Reachability and Control of Affine Hypersurface Systems on Polytopes

Reachability and Control of Affine Hypersurface Systems on Polytopes

Zhiyun Lin and Mireille E. Broucke

Abstract—This paper studies the reachability problem with solved by open-loop control. Here we particularly consider state constraints; that is, to reach a target without leaving a affine hypersurface systems; that is, affine systems with n- polytope. First, we provide a conceptual framework for the dimensional state and n − 1 independent control inputs. The constrained reachability problem. Then we obtain necessary and sufficient conditions for the problem for affine hypersurface contribution of the paper is that we completely solve these systems. In addition, we provide a geometric characterization problems for affine hypersurface systems. Necessary and of the maximal set of initial states from which the objective sufficient conditions are derived for the reachability problem can be met. Finally, algorithms on constructing piecewise affine (using open loop control) with state constraint in a polytope, feedback controls that accomplish the objective are presented. and then we show that if there is an open loop control for I.INTRODUCTION the reachability problem, there is also a piecewise affine state feedback that accomplishes this. Algorithms on constructing Problems of reachability and invariance for dynamical sys- such a piecewise affine state feedback are also provided. tems have been extensively studied in the control literature Throughout the paper, we use the following notations: for a long time. These problems have attracted renewed inter- rank(B) and Im(B) denote the rank and the image of a est due to the emergence of a class of practically important o systems — hybrid systems. The problem of reachability with matrix B. Let A, B be two sets. A , conv(A), vert(A), and state constraints arises in practice for the reason of meeting aff(A) denote the interior of A, the convex hull of A, the safety or performance specifications when synthesizing con- vertices of A, and the smallest affine space containing A, trollers. A typical example of the problem is motion planning respectively. dist(x, A) expresses the distance from a point of multiple vehicles with collision avoidance. x to A and A\B expresses the set difference. More recently, a relevant problem called the control-to- Several proofs are omitted in the paper due to space facet problem was introduced by Habets and van Schuppen limitations. More details can be found in [7]. in [4] and later studied in [2], [3], [5], [8], [10]. In [2], [4], [5], [10], necessary and sufficient conditions are derived for II. PRELIMINARIES AND PROBLEM FORMULATION facet reachability of simplices by affine state feedback, while Consider a control system whose dynamics are affine on in [8], the facet reachability problem of simplices is extended an n-dimensional polytope P, to be solved by any type of feedback control if affine state feedback does not exist. In [3] a similar problem is studied Σ :x ˙ = Ax + a + Bu =: f(x, u), x ∈P, (1) but for rectangular multi-affine systems. The generalization where A ∈ Rn×n, B ∈ Rn×m, a ∈ Rn, and the control of this problem to affine systems on polytopes is more u ∈ Rm lives in the space of piecewise continuous functions. difficult and remains open. We call the above control system an affine system. If in This paper studies the problem on polytopes and the addition rank(B) = n − 1, then we call Σ an affine hy- starting point is reachability with state constraints (using persurface system. Given any piecewise continuous function open-loop control). The purpose of the paper is to provide a u u : t 7→ u(t) and any initial state x0 ∈P, let φt (x0) denote systematic view and methodology for the problem. Consider the solution of Σ starting from x0 with the control u(t). an affine system and consider a polytope in the state space Let F be an (n−1)-dimensional polytope on the boundary and a target set on the boundary of the polytope. The first of P, which will be the target set in our reachability problem. problem is to determine whether all the states in the polytope Notice it is not necessarily a facet of P. can be steered by an open-loop control to reach the target set while the trajectories remain in the polytope before reaching A. Reachability with State Constraints and P-Invariant Sets the target. The second problem is to find the maximal set (or In this section we establish two preliminary results which a well-approximation of it) of initial states in the polytope provide a conceptual framework for the problems that will from which it is possible to reach the target, assuming the be studied in the paper. First we show that any closed set entire polytope cannot. Finally, find a state feedback control can be partitioned into states that can reach a target set and (of any type) that accomplishes the reachability specification states that cannot. Second, we define P-invariant sets, which when the reachability problem with state constraint can be will be a key conceptual tool in our development. The authors are with the Department of Electrical and Computer Engi- Definition 2.1: (a) A point x ∈ P can reach F with P neering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: constraint in P, denoted by x −→ F, if there exist a [email protected]; [email protected]). Supported by the Natural Sciences and Engineering Research Council of Canada. Zhiyun Lin piecewise continuous control u : t 7→ u(t) and T > 0 u u is currently in Zhejiang University, Hangzhou, China. such that the solution φt (x) satisfies φt (x) ∈P for all u Rn t ∈ [0,T ] and φT (x) ∈F. Otherwise, we say x cannot [6]). Define O = {x ∈ : Ax + a ∈ Im(B)}. It is P reach F with constraint in P, denoted by x 6−→ F. fairly easy to prove that O = ∅ when Im(A) ⊆ Im(B) and n (b) A set P′ ⊆ P can reach F with constraint in P, a∈ / Im(B); O = R when Im(A) ⊆ Im(B) and a ∈ Im(B); P P denoted by P′ −→ F, if x −→ F for every x ∈P′. and O is a hyperplane, otherwise. Notice that the vector field (c) Aset P′ ⊆P is said to be a failure set to reach F with f(x, u) on O can vanish for an appropriate choice of u, so P P is the set of all possible equilibrium points of the system. constraint in P, denoted by P′ 6−→ F, if x 6−→ F for O When is a hyperplane (called the dividing hyperplane in all x ∈P′. O [9]), it divides the state space into two disjoint regions inside Proposition 2.1: Let P′ be the maximal set in P such P of which the normal component of x˙ to Im(B) points to the that P′ −→ F and let P′′ be the maximal set in P such that P opposite direction and does not vanish. When O is empty, the P′′ 6−→ F. Then (a) P = P′ ∪P′′ and P′ ∩P′′ = ∅, (b) ′ normal component of x˙ to Im(B) points to a same direction ′ P ′′ P ′ P −→ F and P 6−→ P . and does not vanish on the entire state space. Definition 2.2: A set A⊆P is called P-invariant if for In the following, we consider two cases depending on O: o o all x0 ∈ A and for all piecewise continuous functions u : t 7→ u (A) P ∩ O = ∅; (B) P ∩ O= 6 ∅. u(t), each continuous trajectory segment φt (x0) including x0 that is in P is also in A. III. REACHABILITY ON POLYTOPES – CASE A Note that the time interval that a continuous trajectory For notational convenience, denote by B the subspace u segment φt (x0) including x0 is in P is either a closed Im(B) and denote by B the hyperplane parallel to B and u x interval [0,T ] with T < ∞, or it is [0, ∞); i.e., φt (x0) ∈P going through a point x. For case A, let β be the unit for all t ∈ [0,T ] or [0, ∞). The definition means that the normal vector to B satisfying βT (Ax + a) ≤ 0 for all trajectories cannot leave A before leaving P. Rn + x ∈ P. For any z ∈ , define half spaces Hz = {x ∈ Proposition 2.2: A set A⊂P is P-invariant if and only Rn T T − Rn T T P : β x ≥ β z} and Hz = {x ∈ : β x ≤ β z}, T if A 6−→ P \ A. respectively. Also, let v− (v+) be a point in arg min{β x : T We now present a fundamental result which characterizes x ∈ F} (argmax{β x : x ∈ F}) and denote ∂Pmax = T T the set of states that can reach F with constraint in P in arg max{β x : x ∈ P} and ∂Pmin = arg min{β x : x ∈ terms of P-invariant sets. P P}. Proposition 2.3: P −→ F if and only if no P-invariant set is in P\F. A. Necessary and Sufficient Conditions From above, it is clear that if a set A⊆P\F is P- In this subsection we give the solution of Problem 2.1. o P invariant, then A is a failure set to reach F with constraint Theorem 3.1: Suppose P ∩O = ∅. Then P −→ F if and in P. But the converse is not true: trajectories starting in a only if none of the following conditions holds: o failure set A do not reach F in finite time, but they need − + (a) (Hv− ∩P) \ (F ∪ O) 6= ∅; (b) ∂Pmax ⊂ O∩ H v+ ; not be in A for the same time that they are in P. However, (c) B is parallel to O when F ⊂ O. the following statement is true: The maximal failure set to reach F with constraint in P is P-invariant. This follows B O from Proposition 2.1 and Proposition 2.2. β

B. Problem Formulation ∂Pmax P P This paper addresses the following problems for affine hypersurface systems. F F O Problem 2.1: Find necessary and sufficient conditions v− v − + + P H − B v Hv+ such that P −→ F (using open loop control). β Problem 2.2: Find P′ ⊆ P, the maximal set of initial P states, such that P′ −→ F. Fig. 1. Illustration for Theorem 3.1. One difficulty that arises is that the maximal reachable set may not be closed, and this results in two practical Fig. 1 illustrates the three failure cases of the theorem. In − problems. One is that the time to reach the target set F, the left figure, (Hv− ∩P)\(F ∪O) 6= ∅ (shaded region) and o while finite for each initial condition, may not be uniformly + ∂Pmax ⊂ O∩ H where ∂Pmax is a point. In the right upper bounded on P′. Second, the control effort can tend to v+ figure, F ⊂ O and B is parallel to O. infinity as the initial condition approaches the boundary of ′ We provide two lemmas to prove Theorem 3.1. P . To bypass these problems, we develop in Section III- o Lemma 3.1: Suppose P ∩ O = ∅. Let z be a point in P. B a procedure to form an arbitrarily good, closed, full- o ′ − − dimensional approximation of P . (a) The sets Hz ∩P and H z ∩P are P-invariant; T Problem 2.3: Find a state feedback u = h(x) that accom- (b) If β (Ax + a)=0 for all x ∈ Bz ∩P, then Bz ∩P P and P\Bz are P-invariant. plishes P −→ F. o We first introduce a set that plays an important role in our Lemma 3.2: Suppose P ∩ O = ∅. Let y,z be distinct reachability problem (as well as in controllability problems points and let l be the line segment joining them. o − l (a) If z,y 6∈ O and y ∈H z , then z −→ y. reach F through a line. (IV): Suppose Bx ∩P ⊂O (see l (b) If z,y ∈ O and y ∈Bz, then z −→ y. Fig. 3). Then it follows from the assumptions that Bx ∩F= 6 Sketch Proof of Theorem 3.1 (=⇒) Define ∅. Now we select a point y ∈Bx ∩F. Clearly, x and y satisfy l the assumption in Lemma 3.2(b). Thus, x −→ y, where l is D1 if Thm. 3.1(a) holds and Bv− ∩F∩O = ∅, P the line segment joining from x to y, and so x −→ F.  A− =  D2 if Thm. 3.1(a) holds and Bv− ∩ F ∩ O= 6 ∅,  ∅ otherwise, B. Failure Sets and Partition of the Polytope  − where D1 = (Hv− ∩P) \F and D2 = D1 \ (Bv− ∩ O). Theorem 3.1 gives necessary and sufficient conditions for P the reachability problem P −→ F. This result, in turn, ∂P if Thm. 3.1(b) holds, A = max (2) can be tied to failure sets apropos Proposition 2.3. In this + ∅ otherwise.  subsection we classify all failure sets. First, if Theorem 3.1(c) P \ O if Thm. 3.1(c) holds, holds, then no point in Ao can reach F. We call this a Ao =  ∅ otherwise. total failure. If Theorem 3.1(b) holds, then the failure set is A = ∂P , which is always a face of P. We call Applying Lemma 3.1, one obtains that these sets are P- + max this a face failure. If Theorem 3.1(a) holds, there are several invariant and also in P\F. Thus, the conclusion follows cases. If the failure set A− is a full-dimensional subpolytope from Proposition 2.3. in P we call this a region failure. There are also two extreme O situations. One is a total failure when the closure of A− is B B P which occurs when B is parallel to F. The other situation o − β β is a face failure when A− is a face when H v− ∩P = ∅. P x P x We arrive at the following corollary of Theorem 3.1. o ′ l l Corollary 3.1: Suppose P ∩ O = ∅. Let P = P \ A P′ P z ′ y y where A = A− ∪ A+ ∪ Ao. Then P −→ F and A 6−→ F. Remark 3.1: It can be easily checked that Theorem 3.1 v− F v− F O (b) and (c) are mutually exclusive. This means only one of Fig. 2. Illustration for case I (left) and case II (right). the sets A+ and Ao is not empty. If Theorem 3.1(c) holds, we do not need to check (b) as A+ must be empty. Also we (⇐=) Let x be a point in P\(F ∪O). (I): Suppose F is not do not need to check (a) as A− equals to Ao or is empty. ′ in O. Since the assumption (a) does not hold, it follows that We have identified the maximal set P ⊆ P for Prob- there is a point y in F \ O and in the small neighborhood of lem 2.2. This set, in general, is not closed. This leads to v− satisfying the assumption in Lemma 3.2(a) (see Fig. 2), so difficulties with unbounded control effort and unbounded l x −→ y, where l is the line segment joining x and y. Hence, time to reach F. Consequently, once failure sets have been P identified, it is desirable to remove them via a procedure x −→ F. (II): Suppose F is in O. Then by assumption, we that both well-approximates the set of states that can reach know that B is not parallel to O. Let y be the point selected F and also yields a closed full-dimensional polytope P′′ as in case (I). Then for a proper control with f(y,u) pointing ′′ ′′ P outside of P, there is a point z on the backward trajectory such that P −→ F. The approach is to “cut off” failure o − sets from P by one of two procedures. One procedure is of y satisfying z ∈H x (see Fig. 2). Thus, by Lemma 3.2(a) l P for removing region and face failures A− by cutting along a we have x −→ z and therefore x −→ F. hyperplane which is parallel to a slightly shifted version of O B. The second procedure is for removing face failures A+ B B O by cutting exactly along a hyperplane parallel to B. These β x β P cuts are chosen arbitrarily close to the failure set and so that P the remaining polytope has no failure sets. Of course, if there x is a total failure, there is no need to partition P. l The following procedure partitions P into a disjoint union y of a closed full-dimensional polytope and at most two other v− F v− F polytopes that over-approximate the failure sets A− and A+. Fig. 3. Illustration for case III (left) and IV (right). Algorithm 1: (Let ǫ> 0 be sufficiently small.)

1) If A− 6= ∅, select enough number of points z1,...,zk o o Next, let x be a point in (P ∩ O) \F. Clearly, x is on the + o in P ∩ H v− such that G := aff{z1,...,zk, F∩Bv− } boundary of P by assumption O∩ P = ∅. (III): Suppose is of dimension n − 1 and max dist(x, Bv− ) = ǫ. B ∩P 6⊂ O. Then for a proper control with f(x, u) pointing x∈P∩G x Then divide P along G. inside of P, the trajectory starting from x instantaneously 2) If A 6= ∅, select a point z ∈ P such that enters the interior of P (see Fig. 3), which is not in O any + max dist(x, Bz)= ǫ. Then divide P along Bz. more. Then by the previous argument, it can be driven to x∈A+ P P −→ F (using open-loop control) then there exists a piece-

Let Aǫ− , Aǫ+ , and Pǫ be the collection of sets after the wise affine feedback solving the reachability problem. The application of the division rules in Algorithm 1, where Aǫ− idea is to triangulate the polytope, transform the reachability contains A−, Aǫ+ contains A+, and Pǫ is the remainder. problem within a polytope into a set of reachability prob- Clearly, these three sets (if not empty) are n-polytopes and lems for simplices, and then devise appropriate piecewise ′ F⊂Pǫ ⊆P . Then we have the following corollary which affine controllers on each simplex. The triangulation must follows directly from Algorithm 1 and Theorem 3.1. be performed properly otherwise the procedure may fail. o Pǫ First we present a result on the existence of a piecewise Corollary 3.2: Suppose P ∩ O = ∅. Then Pǫ −→ F. affine feedback that solves the reachability problem on We call Pǫ the ǫ-approximation of maximal reachable set of F. To generalize this terminology, we say that the ǫ- simplices. The result can be obtained from [8] and more details on how to construct such controllers can also be found approximation of maximal reachable set Pǫ = ∅ when P P in [8]. Consider a simplex S and a facet E of S. We have is a total failure, and Pǫ = P when P −→ F. the following lemma. A simple example is presented to illustrate the possible o S failure sets and how Algorithm 1 cuts them off. Consider Lemma 4.1: Suppose S ∩O = ∅. If S −→ E (using open loop control) then there exists a piecewise affine feedback the system S x˙ 1 = x2, u = Fσ(x)x + gσ(x) that accomplishes S −→ E, where σ : x˙ 2 = u. S→{1, 2}. Next we present a lemma that is useful to find a proper It can be easily verified that the system has a dividing triangulation. hyperplane; i.e., O = {(x1, x2) : x2 = 0}, the x1 axis, o P Lemma 4.2: Suppose P ∩O = ∅. If P −→ F, then there and that B is just the x2 axis. Suppose that the polytope P exists a vertex v∗ of P in ∂Pmax such that either v∗ ∈/ O

′ or v∗ ∈F. x2 A− P A+ Aǫ− Pǫ Aǫ+ Proof: Suppose by contradiction that for any vertex v ∈ O O x1 ∂Pmax we have v ∈ O and v∈F / . Note that v∈F / for all o + vertices v ∈ ∂Pmax implies, by convexity, ∂Pmax ⊂H . B v+ β Moreover, since v ∈ O for all v ∈ ∂Pmax, it follows v− v+ v− v+ F F from the convexity of O that ∂Pmax ⊂ O. Hence, by P Fig. 4. An example. Theorem 3.1, this contradicts P −→ F.  Now we present an algorithm for control synthesis. But first, we introduce some concepts on triangulation; for more and the target set F are as shown in Fig. 4. The dividing details, see [1]. hyperplane touches the polytope but has empty intersection Suppose V is a finite set of points such that conv(V) is with its interior. From (2), we get, Ao = ∅ since B is not n-dimensional. A subdivision of V is a finite collection P = − parallel to O; A− = (H ∩P) \F is the patterned region v− {P1,..., Pm} of n-polytopes such that the vertices of each in Fig. 4; and A is just a point. The set of initial states for + Pi are drawn from V; conv(V) is the union of P1,..., Pm; which it is possible to reach F with constraint in P is the set and P ∩P (i 6= j) is a common (possibly empty) face of P ′ i j i P not including the boundary of A− and A . This set is not + and Pj . A triangulation of V is a subdivision in which each closed. Moreover, if an initial state x ∈P′ approaches the 0 Pi is a simplex. Consider a triangulation S = {S1,..., Sq}. boundary of A−, the control input u(x0) tends to infinity in We say S and S are adjacent (denoted by S ∼ S ) if ′ i j i j order to reach F. Also, if x0 ∈P approaches the boundary Si ∩Sj is a facet. A sequence (Sik ,..., Si0 ) is called a path S of A+, the time to reach F tends to infinity. Applying ij ′ to reach Si if Si ∼ Si − and Si −→ Si − for each Algorithm 1, a good closed ǫ-approximation Pǫ of P is 0 j j 1 j j 1 given on the right of Fig. 4. 1 ≤ j ≤ k. The length of such a path is k. Finally, we present a lemma for the reachability problem In what follows we denote F by S0 for notation simplicity. with two target sets which will be used in the next section. Algorithm 2: Let F and F be two (n − 1)-dimensional polytopes on the 1 2 1) Triangulation: boundary of P. o (a) If F is a facet of P then select v∗ as in Lemma 3.3: Suppose P ∩ O = ∅ and suppose B is not P Lemma 4.2. If F is not a facet of P then select v∗ parallel to O. If P −→ F1 ∪ F2 but it does not satisfy as in Lemma 4.2 and in addition satisfying that P 1 2 1 2 P −→ F1, then P \Pǫ ⊆ Pǫ , where Pǫ and Pǫ are v∗ is not in the facet containing F, if it exists. the ǫ-approximation (ǫ > 0 sufficiently small) of maximal (b) For the facet Fj of P that contains F, make a reachable set of F1 and F2, respectively. triangulation of vert(Fj ) ∪ vert(F) such that the interior of each resulting simplex is entirely either IV. CONTROL SYNTHESIS ON POLYTOPES – CASE A in F or not in F; for the remaining facet Fj This section investigates feedback synthesis on polytopes of P, make a triangulation of vert(F ). Denote o j i the triangulation for . for case A, namely, P ∩ O = ∅. We want to show that if {SFj : i =1,...,kj } Fj S i ¯ (c) Let = {S1,..., Sq} := {conv(v∗, SFj ) : v∗ satisfying the property of Lemma 4.2 are in F but none Fj is any facet of P not containing v∗}. of them is in F. So by Lemma 4.2 v∗ ∈/ O. Then, there is a 2) Path Generation: hyperplane G partitioning P into two subpolytopes P1 and P2 such that (1) v∗ ∈P2 \G and F⊂P1 \G, (2) E := G∩P (a) Initialization: Rf := {S0}, Ru := {S1,..., Sq}; satisfies E ∩ O = ∅, (3) argmax{βT x : x ∈ E} 6⊂ F¯ and (b) While (Ru 6= ∅), do arg min{βT x : x ∈E}⊂ F¯. Thus, from Theorem 3.1 and if ∃(Si, Sj ) ∈ Ru × Rf such that Si ∼ Sj and P P S 2 i the assumption P −→ F, we know P2 −→ E ⊂ P1 and Si −→ Sj , then move Si from Ru to Rf . P1 ′ T 3) Controller Synthesis: P1 −→ F. Furthermore, a point v∗ ∈ arg max{β x : x ∈ E} satisfies the property in Lemma 4.2 for the polytope P (a) Let {..., (..., S , S ,..., S ),... } be the collec- 1 i j 0 and also it is not in F¯. Thus, applying Algorithm 2, it gives tion of paths to reach S0 generated from step 2); P1 i a piecewise affine controller that achieves P1 −→ F as we (b) Find u (x) := Fσi(x)x + gσi(x), i = 1,...,q, Si just showed. For the polytope P2, clearly the target set E is that solves Si −→ Fij , where Fij is the common a facet of P2. So again by Algorithm 2, we have a piecewise facet of Si and the followed simplex in the path; P2 i affine controller achieving P2 −→E ⊂P1. (c) For any simplex Si ∈ S, let u(x) = u (x) for all x ∈ Si. If x ∈ P belongs to more than one v5 simplex, set u(x) = uj (x) where j is the index

of a simplex that has the minimum length path v1(v−) v4 to reach S0. v6 β F Lemma 4.3: The collection S obtained in Algorithm 2 is v v (v∗) ∈ O a triangulation of vert(P) ∪ vert(F) such that every simplex 2 3 in S contains v∗ as a vertex. Fig. 7. Illustration for case (III). o Theorem 4.1: Suppose P ∩ O = ∅. There exists a P piecewise affine feedback that accomplishes P −→ F if and (III): Consider the case when is not a facet of and P F P only if P −→ F (using open-loop control). all v∗ satisfying the property of Lemma 4.2 are in F. Then ¯ T T T Sketch of proof: (=⇒) Obvious. (⇐=) Denote by F the we have β v∗ ≥ β x ≥ β v− for all x ∈P. Construct any facet of P containing F. There are three cases. hyperplane such that it goes through the points v− and v∗, and partitions P into two full-dimensional subpolytopes P v v (v∗) 1 5 5 v5(v∗) and P2. Let F12 := P1 ∩P2 and let P3 be the convex hull of v1 v4 v1 v4 v1 v 4 F and F12. Thus, P3 is a full-dimensional polytope in P and F F β P F F12 is in P3. By Theorem 3.1 and the assumption P −→ F, P3 v2 v3(v∗) v2 v3 v2 v3 it can be easily verified that P3 −→ F. Furthermore, F is a facet of P3, so by what we just showed, Algorithm 2 gives Fig. 5. Illustration for case (I). a piecewise affine control, say u(x) = Fσ3(x)x + gσ3(x), P3 x ∈ P3, that achieves P3 −→ F. For the polytopes P1 (I): Suppose F is a facet of P or when F is not a facet P1 and P2, again by Theorem 3.1 we know P1 −→ F12 and of P, there exists a v∗ satisfying the property in Lemma 4.2 P2 P2 −→ F12. Moreover, F12 is a facet of both P1 and P2. and in addition, v∗ 6∈ F¯ (see Fig. 5). For this case, it suffices So there are piecewise affine controllers, u(x)= Fσ (x)x + to show that at every step of the algorithm such that Ru 6= 1 gσ1(x), x ∈ P1 and u(x) = Fσ2(x)x + gσ2(x), x ∈ P2, that ∅, there exists a pair (Si, Sj ) ∈ Ru × Rf such that Si ∩ P P S 1 2 i achieve P1 −→ F12 and P2 −→ F12, respectively. Since Sj =: Fij is a facet and Si −→ Sj . This can be verified by checking conditions (a)-(c) of Theorem 3.1 when the pair F12 is in P3, it means that the controllers can drive all the states not in P3 to P3. Thus, the following controller (Si, Sj ) ∈ Ru × Rf is chosen such that Fij has a vertex T T v with minimum β-coordinate (namely, β v ≤ β w for all Fσ3 (x)x + gσ3(x) x ∈P3 w ∈ Si ∈ Ru). u(x)=  Fσ1 (x)x + gσ1(x) x ∈P1 \P3  F x + g x ∈P \P v5 σ2 (x) σ2(x) 2 3 P  achieves P −→ F.  v1 v4 E β V. REACHABILITY AND CONTROL SYNTHESIS ON F POLYTOPE – CASE B v2 v3(v∗) o Now we come to the case B, (namely, P ∩O= 6 ∅). Recall Fig. 6. Illustration for case (II). that this case means either O is a hyperplane that partitions P into two full-dimensional polytopes, or O is the entire (II): Consider the case when F is not a facet of P and all state space. We only deal with the first one due to space 1 limitations. For this case, we divide P along O that results small, so F21 := P2 ∩ Q1 must not be empty and the in two polytopes, denoted by P1 and P2. The following states in P2 that cannot reach F20 must be able to reach 1 2 algorithm and theorem present solutions to the reachability F21. Then by Lemma 3.3 P2 \Q2 is contained in Q2 ⊆ problem and control synthesis problem. P2, the ǫ-approximation of maximal reachable set of F21. 1 Furthermore, F21 ∩ (P2 \Q2) is of (n − 1)-dimension since 1 P2 Algorithm 3: otherwise F21 is in Q2 meaning that F21 −→ F20 and P2 1 2 1) Initialization: k := 1, Rf := {Q1 = ∅, Q2 = ∅}, therefore P2 −→ F20, a contradiction. Now Q2 ∪Q2 = P2, F Ru := {P1, P2}, bd := {Fi0 = Pi ∩F : i =1, 2; implying P2 ∈/ Ru. Repeat the argument for P1. Algorithm 3 Fi0 is of (n−1)-dimension and Pi is not a total failure will succeed with Ru = ∅. to reach Fi0}. (b) =⇒ (c) Suppose there is a ǫ> 0 such that Algorithm 3 2) while (Fbd 6= ∅), do halts with Ru = ∅. Then the algorithm may produce one (or F k two) of the following pathes: a) for every Fij ∈ bd, find Qi ⊆ Pi, the ǫ- approximation of maximal reachable set of Fij k k−1 k Q1 −→ Q2 −→··· −→F, and update Qi := Qi ∪Qi ; k k−1 b) for every Qi ∈ Rf , if Qi = Pi, then remove Pi Q2 −→ Q1 −→··· −→F, from Ru; m F where ∪m=1,...,kQi = Pi, i =1, 2. For any m =1,...,k, c) update bd := {Fij = Pi ∩Qj : j 6= i; Pi ∈ Ru; m since every Qi is a closed full-dimensional polytope and Qj ∈ Rf ; Fij ∩(Pi \Qi) is of (n−1)-dimension o m and Pi is not a total failure to reach Fij }; Q i ∩O = ∅ from the algorithm, applying Theorem 4.1, we Qm m i m−1 d) k := k +1. have a piecewise affine control that achieves Qi −→ Qj Qm m i k 1 o or Qi −→ F. Note that Qi ,..., Qi may overlap each other. Theorem 5.1: Suppose and is a hyperplane. P ∩O= 6 ∅ O So for any x ∈Pi belonging to more than one polytopes, set The following are equivalent: m∗ the controller at x to be the one defined for Qi where m∗ P m (a) P −→ F (using open-loop control); is the minimum of the indices of Qi containing x. Thus, P (b) There exits a ǫ > 0 such that Algorithm 3 halts with the piecewise affine control accomplishes P −→ F. Ru = ∅; (c) =⇒ (a) Obvious.  (c) There exists a piecewise affine control that accom- P REFERENCES plishes P −→ F. P [1] J. E. Goodman and J. O’Rourke, Eds., Handbook of Discrete and Proof: (a) =⇒ (b) If B is parallel to O, then P −→ F Computational Geometry. CRC Press, 1997. P1 P2 implies P1 −→ F10 and P2 −→ F20 where both F10 and [2] L. C. G. J. M. Habets, P. J. Collins, and J. H. van Schuppen, “Reach- F are of (n − 1)-dimension. Thus, Algorithm 3 succeeds ability and control synthesis for piecewise-affine hybrid systems on 20 simplices,” IEEE Transactions on Automatic Control, accepted, 2006. with Ru = ∅. [3] L. C. G. J. 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