sample points in these cells are rational nmbers and the comptations associated with them are mch easier than in the general case. We instrct 3736 QEPCAD to do so with the measre-zero-error command. All the information we need to solve problem (1) is the level factor polynomials associated with the CAD of system (4), the sample points, and the knowledge of which cells are tre and which false. D. A simple CAD Let s look at the CAD of the following set of polynomial ineqalities { 4 10 2 } + + 1 (5) 7 + 17 The level factor polynomials for system (5) are L1 2(,) = 4 10 2 + + 1 L2 2 (,) = 7 17 L 1() = 2563 + 12032 2 + 133728 149989 L2 1() = 4 + 68 3 + 1244 2 + 934 49857 (6) Note that the level-two factors are the polynomials as they appear in (5). This is always the case with the last projection level, since no variable has yet been eliminated (projected). As briefly mentioned before, the set of real roots of the level-one factors Li 1 () in (6) will partition the space into zero- and one-dimensional cells. These roots can clearly be seen in Figre 1: lines parallel to the axis mark their position. These positions correspond to points where the two polynomials intersect or the tangent to them becomes parallel to the axis (critical points). Accordingly, the root set of the level-two factors, together with the one of the first level, will define the bondaries of the cells in the joint (,) space. The tre cells of the (,) space are specially marked. In optimization, they wold correspond to the problem feasible region in the same space, the parameter x having been specified. III. THE ALGORITHM In the general case of nonlinear parametric optimization there exists no explicit, closed-form formla giving the optimizer or the optimal vale J as a fnction of the parameter x. For example, there exists no expression that gives the roots of a polynomial of degree greater than for in terms of elementary fnctions of its coefficients. We will however present an algorithm which constittes an efficient comptational procedre to evalate the map from x R m to R n and J R. Or objective is namely to evalate the maps in (2). The CAD for system system (4) associated to the optimization problem at hand has been constrcted in advance and the information contained therein is available to the algorithm presented below. Note that the variables we now deal with are (,...,x m,, 1,..., n ) appearing in (4). The projection steps of the CAD constrction phase will first eliminate n moving from the end of the list to the beginning.
50 r 3 r 3 Tre L m+1 3 (x, ) g 3 Tre L 2 1 (,)=0 Tre r 2 r 2 L m+1 2 (x, ) g 2 L m+1 1 (x, ) -30 L 2 2 (,)=0-3 3.4 Fig. 1. CAD of the polynomials in (5). r 1 r 1 A. The algorithm First step (Initialization) We determine in what cell in the x-space the given x parameter lies. This can be done by checking the cells of the CAD and the corresponding level factor polynomials defining their bondaries. The roots of the level factors of the first level (i.e. the level reslting after the last projection step) partition the space into level-one cells that are either (zero dimensional) points in R or (one dimensional) line segments that may also extend to infinity. Accordingly, level-two factors (together with the root set information of the first level factors) partition the (, ) space into level-two cells that can be points embedded in R 2, one-dimensional crves or two-dimensional sbsets of R 2. Same holds for higher dimensions p to R m. For fixed parameters, the point x R m belongs to a niqe cell C x R m. Second step (Finding J ) We now lift cell C x p to the space of the cost-associated variable (see Figre 2). Some cells in the joint (x,) space will be tre, some will be false. In Figre 2, the srfaces represent the zero sets of the (m + 1)-level factors Li m+1. Fixing the vale of x we obtain zero- and one-dimensional sbcells along the axis. These are depicted in Figre 2 with the thick black line rising from point x. We then look for that tre cell in R m+1 which for the given vale of x contains the smallest vale among all other cells in the cylinder above C x.wemay think of it as the first tre cell conting form the bottom p. We denote it by G R m+1. If no sch cell exists, then problem (1) is infeasible for the given vale of x. If the cell exists bt happens to be nbonded from below, then for the given vale of x optimization problem (1) is nbonded 3737 x k R m C x Fig. 2. Lifting to the space. The Figre is based on Christopher Brown s ISSAC 2004 CAD ttorial slides. (from below). In case it is not, the minimm is attained and its vale is the minimm vale of in G obtained for the fixed vale of x. The reasoning above is depicted in Figre 2. By sbstitting the fixed vale of x into the (m+1)-level factors, we obtain three roots r 1, r 2 and r 3, each one corresponding to a different level factor, which partition the reslting axis rising from x C x into for one-dimensional sbcells g i R. It happens that G is the cell over C x, between srfaces L 2 and L 3. Conseqently, the optimal cost vale will be r 2 which is fond by solving the nivariate polynomial eqation L2 m+1 ()=0. This eqation may have more than one real root. Which one corresponds to is given by the CAD sample point information and can be nambigosly determined. This will become clear in the example to follow. Third step (Finding ) To determine the optimizer R n we have to lift the (x,) R m+1 point in the space of the decision variables. Remember that we restrict or attention to those x that yield a niqe optimizer R n. g 1 x
(, )space partition Determining cell C x (1,7) L 2 1 (, )=0 3632 211 L 2 1 (, )=0 (3,3) L 1 1 ()=0 (1,5) L 1 1 ()=0 (-10,1) 1 sample point (3,2) L 2 1 (, )=0 L 2 2 (, )=0 (1,3) 3632 211 L 2 1 (, )=0 (1,1) L 2 2 (, )=0 (3,1) -10 Fig. 3. Level factors partitioning the x space of problem (7). Fig. 4. Point ( 10,1) lies in cell (1,5). The sample point of this cell is labelled with a sign. The level factor polynomials define the bondaries of the cells. First, we sbstitte the vales of x and in the level factors {Li m+2 (x,, 1 )} i=1...tm+2. What we obtain is t m+2 nivariate polynomials in 1 which we denote by {s 1 i ( 1)} i=1...tm+2. We solve them to calclate their real roots set A 1. We then se the precompted CAD cell information together with the level factor signs of each cell to determine which root in A 1 corresponds to the optimizer 1. This root will be part of the optimizer vector =( 1,..., n). This can be formalized as follows Level m+2: {Li m+2 (x,, 1 )} {s 1 i ( 1)} {s 1 i ( 1)} = 0 = CAD 1 = 1. with x =[, ] being the parameter. After the constrction of the CAD, we obtain the following level factors L 1 1 () = L 2 1 (, ) = 27 2 + 8x3 1 L 2 2 (, ) = L1 3(,,) = 256 3 + 128x1 22 768 2 + 144 x2 2 +16x1 4 256x2 1 + 768 + 27x4 2 + 4x3 1 x2 2 144 x2 2 16x4 1 + 128x2 1 256 Similarly, we now sbstitte optimizer 1 in the level factors {Li m+3 (x,, 1, 2 )} i=1...tm+3 to obtain the polynomials {s 2 i ( 2)} i=1...tm+3. By solving them and sing the associated CAD information and level factor signs (which information is already available) we obtain the next optimizer 2.Same procedre is followed ntil we have calclated the optimizer components p to n. It has to be emphasized that the proposed algorithm, when implemented online, only needs to perform the traversal of a tree (see Figre 6) and solve nivariate polynomial eqations. All other information needed is precompted offline when the CAD is constrcted. B. Illstrative example To illstrate the proposed method let s look at the following parametric minimization problem min 4 + 2 + + 1, (7) 3738 L 4 1 (,,,) = 4 + 2 + + 1. We note that the last level factor is the inpt formla of the Boolean expression (4) the optimization problem is nconstrained. The factor polynomials of the first two levels partition the (, ) space as seen in Figre 3. Let s choose = 10 and = 1. The level factor L 1 1 partitions the space into two (fll-dimensional) cells, namely, (,0) and (0, ). The given vale of belongs in the first cell, which is indexed by 1. The root of L 2 2 ( 10,)=0 is readily = 0 and that of L 2 1 ( 10,)=0 is = ± 3632 211. That means, for the specific vale of, the space is partitioned in for (fll-dimensional) cells. As shown in Figre 4, where all the fll dimensional cells have been labelled, the given x point lies in cell (1,5), since 0 < 1 < 3632 211. Once we determined C x, we obtain its sample point from the CAD. For cell (1,5) it is ( 1, 4 1 ) and it is marked on Figre 4 with a sign. We can then lift this point p to the (,) space. The level factors partition the (,) space as depicted in the left part of Figre 5. Lifting point (, )
50 2 (1,5,5) r 3 r 2 L 4 1 (,)=0 Tre False (1,5,3) (1,5,2) (1,5,1) r 1 Tre False 0.5-1.5 1-30 -3 3.5 Fig. 5. On the left side, sample point ( 1, 1 4 ) is lifted to the (,) space. On the right side, point ( 10,1) is lifted to the same space. It is easily observed that the two figres are topologically the same. This is becase both points ( 1, 4 1 ) and ( 10,1) lie in the same cell C x. prodces similar reslts as shown in the right part of Figre 5. We observe that both parts of Figre 5 are topologically the same. This is always the case. The topology of the stack of cells bilt pon a point x depends only on cell C x, not on its actal coordinates. Therefore, the lifting for the sample points of all the cells the x space is decomposed into, can be done in advance. From this lifting, we can constrct a fnction M that maps every cell C x to the index i that corresponds to the cell labeled (i x1,...,i xm,i ) that is the lowest tre cell of R m+1 in the cylinder above x. Here, i = 3. Cell (1,5,3) is clearly marked on the left part of Figre 5. The fnction M can be extended to the -space as well, giving the corresponding index information for the optimizer. By solving the eqation L1 3 ( 10,1,)=0 we obtain { 26.25, 21.78, 1.03}. The fnction M gives s the index i = 3, therefore we know that is the first, i.e. smaller, root of the three (see Figre 5). Frther sbstitting x and in L1 4 and solving eqation L4 1 ( 10,1, 26.25, 1)=0gives U 1 = { 2.26}. We readily conclde that the optimizer is 1 = 2.26. It also happens that the algebraic mltiplicity of this root is two. This is in agreement with Figre 1, where line = is tangent to the nivariate polynomial 4 10 2 + + 1. The whole algorithm is in effect the traversal of the cell tree shown in Figre 6 modlo the soltion of nivariate polynomial eqations. This tree is the instance of the more generic roadmap the algorithm constrcts based on the CAD information. This roadmap is sed by the algorithm to evalate maps (2). 3739 (,,,) space cell (1,5,3,2) (,,) space (, ) space = 0 + cell(1) = 0 cell(3) Fig. 6. Traversing the cell tree. We repeat the above procedre for varios vales of and. The optimizer as a fnction of (, ) is shown in Figre 7. We observe that the optimizer is discontinos along the line = 0. Sch discontinities are characteristic of nonlinear parametric optimization problems. IV. NONLINEAR MPC A. Problem formlation Assme that we have a nonlinear discrete-time system of the form x k+1 = f (x k, k ) (8) sbject to following ineqality constraints for k = 0..N, N being the prediction horizon: g(x k, k ) 0, k = 0..N, (9)
6 4 2 0-2 is intrinsically not efficient in eliminating variables, becase it does this one variable at a time. There exist more efficient methods than CAD for comptation with semialgebraic sets, sch as the Critical Point Method [6], which has mch better comptational properties. Unfortnately there are no crrently available implementations of these methods. In or approach however, once the CAD has been compted, the algorithm needs only perform a tree traversal and solve some nivariate polynomial eqations. All the information to traverse the tree has already been extracted from the CAD cell strctre and level factor signs in each of the cells. -4-6 -50 0 50 100-50 Fig. 7. Optimizer for problem (7). where g( ) is a mltivariate polynomial fnction in the state and control variables. We consider the problem of reglating system (8) to the origin. For that prpose, we define the following cost fnction J(U0 N 1 N 1,x 0 )= k=0 0 50 100 L k (x k, k )+L N (x N, N ) (10) where U0 N 1 := [ 0,..., N 1 ] is the optimization vector consisting of all the control inpts for k = 0..N 1and x 0 = x(0) is the initial condition of the system. Therefore, compting the control inpt boils down to solving the following nonlinear constrained optimization problem min J(U N 1 0,x 0 ) s.t. g(x k, k ) 0, k = 0,...,N. (11) For ease of notation we will drop from now on the sb- and sperscripts in (11). Problem (11) is written in the more compact form min J(,x) s.t. g(x,) 0, (12) where J(,x) is a polynomial fnction in and x, R n is the decision variable vector and x R m is the parameter vector. This is exactly problem (1), a nonlinear parametric optimization problem. Or goal is to obtain the vector of control moves. VI. CONCLUSIONS AND OUTLOOK The main contribtion of this paper is a new algorithm for performing nonlinear constrained parametric optimization of polynomial fnctions sbject to polynomial constraints. The algorithm ses cylindrical algebraic decomposition to evalate the map from parameter space to the corresponding optimizer and optimal vale. Secondly, the method has been linked to model predictive and optimal control problems. It has to be noted that there exists also the potential of combining dynamic programming with the proposed method to exploit the recrsive strctre of dynamical systems [7]. Finally, it shold be stated that althogh the proposed algorithm is extremely general and can in principle be applied to a wide variety of problems, its application is limited by the comptational cost of the CAD procedre. Unless algorithmic breakthroghs take place or more efficient methods for CAD constrction are implemented, the practical relevance of the proposed scheme will be restricted to problems with a relatively small nmber of variables. REFERENCES [1] S. J. Qin and T. A. Badgwell, An overview of nonlinear mpc applications, in Nonlinear Model Predictive Control: Assessment and Ftre Directions, F. Allgöwer and A. Zheng, Eds. Birkhaser, 1999. [2] C. Garcia, D. Prett, and M. Morari, Model predictive control: theory and practice -a srvey, Atomatica, vol. 25, pp. 335 348, 1989. [3] A. Bemporad, M. Morari, V. Da, and E. N. Pistikopolos, The explicit linear qadratic reglator for constrained systems, Atomatica, vol. 38, pp. 3 20, 2002. [4] B. Caviness and J. Johnson, Qantifier elimination and cylindrical algebraic decomposition. Wien: Springer Verlag, 1998. [5] C. W. Brown, QEPCAD B: a program for compting with semialgebraic sets sing CADs, ACM SIGSAM Blletin, vol. 37, pp. 97 108, 2003. [6] S. Bas, R. Pollack, and M.-F. Roy, Algorithms in real algebraic geometry. New York: Springer Verlag, 2003. [7] I. A. Fotio, P. A. Parrilo, and M. Morari, Nonlinear parametric optimization sing cylindrical algebraic decomposition, Atomatic Control Laboratory, Tech. Rep., Mar. 2005. V. REMARKS The main comptational bottleneck of the proposed method is the offline constrction of the CAD. On the one hand, focsing only on fll dimensional cells greatly redces the comptational brden as compared to the general case since we avoid comptation sing algebraic nmbers. On the other hand, the cylindrical algebraic decomposition 3740