Optimal control and piecewise parametric programming

Proceedings of the Eropean Control Conference 2007 Kos, Greece, Jly 2-5, 2007 WeA07.1 Optimal control and piecewise parametric programming D. Q. Mayne, S. V. Raković and E. C. Kerrigan Abstract This paper deals with the problem of parametric piecewise qadratic programming (in which the cost is a piecewise qadratic fnction of both the decision variable and a parameter) and the problem of parametric piecewise affine qadratic programming (in which both the cost and the constraint depend on a piecewise affine fnction of the decision variable and a parameter). Parametric programming seeks a soltion for each vale of the parameter, and can therefore be sed to obtain explicit soltions of some constrained optimal control problems where the state is the parameter. The techniqe of reverse transformation for parametric programming, introdced in earlier papers, is extended to remove nnecessary overlapping of polytopes on which the soltion is defined. The improved techniqe is then employed for the determination, sing dynamic programming, of explicit control for linear systems with a piecewise qadratic cost and explicit control of piecewise affine systems with qadratic cost. I. INTRODUCTION A conventional optimization problem has the form = min {V () U} where is the decision variable, V () is the cost to be minimized, and U is the constraint set; let 0 denote the soltion (the minimizer) to the problem. A parametric programming problem takes the form (x) = min {V (x,) U(x)} and both the vale fnction x (x) and the minimizer x 0 (x) become fnctions of the parameter x; the minimizer 0 ( ) may be set-valed (for each x, 0 (x) is a set). Optimal control problems often take this form, with x being the state, and, in open-loop optimal control, being a control seqence; in state feedback optimal control, necessary when ncertainty is present, dynamic programming is employed yielding a seqence of parametric optimization problems in each of which x is the state and a control action. In certain problems, it is possible to characterize both the vale fnction ( ) and the 0 ( ) as fnctions of a specific type that can be explicitly compted [1] [13]. When distrbances are present, it is necessary to compte the soltion seqentially sing dynamic programming as in [3], [7], [11], [13]. To solve the parametric problem, we extend the reverse transformation procedres proposed in [14] and tilized in [3], [6], [8], [9], [15]. In this procedre, a condition that is potentially satisfied at the soltion and which, if known, simplifies the soltion is assmed to hold. The simplified soltion, nder this condition, is then obtained, yielding a Research spported by the Engineering and Physical Sciences Research Concil, UK and the Royal Academy of Engineering, UK. D. Q. Mayne is with the Department of Electrical and Electronic Engineering, Imperial College London, S. V. Raković is with the Atomatic Control Laboratory, ETH, Zürich and E. C Kerrigan is with the Department of Aeronatics and the Department of Electrical and Electronic Engineering, Imperial College London minimizer and vale fnction (fnctions of the parameter x). Finally, the region of state space in which the assmed condition holds, is then determined; the minimizer and vale fnction of the simplified problem are the minimizer of the original problem in this region. Assming a different condition yields a different region; in this way, a piecewise affine or piecewise qadratic soltion (covering the whole parameter set) to the original parametric problem may be obtained. In parametric qadratic programming, the condition assmed is the set of active constraints at the soltion; if this is known, the original ineqality constrained problem is converted into an easily solved eqality constrained problem. In the dynamic programming soltion of constrained linear systems with piecewise affine or piecewise qadratic cost [15], [16] the vale fnction at each time is piecewise affine or piecewise qadratic; at each stage the cost to be minimized (the sm of the stage cost and the vale fnction at the sccessor state) is piecewise affine or piecewise qadratic, and the condition assmed is the index that specifies the polytope in which the optimal state-control pair lies (and in which the optimal cost is affine or qadratic). If this condition is known, the original problem is converted into a parametric linear or qadratic program. In control of piecewise affine systems (where the dynamics are affine in each polytope of a polytopic partition of the state-control space) the condition assmed [8], [9] is the time-seqence {s 0,s 1,...,s N 1 } of indices that specifies that the optimal state-control pair at time i lies in the polytope P si ; if this time-seqence is known, the nonlinear optimal control problem is converted into a simpler, linear, time-varying optimal control problem. There is, however, a drawback in the reverse transformation procedre crrently employed arising from the fact that the simpler problem sometimes involves artificial constraints that do not appear in the original problem. Conseqently, a soltion to the simpler problem (corresponding to the assmption that the condition is satisfied) may not be a soltion to the original problem giving rise to sprios soltions. Conseqently, the regions in which the soltion is affine or qadratic may overlap; the optimal soltion in an overlapping region mst then be determined by comparing the soltions corresponding to each of the overlapping regions. The main prpose of this paper is to present a modified version of the reverse transformation procedre that prevents nnecessary overlapping of regions and removes it entirely in convex problems. We develop the new procedre in the context of two problems: parametric piecewise qadratic programming in 2, its application to the dynamic programming soltion of constrained linear systems with qadratic or piecewise qadratic cost in 3, parametric piecewise affine qadratic ISBN: 978-960-89028-5-5 2762

programming in 4 and its application to optimal control of piecewise affine systems in 5. This paper corrects the soltion to the first problem that appears in [16]. A. Introdction II. PARAMETRIC PIECEWISE QP The general parametric programming problem P(x) is defined by: (x) = min{v (x,) (x,) Z} (1) 0 (x) = arg min{v (x,) (x,) Z} (2) where x R n, R m (z = (x,) R n R m ); 0 (x) is the soltion and (x) the vale of problem P(x). Let X Proj X Z {x sch that (x,) Z}. The following reslt is established in [17] and earlier in [18] nder the stronger hypothesis that the gradients of the active constraints are linearly independent; a self-contained proof is given in [16]: Proposition 1: Sppose V : Z R is a strictly convex, continos fnction and that Z is a polytope (compact polyhedron). Then, for all x X = Proj X Z, the soltion 0 (x) to P(x) exists and is niqe. The vale fnction ( ) is strictly convex and continos with domain X, and the soltion 0 ( ) is continos on X. The parametric piecewise qadratic programming problem has more strctre and reqires a few definitions: Definition 1: A set {Z i i I}, for some index set I, is called a polytopic partition of the polytopic set Z if Z = i I Z i and the sets Z i, i I are polytopes with non-intersecting interiors (relative to Z). Definition 2: A fnction V : Z R is said to be piecewise qadratic on a polytopic partition {Z i i I} of Z if it satisfies (for some Q i,q i,r i, i I) V (z) = V i (z) (1/2)z Q i z + q iz + r i, (3) for all z Z i, all i I. The parametric piecewise qadratic programming problem P(x) (we se the same symbol P(x) for simplicity) is defined by (1) where, now, V ( ) is piecewise qadratic and Z is a polytope. The following assmption is assmed to hold in the seqel: Assmption 1: The fnction V : Z R is continos, strictly convex, and piecewise qadratic on the polytopic partition {Z i i I} of the polytope Z in R n R m. Under this assmption, the piecewise qadratic programming problem P(x) satisfies the hypotheses of Proposition 1 so that its vale fnction ( ) is strictly convex and continos and the minimizer 0 ( ) is continos. Assmption 1 implies that Q i is positive definite for all i I. For each x, let the set U(x) be defined by U(x) { (x,) Z}. (4) Ths U(x) is the set of admissible at x and P(x) may be expressed in the form (x) = min {V (x,) U(x)}. Becase of the piecewise natre of V ( ), we reqire: Definition 3: A polytope Z i in a polytopic partition {Z i i I} of a polytope Z is said to be active at z Z if z = (x,) Z i. The set of polytopes active at z Z is S(z) {i I z Z i }. (5) A polytope Z i in a polytopic partition {Z i i I} of a polytope Z is said to be active for Problem P(x) if (x, 0 (x)) Z i. The set of active polytopes for P(x) is S 0 (x) defined by S 0 (x) S(x, 0 (x)). (6) Ths S 0 (x) = {i I (x, 0 (x)) Z i }. To proceed, and to relate or version of reverse transformation to earlier versions, we define, for each i I, the sbproblem P i (x) employed in these versions and defined by Vi 0 (x) = min{v (x,) (x,) Z i } (7) 0 i(x) = arg min{v (x,) (x,) Z i }. (8) As before, the set of feasible for P i (x) is U i (x) { (x,) Z i }. The qestion arises: how are the soltions to P i (x), i S 0 (x) related to the soltion of the original problem P(x)? This is answered by [16]: Proposition 2: Sppose x X is given. The following two statements are eqivalent: (i) is optimal for the original problem P(x) ( = 0 (x)). (ii) is optimal for problem P i (x) for all i S 0 (x) ( = 0 i (x) for all i S0 (x)). A proof of this reslt appears in [16] bt is inclded here since it is simple and motivates the algorithm given later. Proof: (i) Sppose is optimal for P(x); then, (x,) Z i for all i S 0 (x). If is not optimal for P i (x) for some i S 0 (x), there exists a v sch that (x,v) Z i and V (x,v) = V i (x,v) < V i (x,) = V (x,) = (x), a contradiction. (ii) Sppose is optimal for P i (x) for all i S 0 (x). Then V (x,) = V i (x,) V (x, 0 (x)) = (x) for all i S 0 (x). Bt (x) V (x,) (by the optimality of 0 (x)), so that V (x,) = (x) which establishes the optimality of for P(x). For each i I, Z i may be defined by Z i {M i N i x + p i }. (9) Let M i j, Ni j and pi j denote, respectively the jth row of Mi, N i and p i. Let I i (x,) {j M i j = N i jx + p i j} (10) denote the active constraint set for P i (x) at (x,) Z i. Let F i (x,) denote the set of feasible directions at U i (x) { (x,) Z i }: F i (x,) {h R m M i jh 0 j I i (x,)}, (11) 2763

and let PC i (x,) denote the polar cone of F i (x,) at 0: PC i (x,) {v R m v h 0 h F i (x,)} = (M i j) λ j λj 0 j I i (x,). (12) j I i(x,) Using a well-known condition of optimality employed in [8], [19] in this context we have: Proposition 3: is optimal for problem P i (x) if and only if U i (x) and V i (x,) PC i (x,). B. Improved reverse transformation algorithm Proposition 2 shows that solving, as in the original version of the reverse transformation procedre, Problem P i (x) if polytope Z i is active, does not necessarily yield a soltion of the original problem; it does so if Z i is the sole active polytope. For any x X, 0 (x) (the soltion of P(x)) satisfies the eqality constraint M i j = N i jx + p i j, j I 0 i (x), i S 0 (x) (13) where M i j denotes the jth row of Mi, etc, and I 0 i (x) I i (x, 0 (x)) indexes those constraints in (9) that are active at (x, 0 (x)). For simplicity, we rewrite (13) as Hence 0 (x) is the niqe soltion of E x = F x x + g x. (14) (x) = min {V (x,) E x = F x x + g x }. We define, for all x, x X, the (easy) eqality constrained problem P x ( x) by Vx 0 ( x) = min{v ( x,) E x = F x x + g x } (15) 0 x( x) = arg min{v ( x,) E x = F x x + g x }. (16) where V ( x,) = V i ( x,), i S 0 (x) and is, therefore, qadratic, for all ( x,) satisfying (14). The soltion to this problem is x ( x) = (1/2) x Q x x + s x x + r x (17) 0 x( x) = K x x + k x. (18) where Q x,s x,r x,k x,k x are easily compted. The minimizer 0 x( x) to P x ( x) satisfies (14), the same eqality constraints as those satisfied by 0 (x). We know that 0 x( ) is optimal for P x ( x) and is also optimal for P(x); the next reslt shows that 0 x( ) is also optimal for P( x) for all x in some region R x R n. Proposition 4: Let x be an arbitrary point in X. Then: (i) The soltion to P(x) satisfies 0 ( x) = 0 x( x) and ( x) = Vx 0 ( x) for all x R x defined by: { R x x 0 x( x) U i ( x) i S 0 (x) V i ( x, 0 x( x)) PC 0 i (x) i S0 (x) (19) where PC 0 i (x) PC i(x, 0 (x)), (ii) R x is a polytope, and (iii) x R x. } This proposition is an elementary conseqence of Proposition 3 and corrects an earlier version in [16]. Since there are a finite nmber of polytopes Z i and a finite nmber of constraints defining each polytope, there exists a finite set of points {x j,j J } in X sch that X = j J R xj. Hence, we have Theorem 1: There exists a finite set of points {x j,j J } in X sch that R = {R xj j J } is a polytopic partition of X. The vale fnction ( ) is piecewise qadratic and the minimiser 0 ( ) is piecewise affine in R, being eqal, respectively, to x j ( ) and 0 x j ( ) in each region R xj, j J. Propositions 3 and 4 motivate: Improved Reverse Transformation Algorithm 1. Initialize: Set R =. 2. Update: Select x X \ R, solve P(x). Determine the affine minimizer 0 x( ), the qadratic vale fnction x ( ) and the polytope R x. Set ( ) = x ( ) and 0 ( ) = 0 x( ) in R x. Set R = R x R. 3. Iterate: While R X, iterate. The algorithm generates a seqence R xj, j J of polytopes that form a polytopic partition of X and yields the soltion (both the vale fnction and the minimizer) to the parametric piecewise qadratic problem P(x) for all x X. We can compare the original and improved version of reverse transformation by their application to the example shown in Figre 1. Here Z is the rectangle X U. 0 1( ) 0 2( ) X Z 2 Z 1 x (a) Soltions of P 1 (x) and P 2 (x) Fig. 1. U 0 ( ) X Z 2 Z 1 x (b) Soltion of P(x) Improved reverse transformation algorithm The polytope Z is partitioned into two polytopes Z 1 and Z 2 in each of which V ( ) is qadratic (in z = (x,)). The original version yields 0 1( ) and 0 2( ) shown in Figre 1(a); both 0 1(x) and 0 2(x) exists at each x X and frther investigation is reqired to choose the tre minimizer 0 (x). The improved algorithm, in contrast, yields the niqe minimizer 0 ( ) shown in Figre 1(b). III. DYNAMIC PROGRAMMING SOLUTION OF THE CONSTRAINED LQR PROBLEM This problem was considered in [15] sing an earlier version of reverse transformation. Consider the problem of controlling the linear system x + = Ax + B (20) U 2764

where x and are the crrent state and control, and x + is the sccessor state. Let {(0),(1),...,(N 1)}. We pose an optimal control problem P(x) defined by sbject to the constraints VN(x) 0 = min V N (x,) (21) 0 N(x) = arg min V N (x,) (22) x(i) X,(i) U,i = 0,...,N 1, x(n) X f, (23) where x(i) φ(i;x,) is the soltion at time i of (20) with initial state x at time 0 and control seqence. The sets X, U and X f are assmed to be polytopic. The cost fnction V N ( ) is defined by V N (x,) N 1 i=0 l(x(i),(i)) + V f (x(n)) (24) where l( ) and V f ( ) are continos strictly convex piecewise qadratic fnctions defined, respectively, on polytopic partitions of X U and X f. The complexity of the problem, especially when l( ) is piecewise qadratic, sggests that dynamic programming is more efficient when N is large. Let f(x,) Ax+B. The constrained dynamic programming recrsion is i+1(x) = min {l(x,) + i (f(x,)) U, f(x,) X i }, (25) X i+1 = {x X U sch that with bondary conditions f(x,) X i } (26) 0 ( ) = V f ( ), X 0 = X f (27) It is easily seen that each stage of the dynamic programming recrsion is a parametric piecewise qadratic programming problem (even if l( ) is qadratic) to which the algorithm described above may be seflly applied with ( ) = Vi+1 0 0 ( ), V (x,) = l(x,) + Vi (Ax + B) (which is piecewise qadratic in (x,) since l( ) and Vi 0 ( ) are piecewise qadratic) and Z = {(x,) x X, U,Ax + B X i } (which is polytopic since U, X and X i are polytopic). IV. PARAMETRIC PIECEWISE AFFINE QP The parametric piecewise affine qadratic program P(x) is defined by where (x) = min {V (x,) (x,) Z} (28) V (x,) l(x,) + V f (f(x,)), Z {(x,) Z 0 f(x,) X f }. (29) It is assmed that the cost fnctions l( ) and V f ( ) are strictly convex and qadratic, that Z 0 and X f are polytopic and that f( ) is continos and piecewise affine in a polytopic partition P = {P i i I} of Z 0 so that, for each i I f(x,) = f i (x,) A i x + B i + c i (x,) P i (30) The minimizer for P(x) (see (28)) is 0 (x) and may be set-valed. The domain of ( ) is X {x sch that (x,) Z}. The complicating factor (compared with the piecewise qadratic problem) is f( ) which renders the problem non-convex. For each i I, define the set Z i P i by Z i {(x,) P i f i (x,) X f } (31) Since f i ( ) is affine, each Z i is a polytope and Z = i I Z i is a polygon (nion of a finite set of polyhedra). Also, for each i I, define the cost V i : Z i R by V i (x,) l(x,) + V f (f i (x,)) (x,) Z i. (32) Since f i ( ) is affine, V i ( ) is qadratic and is strictly convex. Ths, for each i I, problem P i (x), defined by i (x) = min {V i (x,) (x,) Z i } (33) is a qadratic program with V i ( ) strictly convex so that, for each x Proj X Z i, P i (x) has a niqe global minimizer 0 i (x). Let U i (x) { (x,) Z i }. (34) The domain of i ( ) and 0 i ( ) is X i = {x U i (x) }. Necessary and sfficient conditions for the optimality of for P i (x) are U i (x) and V i (x,) PC i (x,) where, now, PC i (x,) is the polar cone of the set of feasible directions for Z i at 0. Since V (x,) l(x,)+v f (f(x,)), we have V (x,) = V i (x,) for all (x,) Z i. Hence V ( ) is piecewise qadratic and continos bt not continosly differentiable; V ( ) is not necessarily convex so that, for each x, the global minimizer is not necessarily niqe, and V ( ) may have several local minima thogh each local minimizer of P(x) is a global minimizer of P i (x) for some i I. Also let S(x,) be defined as in (5). Becase V ( ) is not necessarily convex, 0 (x) may be set-valed and a local minimm is not necessarily global; however 0 (x) { 0 i (x) i I} and has finite cardinality. Hence Proposition 2 needs modification. We have, instead Proposition 5: Let x X be given. If 0 (x) is optimal for P(x), then is optimal for P i (x) for all i S(x,). The proof of this reslt is similar to the proof of part (i) of Proposition 2. Similarly to (13), any x X, any 0 (x) satisfies M i j = N i jx + p i j, j I i (x,), i S(x,) which may be written as E (x,) = F (x,) x + g (x,). For all x X, all 0 (x), we define as before, for all 2765

x X, the eqality constrained problem P (x,) ( x) by V(x,) 0 ( x) = min {V ( x,v) E (x,)v = F v (x,) x + g (x,) } (35) 0 (x,)( x) = arg min {V ( x,v) E (x,)v = F v (x,) x + g (x,) }. (36) As before, V(x,) 0 ( ) is qadratic and 0 (x,)( ) is affine and niqe (as in (15) and (16)) and are easily compted. For each x X, each 0 (x), let the region R (x,) be defined as before: R (x,) { x 0 (x,) ( x) U i( x), V i ( x, 0 (x,) ( x)) PC i (x,) i S(x,)} (37) where PC i (x,) is defined as in (9) (12). As before, R (x,) is a polytope and x R (x,). For each x R (x,), each ū 0 ( x), ( x,ū) Z i and ū minimizes V i ( x,) in U i ( x) for all i S( x,ū). Becase 0 (x,)( ) is now a local, rather than a global, minimizer, it is possible for the regions R (x,) to overlap. This is illstrated in Figre 2 where P 1 and P 2 are, respectively, the polytopes above and below the diagonal and Z 0 = P 1 P 2. The polytopes Z 1 and Z 2, which are, respectively, sbsets of P 1 and P 2, are disjoint. The sets R (x1, 1) where 1 0 (x 1 ) and R (x2, 2) where 2 0 (x 2 ) overlap. Consider a point x lying in, say, an overlap region R = j K R (xj, j) for some index set K. Then each 0 (x j, j) ( x), j K, is a local minimizer and at least one is a global minimizer for P( x). Hence, if x R, 0 (x j, j ) ( x) 0 ( x) for each j arg min j {V ( x, 0 j ( x)) j K} and we can no longer claim that that the vale fnction ( ) for P(x) is piecewise qadratic and that the minimizer 0 ( ) is piecewise affine on a polytopic partition of Z since the bondaries between the sb-regions of R may be parabolic. Smmarizing: 1 2 Z 1 P 1 P 2 Z 2 x 1 x 2 R (x1, 1) R (x2, 2) R Z 0 Fig. 2. Overlapping regions: R = R (x1, 1 ) R (x2, 2 ) Proposition 6: The set of states for which P(x) has a soltion is a polygon (nion of a finite set of polyhedra). There exists a finite set of points {(x j, j ) j J } sch that X = j J R (xj, j). If x j K R (xj, j) for some K J, then ( x) = V ( x, 0 j ( x)) and 0 j ( x) 0 ( x) for any j arg min j {V ( x, 0 j ( x)) j K}. Despite the non-convexity of V ( ) and the non-polytopic x natre of Z we can employ a modification of the algorithm, described in 2, to obtain a set of polytopes of the form R (x,), each lying in X = Proj X Z. Becase of nonconvexity, regions R (xj, j), j J, may overlap. Also, becase points in R merely satisfy, in general, a necessary condition of optimality, there may exist points x in R k = j Jk R (xj, j) (at the kth iteration of the algorithm) sch that (x) < min j {V(x 0 j, j ) (x) j J k}. Hence the sccessor point x in Step 2 of the algorithm shold not necessarily be soght in X \ R k otherwise these points may be overlooked. However, the nmber of overlaps is considerably redced with the improved algorithm becase false minimizers lying on internal bondaries separating one polytope from another are avoided. See Figre 3 where V (x,) is plotted for a given vale x of the parameter. The original version identifies a, b and c (twice) as local minimizers whereas the improved version identifies a and c. The reslts of this section still hold if l( ) and V f ( ) V (x,) a b c Fig. 3. Non-convex problem are piecewise qadratic provided P is appropriately sbpartitioned. V. OPTIMAL CONTROL OF A PIECEWISE AFFINE SYSTEM WITH QUADRATIC COSTS Consider the problem of controlling the piecewise affine system described by sbject to the constraints x + = f(x,) (38) (x(i),(i)) Z 0, i = 0,1,...,N 1, x(n) X f (39) where x(i) φ(i;x,), Z 0 and X f are polytopic and f( ) is continos and piecewise affine satisfying: f(x,) = A j x + B j + c j (x,) P j (40) where P = {P i i I} is a polytopic partition of Z. The cost is, as before, V (x,) = N 1 i=0 l(x(i),(i)) + V f (x(n)) (41) where N is the horizon, denotes the control seqence {(0),(1),...,(N 1)} and x(i) = φ(i;x,); l( ) and V f ( ) are assmed, for simplicity, to be qadratic. For each s {s(0),s(1),...,s(n 1)} S I N we define Z s {(x,) (φ(i;x,),(i)) P s(i), i = 0,...,N 1; φ(n;x,) X f }. (42) 2766

and define Z to be the polygon s S Z s. We now observe that, if (x,) Z s, then the (nonlinear) piecewise affine system behaves like the time-varying system x(t + 1) = A s(t) x(t) + B s(t) (t) + c s(t) (43) so that V ( ) is qadratic (V (x,) = V s (x,) where V s ( ) is qadratic) for all (x,) Z s. Ths, for each s S, problem P s, defined by s (x) = min {V s (x,) (x,) Z s } (44) is a qadratic program. It can now be seen that the piecewise affine optimal control problem is identical to the problem discssed in 4 with replacing and Z s, s S replacing Z i, i I. Hence the reslts obtained in 4 apply with obvios modifications to the piecewise affine optimal control problem considered here. VI. CONCLUSION A procedre for parametric piecewise qadratic programing that is an improvement on earlier versions of reverse transformation is described. 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