Identification of the Symmetries of Linear Systems with Polytopic Constraints

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1 2014 American Control Conference (ACC) Jne 4-6, Portland, Oregon, USA Identification of the Symmetries of Linear Systems with Polytopic Constraints Clas Danielson and Francesco Borrelli Abstract In this paper we consider the problem of finding all the state-space and inpt-space transformations that preserve the parameters of a constrained linear system. Sch transformations are called symmetries. For systems constrained by bonded polytopes the set of all symmetries is a finite grop and reqires techniqes from discrete mathematics to find. We transform the problem of finding the symmetries of a constrained linear system into the problem of finding the symmetries of a vertex colored graph. The symmetries of a vertex colored graph can be fond efficiently sing graph atomorphism software. We demonstrate or symmetry identification procedre on a qadcopter example. I. INTRODUCTION This paper presents a procedre for identifying the symmetries of linear systems with bonded polytopic constraints. Symmetries of linear systems are state-space and inpt-space transformation the preserve the system parameters. In particlar the state-space matrices and constraint sets are mapped to themselves. Symmetry has been sed extensively in nmeros fields to redce comptational complexity. In recent years symmetry has been applied to optimization to solve linear-programs 3, semi-definite programs 12, and integer-programs 4. In 5 symmetry was sed to find the transition probabilities for Markov-Chains that prodce the fastest convergence. In 8 symmetry was exploited to redce the comptational complexity of H 2 and H control design. In 17 symmetry was sed to simplify the analysis and control design for large-scale distribted parameter system. In previos work the systems considered were nconstrained. For nconstrained systems linear algebra techniqes can be sed to find the system symmetries. However when the system is constrained the problem is more difficlt. In this case discrete-mathematics techniqes mst be sed to find the symmetries. In this paper we consider systems with bonded polytopic constraints. We show how the problem of finding the symmetries of the constrained system can be transformed into the problem of finding the symmetries of a vertex-colored graph. Graph symmetries can be efficiently fond sing graph atomorphism software 16, 11, 14. We begin this paper by motivating or work with an application to model predictive control. In Section II we formally define the symmetry identification problem for constrained systems and compare with the nconstrained case. In Section III we show that the symmetry identification problem is eqivalent to finding the set of permtation matrices that commte with a specifically designed set of matrices. In Section IV we show how the set of all permtation matrices that commte with a set of matrices can be identified sing graph theory. Finally in Section V we smmarize or reslts and present an example to demonstrate or software. A. Motivation: Symmetric Explicit Model Predictive Control In 9 we presented a method for exploiting the symmetries of a constrained dynamic system to redce the memory reqirement for explicit model predictive control. In this section we briefly smmarize or reslts from that paper. This research was spported by the National Science Fondation nder Grant n clas.danielson@me.berkeley.ed Consider the following model predictive control problem for a constrained linear time-invariant system with qadratic cost N minimize x T 0,..., N Px N + x T k Qx k + T k R k N k0 sbject to: x k Ax k + B k x k X, k U for k 0,..., N (1a) (1b) (1c) where x 0 x(t) R nx is the measred state, x k R nx is the predicted state nder control action k R n over the horizon N, X and U are the polytopic state and inpt constraint sets, and P P T 0, Q Q T 0, R R T 0. In 1 it was shown that (1) can be solved parametrically to prodce a continos piecewise affine on polytope controller F 1 x(t) + G 1 for x(t) R 1 0 κ(x(t)). (2) F px(t) + G p for x(t) R p where R i X are polytopes called critical regions, F i R n nx are the feedback matrices and G i R n are the feedforward vectors. The triple (F i, G i, R i) is called the i-th controller piece and I 1,..., p} is the set of all controller pieces. A symmetry of the explicit model predictive controller (2) is a state-space Θ R nx nx and inpt-space Ω R n n transformation that permtes the controller pieces I. The transformation pair (Θ, Ω) preserves the control law Ω κ(θx) κ(x). Definition 1: A pair of invertible matrices (Θ, Ω) is a symmetry of (2) if for every i I there exists a j I sch that ΩF i Θ F j (3a) ΩG i G j (3b) Θ R i R j. (3c) The set of all symmetries of the controller (2) form a grop called the symmetry grop At(κ) of the controller. In 9 we showed how controller symmetry can be sed to redce the storage reqirements for explicit model predictive control by eliminating symmetrically redndant controller pieces. In addition we proved that the controller (2) is symmetry with respect to the intersection of the symmetry grops of the cost fnction (1a), dynamics (1b), and constraints (1c), of the model predictive control problem. In this paper we show how to find the symmetry grops of the dynamics and constraints. Moreover we provide methods for both finding the symmetry of the cost fnction (1a) or for modifying the cost fnction (1a) so that it is symmetric with respect to the symmetry grop of the dynamics (1b), and constraints (1c). B. Mathematical Backgrond A half-space is the set x R n : hx k} where h T R n and k R. A polytope P R n is the intersection of a finite nmber of half-spaces P x R n : h ix k i, i 1,..., m} where m is the nmber of half-spaces. The half-spaces parameters h i and k i for i 1,..., m can be collected in the matrix H R m n and vector K R m. A half-space is redndant if removing it from the description of P does not change the set. If the polytope P /$ AACC 4218

2 contains the origin in its interior then it can be normalized P x R n : Hx 1} where K 1 R m is the vector of ones. Let M R n n and P R n then M P Mx : x P}. Let N 1,..., n} N be a finite set. A permtation σ is a bijective fnction from N to itself. Permtations will be denoted sing the Cayley notation. For instance the permtation σ on N 1, 2, 3} sch that σ(1) 2, σ(2) 3, and σ(3) 1 will be denoted σ. (4) A permtation matrix P R n n is a sqare binary matrix that has exact one 1 in each row and colmn. Each permtation σ corresponds to a permtation matrix P R n n given by 1 if i σ(j) P ij (5) 0 otherwise. A grop (G, ) is a set G along with a binary operator sch that the operator is associative, the set G is closed nder the operation, the set G incldes an identity element e G, and the inverse g G of each element g G. A grop is called trivial if it contains only the identity element e. A grop is called finite if it has a finite nmber of elements G. A set S G is called a generating set if every element of G can be expressed as a prodct of the elements of S. The grop generated by the set S is denoted by S. Let G and H be grops. A grop homomorphism f : G H is a fnction from G to H satisfying f(g h) f(g) f(h) for all g, h G. A grop isomorphism is a bijective grop homomorphism. Grops G and H are isomorphic if there exists an isomorphism between them. Let S G be a generating for G and f a grop isomorphism from G to H then f(s) is a generating set for H. In this paper we consider two types of grops; matrix grops and permtation grops. In matrix grops the elements of G are matrices. With abse of terminology the elements of a permtation grop G can be permtations or the corresponding permtation matrices. An ndirected graph Γ (N, E) is a set of vertices N together with a list of nordered pairs E N N called edges. A graph coloring C : N C assigns a color C(i) C to each vertex i N of the graph. A graph symmetry is a permtation σ of the vertex set N that preserves the edge set σ(e) E and vertex coloring C(σ(i)) C(i) where σ(i, j}) σ(i), σ(j)}. II. PROBLEM STATEMENT We consider a constrained dynamic system Σ modeled by x(t + 1) Ax(t) + B(t) x(t) X, (t) U (6a) (6b) where A R nx nx is the dynamics matrix, B R nx is the inpt matrix, X R nx is the state constraint set and U R n is the inpt constraint set. The constraint sets are fll-dimensional, bonded polytopes containing the origin in their interior Definition 2: The symmetry grop At(A, B) of the dynamics (6a) is the set of all pairs of invertible matrices (Θ, Ω) satisfying ΘA AΘ (8a) ΘB BΩ. (8b) Definition 3: The symmetry grop At(X, U) of the constraints (6b) is the set of all pairs of invertible matrices (Θ, Ω) satisfying Θ X X (9a) Ω U U. (9b) The symmetry grop At(Σ) At(A, B) At(X, U) of the constrained dynamic system (6) is the intersection of the symmetry grops of the dynamics At(A, B) and constraints At(X, U). The symmetry grop At(Σ) is the set of all transformations of the state-space Θ and inpt-space Ω that preserve the system parameters A, B, X, and U. The symmetry identification problem is the problem of finding a set of generators for the symmetry grop At(Σ). We formally define this problem below. Problem 1 (Symmetry Identification): Let At(Σ) be the set of all pairs of matrices (Θ, Ω) satisfying (8) and (9). Find a set Gen(At(Σ)) that generates the symmetry grop At(Σ). The related problem of finding the symmetry grop At(A, B) for the nconstrained linear system (6a) is mch simpler since the closre of At(A, B) is a vector-space 13. On the other hand the symmetry grop At(X, U), and therefore At(Σ) At(X, U), is typically discrete. Therefore we cannot se linear algebra to find the symmetry grop At(Σ). In this case methods from discretemathematics mst be sed to find the symmetry grop At(Σ). Proposition 1: Let X and U be fll-dimensional and bonded sets. Then At(Σ) is a finite grop. Proof: See 10 In the case where the symmetry grop At(Σ) is finite we se discrete mathematics techniqes to find the grop At(Σ). III. SYMMETRY IDENTIFICATION In this section we present or methodology for solving Problem 1 (Symmetry Identification). The symmetry grop At(Σ) of the constrained dynamic system (6) is a sbgrop of the symmetry grop At(X, U) of the constraint sets X and U. For bonded sets X and U the symmetry grop At(X, U) is isomorphic to a permtation grop acting on the non-redndant half-spaces of X and U. In other words Θ and Ω permte the facets of X and U respectively. We will denote this permtation grop by Perm(X, U). The symmetries (Θ, Ω) At(X, U) and (P x, P ) Perm(X, U) are related by the expression Hx 0 Hx 0 Θ 0 (10) 0 H 0 H 0 Ω where the rows of H x and H are the half-spaces of X and U respectively, and P x R mx mx and P R m m. Taking the psedo-inverse of H x and H we can express Θ and Ω in terms of P x and P by X x R nx : H xx 1} U R n : H 1} (7a) (7b) Θ ( H T x Hx ) H T x P xh x Ω ( H T H ) H T P H (11a) (11b) where H x R mx nx and H R m n. We assme that the half-spaces that define X and U are not redndant. The symmetry grop of a constrained dynamic system is the intersection of the symmetry grop of the dynamics and constraints. These symmetry grops are defined below. where the matrices H T x H x and H T H are invertible since X and U are bonded sets. In 7 it was shown that (11) is a grop isomorphism from Perm(X, U) to At(X, U). Therefore the problem of identify the symmetry grop At(X, U) is eqivalent to finding the grop of permtation matrices Perm(X, U). 4219

3 In 7 it was shown that the permtation grop Perm(X, U) is the set of all matrices P x and P that satisfy the commtative relation Hx(H T x Hx) H T x 0 0 H (H T H ) H T Hx(H T x H x) H T x 0 0 H (H T H) H T (12) In Section IV we will describe the procedre for finding all permtation matrices P x and P that commte with some matrix. For the matrices (Θ, Ω) At(X, U) to be elements of the symmetry grop At(Σ) At(X, U) they mst also map the dynamics matrices A and B to themselves. This sggests that we can constrct At(Σ) by selecting the elements of At(X, U) that satisfy (8). However this is impractical since the grop At(X, U) can be very large At(X, U) O(m x!m!). For comptational tractability we need to restrict orselves sing the generators Gen(At(X, U)) of the grop At(X, U) which have bonded cardinality Gen(At(X, U)) (m x)(m ). Unfortnately the grop At(Σ) At(X, U) At(A, B) can be non-trivial while none of the generators (Θ, Ω) Gen(At(X, U)) satisfy (8). This is demonstrated by the following example. Example 1: Consider the constrained atonomos system x(t + 1) Ax(t), x(t) X (13) where X is the sqare X x R 2 : Hx 1} with H T I, I, and A (14) The symmetry grop At(X ) of the sqare X is the dihedral-4 grop; the eight element set of all rotations by increments of 90 and the reflections abot the horizontal, vertical, and both diagonal axes. The grop At(X ) can be generated by the horizontal and diagonal reflections Θ and Θ (15) The symmetry grop At(Σ) contains for elements; the rotations by increments of 90 degrees. However neither of the generators Θ 1 or Θ 2 commte with the matrix A. Or method for finding At(Σ) is similar to the method for finding At(X, U). We find a permtation grop Perm(Σ) isomorphic to At(Σ). The generators of At(Σ) are then the image of the generators of Perm(Σ) mapped throgh the isomorphism (11). Since At(Σ) At(X, U) is a sbgrop of At(X, U) it is also isomorphic to a permtation grop Perm(Σ) Perm(X, U). If we combine the condition (8) that Θ and Ω commte with A and B with eqation (10) which relates (P x, P ) Perm(X, U) and (Θ, Ω) At(X, U) we obtain HxA H xb HxA H xb Θ 0 0 Ω. (16) This expression allows s to characterize Perm(Σ) as the set of all permtation matrices (P x, P ) Perm(X, U) that satsify the commtative relation (17) HxA(H T x Hx) H T x H xb(h T x Hx) H T x HxA(H T x Hx) H T x H xb(h T x Hx) H T x. In Section IV we will describe the procedre for finding the permtation grop Perm(Σ). We now present the main theorem of this paper that relates the permtation grop Perm(Σ) to the symmetry grop At(Σ). Theorem 1: Let Perm(Σ) be the set of all permtation matrices P x R mx mx and P R m m that satisfy (12) and (17). Then the grop Perm(Σ) is isomorphic the symmetry grop At(Σ) of the constrained dynamic system (6) with isomorphism (11). Proof: See 10 Or procedre for identifying the symmetry grop At(Σ) (Problem 1) can be smmarized by Procedre 1. In the next section we will describe the procedre for find the grop Perm(Σ). Procedre 1 Calclation of At(Σ) 1: Find a generating set Gen(Perm(Σ)) for the grop Perm(Σ) 2: Calclate the generators Gen(At(Σ)) for the symmetry grop At(Σ) from Gen(Perm(Σ)) sing the isomorphism (11) A. Symmetry of the Cost The symmetry grop At(MPC) of the model predictive control problem (1) is the intersection of the symmetry grops of the cost (1a), dynamics (1b), and constraints (1c). So far we have shown how to find the symmetry grop At(Σ) At(MPC) of the dynamics and constraints. There are two ways we can address the symmetry of the cost fnction; we can find the symmetries At(J) of the cost fnction or we can modify the cost fnction so that it is symmetric with respect to the symmetries At(Σ) of the dynamics and constraints. Definition 4: The symmetry grop At(J) of the qadratic cost fnction (1a) is the set of all pairs of invertible matrices (Θ, Ω) satisfying Θ T PΘ P Θ T QΘ Q (18a) (18b) Ω T RΩ R. (18c) The symmetry grop At(MPC) At(Σ) At(J) can be fond by finding the set of all permtations (P x, P ) Perm(Σ) that satisfy the commtative relations Px 0 HxQ H T x 0 HxQ 0 HR H H T T x 0 Px 0 0 HR H T (19) and Px 0 HxP H T x 0 P 0 HxP H T x 0 Px 0. (20) where H x H x(h T x H x) and H H (H T H ). Corollary 1: Let Perm(MPC) be the set of all permtation matrices P x and P that satisfy (12), (17), (19), and (20). Then the grop Perm(MPC) is isomorphic to the symmetry grop At(MPC) At(J) At(Σ) of the model predictive control problem (1) with isomorphism (11). Alternatively we can modify the cost fnction (1a) so that it shares the symmetries of the system At(Σ). The cost (1a) can be made symmetric with respect to At(Σ) by taking its barycenter N J(X, U) x(n) T Px(N) + k0 x(k) T Qx(k) + (k) T R(k) (21) where U (k)} N k0 is the inpt trajectory, X x(k)}n k0 is the state trajectory, P 1 G Θ G ΘT PΘ (22) and likewise for Q and R, and G At(Σ). 4220

4 Proposition 2: The cost fnction (21) is symmetric with respect to the symmetries At(Σ) of the dynamics and constraints. Proof: Follows directly from the closre of grops 5, 4 i.e. (Θ, Ω) At(Σ) At(Σ) for all (Θ, Ω) At(Σ). IV. MATRIX PERMUTATION GROUPS In this section we present the procedre for finding the set of all block strctred permtation matrices that commte with a set of matrices M 1,..., M r. We reformlate this problem as a graph symmetry problem which can be efficiently solved sing software packages sch Naty 16, Sacy 11, and Bliss 14. This procedre can be sed to find the generators of the permtation grops Perm(Σ), Perm(X, U) and Perm(MPC) which are sed to find the generators of the symmetry grops At(Σ), At(X, U) and At(MPC) respectively. A. Permtations that Commte with a Single Matrix We begin by presenting the procedre for finding the set of all permtation matrices that commte with a single matrix M. This problem was first addressed in 16. Or method is taken from 4. The grop Perm(M) is defined as the set of all permtation matrices that commte with a matrix M R m m PM MP. (23) We find the grop Perm(M) by constrcting a graph Γ m(m) whose symmetry grop At(Γ m(m)) is isomorphic to the permtation grop Perm(M). The generators of At(Γ m(m)) can then mapped to generators for Perm(M). Procedre 2 prodces a vertex colored graph Γ m(m) whose symmetry grop At(Γ m(m)) is isomorphic to Perm(M). The generators of the grop At(Γ m(m)) are fond sing graph atomorphism software 16, 11, 14. The graph Γ m(m) contains a vertex for each row r i for i 1,..., m, each colmn c j for j 1,..., m, and each element e ij for i, j 1,..., m of the matrix M. For each niqe vale s M ij : i, j 1,..., m} of the matrix we create a vertex s. The rows, colmns, and element vertices are different colors. Each vale vertex s M ij : i, j 1,..., m} has a different color. The graph Γ m(m) contains edges that connect each element vertex e ij to the row r i and colomn c j vertices and the vertex for the vale s M ij. If M ij M kl then the element vertices e ij and e kl are both connected to the same vale vertex s M ij : i, j 1,..., m}. The graph Γ m(m) (N, E) has N m 2 +2m+m s vertices where m s M ij : i, j 1,..., m} is the nmber of niqe vales in the matrix M. There are 3 + m s vertex colors. The graph Γ m(m) has E 3m 2 edges; each element vertex e ij is neighbors with the row vertex r i, colmn vertex c j, and the vale s M ij vertex. The graph atomorphism software searches for permtations σ At(Γ m(m)) of the vertex set N the preserves the edge list E and vertex coloring. The graph atomorphism software prodces a set Gen(At(Γ m(m))) of at most m 1 permtations that generate the symmetry grop At(Γ m(m)). The graph Γ m(m) is constrcted so that the symmetries σ At(Γ m(m)) have a particlar strctre. The graph atomorphism software will not find permtations σ that transpose row vertices with non-row vertices since they have different colors. Therefore each σ At(Γ m(m)) will map row vertices r i to row vertices. Likewise each σ At(Γ m(m)) maps colmn vertices c j to colmn vertices and element vertices e ij to element vertices. The vale vertices s M ij : i, j 1,..., m} can only be mapped to themselves since each has a different color. Let σ m denote the restriction of the permtations σ At(Γ m(m)) to the set of row vertices N r r 1,..., r m} N. The edges of the graph Γ m(m) are constrcted sch that σ m is also the restriction of σ At(Γ m(m)) to the colmn vertices. The element vertices e ij are permted according to the expression σ(e ij) e σm(i)σm(j). And the vale vertices s M ij : i, j 1,..., n} are not permted by any σ At(Γ m(m)). Therefore each row permtation σ m niqely defines the permtation σ At(Γ m(m)). Ths withot loss of generality we can define the following isomorphism from At(Γ m(m)) to Perm(M) by 1 for r j σ m(r i) P ij (24) 0 otherwise where σ m At(Γ m(m)). This reslt is formally stated in the following theorem. Theorem 2: The symmetry grop At(Γ m(m)) of the vertex colored graph Γ m(m) is isomorphic to the permtation grop Perm(M) of the sqare matrix M with isomorphism (24). Proof: See 4. This procedre is demonstrated by the following example. Example 2: The graph Γ 2(I) for the 2 2 identity matrix I R 2 2 is shown in Figre 1. The green diamonds represent the row vertices r 1 and r 2 and the ble sqares represent the colmn vertices c 1 and c 2. The white circles represent the matrix element vertices e ij. The prple and yellow hexagons represent the niqe matrix vales I 1,1 I 2,2 1 and I 1,2 I 2,1 0 respectively. The graph atomorphism software will search for permtations σ of the vertex set N that preserve the edge list E and vertex coloring. For instance the permtation r1 r σ 2 c 1 c 2 e 11 e 12 e 21 e 22 s 1 s 0 (25) c 1 r 2 r 1 c 2 e 11 e 12 e 21 e 22 s 1 s 0 is not a symmetry of Γ m(m) since it transposes vertices of different colors; row vertex r 1 and colmn vertex c 1. The permtation r1 r σ 2 c 1 c 2 e 11 e 12 e 21 e 22 s 1 s 0 (26) r 1 r 2 c 2 c 1 e 11 e 12 e 21 e 22 s 1 s 0 is not a symmetry of Γ m(m) since it changes the edge list; c 2, e 22} E bt σ(c 2, e 22}) c 1, e 22} E. However the permtation r1 r σ 2 c 1 c 2 e 11 e 12 e 21 e 22 s 1 s 0 (27) r 2 r 1 c 2 c 1 e 22 e 21 e 12 e 11 s 1 s 0 and the identity permtation are elements of At(Γ m(m)) since they preserve the edge list E and vertex coloring. In fact these are the only of the 10! possible permtations of N that preserve the edge list E and vertex colors. Using the isomorphism (24) we find that the permtation grop of the matrix M I is Perm(I) , } (28) Alternative graph constrctions Γ m(m) can be fond in the literatre 6, 16. Using Procedre 2 we can find the permtation grop Perm(M) for matrices M R m m in thosand of dimensions O(m) This means we can find the symmetries of systems with thosands of constraints. 4221

5 Fig. 1. Graph Γ 2 (I) for the 2 2 identity matrix I. B. Permtations that Commte with Mltiple Matrices In this section we present the procedre for finding the set of all permtation matrices that commte with mltiple matrices M 1,..., M r. This reslt is needed to calclated Perm(Σ) and Perm(MPC). For notational compactness we will often denote M 1,..., M r by M k } r k1 The obvios way to calclate the set of all permtation matrices that commte with mltiple matrices M 1,..., M r is to calclate the grops Perm(M 1),..., Perm(M r) and then compte their intersection. However this method is impractical since the grops Perm(M i) can have exponential size in m. Calclating the intersection of grops sing the generators is non-trivial since the intersection of the grop generators r i1 Gen(Perm(Mi)) does not necessarily generate of the intersection of the grops r i1 Perm(Mi). To calclate the grop Perm(M k } r k1) r k1 Perm(M k ) we constrct a matrix M that captres the symmetry strctre of the set of matrices M k } r k1. The matrix M has a distinct vale M ij for each niqe r-tple (M ij1,..., M ijr). In other words M i1 j 1 M i2 j 2 if and only if M i1 j 1 k M i2 j 2 k for k 1,..., r. The following corollary shows that Perm(M) r k1 Perm(M k ). Corollary 2: Let M satisfy M i1 j 1 M i2 j 2 if and only if M i1 j 1 k M i2 j 2 k for k 1,..., r. Then Perm(M) r k1 Perm(M k). Proof: If P commtes PM k M k P with each matrix M k for k 1,..., r then is mst commte with M and conversely. Ths we can se Procedre 2 to find generators of the grop Perm(M k } r k1) r k1 Perm(M k ). This is demonstrated in the following example. Example 3: Consider the atonomos system (13) from Example 1. In order to find the symmetry grop At(Σ) we mst find the grop Perm(Σ) of all permtation matrices P that commte with the matrices M 1 H(H T H) H T and M 2 HA(H T H) H T M where and M (29) The are for niqe pairs (M ij2, M ij2) of vales for these two matrices (M ij2, M ij2) ( - 1, 0), ( 1, 0), } 0, , 0, 1 2. (30) Let s 1, s 2, s 3, and s 4 be arbitrary distinct vales. Define the matrix M R 4 4 sch that s 1 if (M ij2, M ij2) ( 1 2, 0) s M ij 2 if (M ij2, M ij2) ( 0, - 1 ) 2 s 3 if (M ij2, M ij2) ( - 1 2, 0) (31) s 4 if (M ij2, M ij2) ( 0, 1 ) 2. Then we get the matrix s1 s 2 s 3 s 4 M. (32) s 4 s 1 s 2 s 3 s 3 s 4 s 1 s 2 s 2 s 3 s 4 s 1 The grop Perm(M) of permtation matrices that commte with the matrix M is identical to the grop Perm(M 1, M 2) of permtation matrices that commte with M 1 and M 2. The matrix M is a circlant 4 4 matrix. Ths its symmetry grop is the cyclic-4 grop C 4. Indeed sing graph atomorphism software to identify the symmetries of Γ(M) we find At(Γ(M)) σ m is generated by the 4-cycle r1 r 2 r 3 r 4 σ m. (33) r 4 r 1 r 2 r 3 Using isomorphism (24) we find that Perm(M) Perm(M 1, M 2) P is generated by the single permtation matrix 0 1 P (34) Using isomorphism (11) from Perm(Σ) to At(Σ) we find that At(Σ) Θ is generated by 0 1 Θ (35) 0 where Θ is the 90 degree rotation matrix. Ths At(Σ) Θ is the set of all rotations of the state-space by increments of 90 degrees. C. Block Strctred Permtation Matrices The permtation matrices in the permtation grops Perm(X, U), Perm(Σ), and Perm(MPC) have a block strctre: The state-space and inpt-state constraints are permted independently P R m m. (36) where P x R mx mx and P R m m We can encode this desired block strctre for the permtation matrix P into the graph Γ(M). We color the first β m x nodes of the graph differently then the last m nodes. The graph atomorphism software only retrns permtations σ m that do not switch vertices of different colors. Ths by isomorphism (24) the permtation matrix P will have the desired block strctre. Procedre 2 Create graph Γ β (M) whose symmetry grop At(Γ β (M)) is isomorphic to the permtation grop Perm β (M). 1: Inpt: Sqare matrix M and block dimension β 2: Otpt: Vertex colored graph Γ β (M) (N, E) 3: Constrct the vertex set N : a. Add a vertex r k i for i 1,..., n for each row with color R 1 for first β rows and R 2 for remaining rows. b. Add a vertex c k j for j 1,..., n for each row with color C 1 for first β rows and C 2 for remaining rows. c. Add a vertex e ij for i, j 1,..., n for each element. d. Add a vertex s for each niqe matrix element M ij. 4: Constrct the edge set E: a. Connect each row vertex r i with the element vertex e ij. b. Connect each colmn vertex c j with the element vertex e ij. c. Connect each element vertex e ij with vale vertex s. V. SYMMETRY IDENTIFICATION SOFTWARE We have implement or symmetry identification procedre in Matlab. The symmetry identification software reqires as an inpt the dynamics matrices A and B, and the sets X and U to calclate the symmetry grop At(Σ) of the constrained dynamics system 4222

6 4 3 Fig. 2. z y x 2 Qadcopter layot. (6). The sets X and U can be provided as normalized half-space matrices H x and H or as mpt-toolbox 15 polytope objects. In addition the cost matrices P, Q, and R are optional inpts for calclating the symmetry grop At(MPC) of the model predictive control problem (1). Or symmetry identification software calclates the matrices Hx(H M 1 T x H x) H T x 0 (37) M H (H T H) H T HxA(H T x Hx) H T x H xb(h T H) H T. (38) In addition if the optional inpts P, Q, and R are inclded the software will also calclate the matrices HxQ M 3 H T x 0 HxP 0 HR H T and M 4 H T x 0. (39) The symmetry identification software prodces a m x + m by m x + m matrix M with distinct vales M ij for each niqe pair (M ij1, M ij2) or each niqe 4-tple (M ij1, M ij2, M ij3, M ij4) when the optional inpts P, Q, and R are inclded. The symmetry identification software ses Procedre 2 to prodce the graph Γ mx (M). The symmetry of the graph Γ mx (M) are identified sing Sacy v Sacy can find the symmetry grop of graphs with millions of vertices in seconds 11. For or application this means we can analyze constrained dynamics systems with thosands of constraints m x and m. The symmetrys σ At(Γ mx (M 1, M 2)) are transformed into permtation matrices P x and P sing (24) to prodce the grop Perm(Σ) Perm mx (M 1, M 2) (respectively Perm(MPC) Perm mx (M 1, M 2, M 3, M 4)). Finally the symmetry identification software applies isomorphism (11) to prodce the symmetry grop At(Σ) (respectively At(MPC)) of the dynamic system (6) (respectively the model predictive control problem (1)). A. Example: Qadcopter We applied or symmetry identification software to the qadcopter system described in 2 and shown in Figre 2. The qadcopter has 13 states; 6 cartesian position and velocity states, 6 anglar position and velocity states, and 1 integrator state to conter-act gravity. The qadcopter has 4 inpts corresponding to the voltage applied to the for motors. The symmetry identification software fond 16 symmetries with 3 generators. The generators of the state-space transformations are } Θ 1 blkdiag I 2, I 2, (40a) Θ 2 blkdiag I 2 Θ 3 blkdiag I 2, I 2, I 2 }, }, (40b) (40c) where is the Kronecker prodct and blkdiag represents a blockdiagonal matrix. The generators of the inpt-space transformations are Ω , Ω , Ω (41a) 1 0 The generator pair (Θ 1, Ω 1) flips the qadcopter abot the x- axis. The inpt-space transformation Ω 1 swaps motors 1 and 3 and reverse the direction of thrst. The state-space transformation Θ 1 reverses the y- and z-axes and reverses rotations abot the x- and z-axes. The generator pair (Θ 2, Ω 2) rotates the qadcopter 90 degrees abot the z-axis. The inpt-space transformation Ω 2 maps motor 1 to 2, motor 2 to 3, motor 3 to 4, and motor 4 to 1. The state-space transformation Θ 2 maps the x-axis to the y-axis and the y-axis to the negative x-axis. Rotations abot the x- and y axis are swapped and rotations abot the x-axis are reversed. The generator pair (Θ 3, Ω 3) flip the qadcopter abot a diagonal axis. The inpt-space transformation Ω 3 swaps motors 1 and 4, and 2 and 3. The state-space transformation Θ 2 swaps and reverses the x- and y-axes. Rotations abot these axes are also swapped and reversed. We generated an explicit model predictive control for the qadcopter system sing mpt-toolbox 15. Withot exploiting symmetry the explicit MPC had 10, 173 regions and reqired 53.7 megabytes of storage. By exploiting symmetry the nmber of region is redced to 772 regions and 4.2 megabytes of storage. Frthermore the comptational complexity of implementing the controller has not be increased 10. REFERENCES 1 A. Bemporad, M. Morari, V. Da, and E. Pistikopolos. The explicit linear qadratic reglator for constrained systems. 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