Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeA02.5
Necessary and Sufficient Conditions for Reachability of Discrete Time Affine Systems on Simplices Min Wu, Gangfeng Yan, and Zhiyun Lin In the paper, we study reachability problems in discrete time affine systems on simplices—that is, to synthesize an affine control such that all the states in a simplex are driven to a target polytope in finite time. The work has been greatly influenced by the control-to-facet problem in a continuous time setting. However, reachability problems in a discrete time setting have several distinct features. First, a trajectory of a discrete time system is a discrete sequence. Thus, so-called restricted facets are irrelevant to the problem of reaching an adjacent polytope. But in continuous time, if a trajectory exits through a restricted facet, it may reach another adjacent polytope rather than the desired one. Thus restricted facets are actually one of the control specifications for the reachability problem. Second, the solvability of the problem in a continuous time setting only depends on the simplex itself. In other words, as long as the target polytope has a common facet with the simplex, it can be uniquely determined from the simplex and the system dynamics whether there exists a feasible affine feedback or not. But in a discrete time setting, the solvability depends not only on the simplex but also on the target polytope. Different shapes of the target polytope may lead to different solutions to the reachability problem. Based on these observations, we propose the reachability problems for discrete time affine systems and present solutions to the reachability problems. First, we study the case where the union of the simplex and the target polytope is convex. Next we generalize the results by dropping the convex assumption. For the convex case, necessary and sufficient conditions are obtained in the form of linear inequalities together with a bilinear matrix inequality (BMI). Alternatively, we explore checkable conditions with less computation complexity. For the general case, necessary and sufficient conditions are presented in the form of set inclusion. Moreover, an algorithm is provided to check the conditions. For theoretical completeness, the problem of making a polytope positive invariant is considered in the paper as well. Finally, a temperature control system is presented to illustrate our results.
Abstract— This paper studies reachability problems in discrete time affine systems on simplices—that is, how to steer the states in a simplex to a target polytope in finite time. Reachability problems in a discrete time setting have several distinct features from those in a continuous time setting and the solvability depends not only on the simplex itself but also on the target polytope. In this paper we first study the case where the union of the simplex and the target polytope is convex. Then we generalize the results by dropping the convex assumption. For theoretical completeness, the problem of making a polytope positive invariant is considered as well. Finally, a temperature control system is presented to illustrate our results.
Key words: Reachability, affine system, simplex, invariance I. INTRODUCTION A piecewise affine hybrid system consists of a partition of the state space and a collection of affine dynamics valid on each region. Since many physical systems in a first approximation can be described by piecewise affine hybrid systems, this class of systems has been widely studied. In the last five years, it has regained considerable research attentions due to some promising new ideas appeared in reachability analysis. The problem of reaching target sets for (piecewise) affine systems defined on simplices and polytopes is first introduced and investigated by Habets and van Schuppen [1], and is referred to as the control-to-facet problem. Further results are given in [2], [3]. For an affine system in continuous time, the problem is to synthesize an affine control to reach an exit facet of a simplex in finite time. Necessary conditions called invariance conditions are presented in the form of linear inequalities defined at the vertices of the simplex, which restrict the trajectories not to leave the simplex from so-called restricted facets. Sufficient conditions are also presented to ensure that all closed-loop trajectories reach an exit facet and then leave the simplex. This problem is also studied by Roszak, Lin, and Broucke [4]–[7] by characterizing explicitly the states in the simplex failing to reach the exit facet. On the other hand, Hodrus et al. [8]–[11] consider reachability problems in discrete time and some sufficient conditions are presented. Reachability problems are related to set invariance problems, which are studied quite extensively [12], [13]. For example, invariance properties of polyhedral sets are investigated in [14] and [15].
II. PRELIMINARIES A. Terminologies Notations R and Z+ 0 are used to represent the set of real numbers and non-negative integers, respectively. We use co(v1 , . . . , vn ) to denote the convex hull of points v1 , . . . , vn . Let P be an n-dimensional polytope in Rn . It can be written as an intersection of d half spaces, where d is the least
The authors are with the Department of Systems Science and Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027 P. R. China. Corresponding author: Zhiyun Lin (
[email protected] ) The work is supported by National Natural Science Foundation of China under Grant 60875074.
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
63
WeA02.5 number required. That is, P=
d \
{x ∈ Rn |nTi x ≤ γi },
other regions (see Fig. 1). However, in the continuous time setting, both would be true as long as both P1 and P2 have the same common facet with S.
(1)
i=1
where ni is a unit normal vector and γi is a constant. The set {x ∈ Rn |nTi x = γi } is called its supporting hyperplane. A facet of polytope P is the intersection of P and one of its supporting hyperplanes, and is of (n − 1)-dimension. That is, Fi = {x ∈ P|nTi x = γi }, i = 1, . . . , d. PSfrag replacements Specifically, a simplex is an n-dimensional polytope with n + 1 vertices. We denote vert(P) the set of vertices of P. B. Comparison of Reachability Problems in Continuous and Discrete Time Setting
Fig. 1. S.
For the control-to-facet problem in continuous time, a facet of a polytope is called restricted if it is expected that no trajectory of the closed-loop system exits through that facet. The control objective (reachability specification) is to make trajectories exit the polytope not through the restricted facets but other ones. In doing so, conditions are imposed by only looking at the inner product of the vector fields at vertices and the normal vectors of restricted facets. But this method can not be extended to the discrete time case. In other words, it is not enough to ensure that no trajectory leaves the polytope through the restricted facets by only checking the inner product of the normal vectors of restricted facets and the difference ∆x = x(k + 1) − x(k) at the vertices on the restricted facets. To see this, we consider an example with all the facets being restricted. The constraints are satisfied on vertices but trajectories still leave the polytope. Consider a discrete time linear system x1 (k + 1) −3 0 x1 (k) = x2 (k + 1) 0 −3 x2 (k)
S
P1
P2
P1 (dotted) and P2 (dashed) have the same common facet with
These observations lead to the study of reachability problems in discrete time. C. Problem Formulation Consider a discrete time affine system x(k + 1) = Ax(k) + a + Bu(k),
(2)
where A ∈ Rn×n , a ∈ Rn , and B ∈ Rn×m . We let F (x) := {Ax + a + Bu| u ∈ Rm } denote the set of possible next state of x for some control input u. In the paper, we first address a set-invariance problem. Problem 1: Let Q be an n-dimensional polytope. Find, if possible, a continuous feedback control u(x) for system (2) such that Q is positive invariant. Next, we consider reachability problems. Let S be a simplex and let P be a polytope in Rn . Problem 2: Find, if possible, an affine feedback u(k) = F x(k) + g, where F ∈ Rm×n and g ∈ Rm , such that for any x0 ∈ S, there is a T ∈ Z+ 0 satisfying (i) x(k, x0 ) ∈ S for k = 0, . . . , T, (ii) x(T + 1, x0 ) ∈ P. Problem 3: Find, if possible, an affine feedback u(k) = F x(k) + g, where F ∈ Rm×n and g ∈ Rm , such that for any x0 ∈ S \ P, there is a T ∈ Z+ 0 satisfying (i) x(k, x0 ) ∈ S for k = 0, . . . , T, (ii) x(T + 1, x0 ) ∈ P\S, and for x0 ∈ S ∩ P, x(1, x0 ) ∈ P\S. There is a slight difference between these two problems. Problem 3 means that as long as a trajectory reaches the common set of S and P, it will immediately enter P at the next step, but for Problem 2, even when trajectories arrive on S ∩ P (if exists), it may go back to S and then enter P some time later. It is worth pointing out that the solution to Problem 3 is easier to be solved and if it exists, it is a solution to Problem 2 as well.
defined in the simplex ABC with vertices A = [−0.1, −0.1]T , B = [0.2, −0.1]T , and C = [−0.1, 0.2]T . It can be checked that the difference x(k + 1) − x(k) on each vertex points inside the simplex, which is similar to the constraints in the continuous time setting and is used in [8] as a condition to ensure that no trajectory leaves the simplex in the discrete time setting. However, this is not enough. As we can check that the next steps of points A, B, and C will be outside of the simplex. In addition, the concept of restricted facet is not well defined in the discrete time setting because trajectories for discrete time systems are discrete sequences in the state space. It is not trivial to tell whether the trajectories leave the polytope through this facet or another. Actually, in the discrete time setting, the triggered discrete event does not depend on the reached facet as in the continuous time setting, but depends on the target polytope. For example, for two different polytopes P1 and P2 , both have the same common facet with S, all the states in S may reach P1 without entering into other region, but some states may not be able to reach P2 without entering into
64
WeA02.5 III. F INITE T IME E SCAPE
AND
S ET I NVARIANCE
IV. R EACHABILITY A NALYSIS
In this section, we first present checkable conditions for autonomous affine systems such that all trajectories leave a simplex in finite time. The following lemma is similar to the continuous counterpart, which can be easily obtained using the same idea as in [2]. Lemma 1: For an autonomous affine system
In this section, we address Problems 2 and 3. Necessary and sufficient conditions for solvability of both problems will be presented for several cases. Some techniques below are analogue to the ones used in continuous time analysis [3]– [5]. A. S ∪ P is convex
x(k + 1) = Ax(k) + a
First, we consider the case where P is an adjacent polytope of S such that S∪P is convex and F := S∩P is the common facet. Theorem 2: Suppose that S ∪ P is convex. Problem 2 is solvable if and only if there exists a control u(x) = F x + g, where F ∈ Rm×n and g ∈ Rm , such that the closed loop system has no fixed point in S and
in a simplex S, the following three statements are equivalent: (i) there is no fixed point in S; (ii) there exists a ξ ∈ Rn such that ξ T ((A − I)x + a) < 0 for all x ∈ S; (iii) all trajectories starting in S leave it in finite time. Next, we present results on set invariance and address Problem 1. Discrete-time Nagumo’s Theorem (Lemma 2) is applied. Lemma 2: [13] For a discrete time system x(k + 1) = f (x(k)), a set X is positive invariant if and only if f (X ) ⊂ X. Notice that if X is a polytope and the dynamics is affine, then every trajectory initiating in X is a convex combination of the trajectories starting from the vertices of X . Therefore, we have the following simple condition for the positive invariance property of a polytope. Lemma 3: For an autonomous affine system
Avi + a + B(F vi + g) ∈ S ∪ P, ∀vi ∈ vert(S). (4) Proof: (⇒) If the closed loop system has a fixed point, then the trajectory starting at the fixed point will remain in this point, which is a contradiction. If condition (4) fails at some vertex vi , then the trajectory initiating at vi will leave S ∪ P at the next step. It contradicts with the assumption that Problem 2 is solvable. (⇐) Since S has no fixed point for the closed loop system, all trajectories starting in S will leave S in finite time by Lemma 1. That is, for any point x0 ∈ S, there is a T ∈ Z+ 0 such that x(k, x0 ) ∈ S, k = 0, . . . , T , and x(T +1, x0 ) 6∈ S. On the other hand, condition (4) guarantees that x(T + 1, x0 ) ∈ S ∪ P by convex argument. Therefore, x(T + 1, x0 ) ∈ P. Using Lemma 1, we have the following corollary. Corollary 1: Suppose that S ∪ P is convex. Problem 2 is solvable if and only if there are vectors u1 , . . . , un+1 , and ξ ∈ Rn such that
x(k + 1) = Ax(k) + a, a polytope Q is positive invariant if and only if Av + a ∈ Q for all v ∈ vert(Q). Before presenting a necessary and sufficient condition for the solvability of Problem 1, we introduce a lemma on the existence and uniqueness of an affine feedback u = F x + g providing the control inputs u1 , . . . , un+1 at the vertices of a simplex. Lemma 4: [1] Consider two sets of points {v1 , . . . , vn+1 } and {u1 , . . . , un+1 }. Suppose that v1 , . . . , vn+1 are affinely independent. Then there exist a unique matrix F ∈ Rm×n and a unique vector g ∈ Rm such that for each vi , we have ui = F vi + g. Theorem 1: Problem 1 is solvable if and only if there exist vectors ui , i = 1, 2, . . . , such that
Avi + a + Bui ∈ S ∪ P,
(5a)
ξ T ((A − I)vi + a + Bui ) < 0,
(5b)
for all vi ∈ vert(S). The affine feedback F x + g solving Problem 2 can be constructed uniquely when u1 , . . . , un+1 are solved from (5). Notice that (5a) results in a linear inequality problem, while (5b) is a Bilinear Matrix Inequality (BMI) problem since both ui and ξ need to be determined. Generally, it is not easy to find the solution of (5) (some BMI problems are NP-hard). In the following, we present a simple method to find the solution of Problem 2 with the idea inspired from [5]. For vi ∈ vert(S), notice that if F (vi ) ∩ (S ∪ P) = ∅, then Problem 2 is not solvable. Therefore, in the following, we assume that F (vi ) ∩ (S ∪ P) 6= ∅, which is equivalent to the condition (5a). It is observed that the set F (vi ) ∩ (S ∪ P) is also a polytope (possibly lower-dimensional). Denote I(vi ) the set of vertices of F (vi ) ∩ (S ∪ P). Then we let
Avi + a + Bui ∈ Q, ∀vi ∈ vert(Q). (3) This theorem can be deduced from Lemma 3 as did in [3] and the proof is omitted. Notice that if we represent Q in the form of (1), then condition (3) can be written as a set of linear inequalities. The solvability of linear inequalities can be determined by Farkas’ Lemma. So it is polynomial hard in finding a solution ui satisfying (3). There is some literature related directly to this result. For example, the invariant polyhedral set of linear discrete time system is analyzed in [14], [15], and in [12], [13] the invariant C-set (polytope containing the origin) is considered.
N (vi ) := {w − vi | w ∈ I(vi )} .
65
(6)
WeA02.5 for all ui ∈ Rm , which means x(1, vi ) ∈ Hv−i . Consider vi ∈ F. Then x(1, vi ) ∈ Hv−i means x(1, vi ) ∈ / P\S. It contradicts to the requirement that for x0 ∈ S ∩ P, x(1, x0 ) ∈ P \ S in Problem 3. Next consider vi 6∈ F. Notice that for any simplex there is only one vertex not on the facet F. So we have Hv−i ∩S = vi . Hence, from the assumption that Avi + a + Bui ∈ S ∪ P, we obtain x(1, vi ) = vi . Thus, vi is a fixed point and the problem is not solvable, which is a contradiction. (⇐) For x0 ∈ S ∩ P, it can be easily obtained from conditions (i) and (ii) that x(1, x0 ) ∈ P \ S. For x0 ∈ S \ P, it is similar to the sufficiency proof of Theorem 2.
Clearly, it has a finite number of elements. Moreover, it can be checked that for any ui satisfying condition (5a), (A − I)vi + a + Bui ∈ co{N (vi )}, where co{N (vi )} denotes the convex hull of the vectors in N (vi ). It follows that BMI condition (5b) can be transformed to a set of linear inequalities and thus a solution of ξ can be solved easily. Theorem 3: Suppose that S ∪ P is convex and that for every vi ∈ vert(S), F (vi ) ∩ (S ∪ P) 6= ∅. Then Problem 2 is solvable if and only if there exist ei,ki ∈ N (vi ) (i = 1, . . . , n + 1) such that T ξ e1,k1 < 0 T ξ e2,k2 < 0 (7) . .. T ξ en+1,kn+1 < 0
B. S ∪ P is not convex In this subsection, we assume that P is an adjacent polytope such that F = S ∩ P is a common facet, but S ∪ P may not be convex. Let P 0 be the largest subset of P such that S ∪ P 0 is convex. Note that if there is a solution for the reachability problem from S to P 0 , the reachability problem from S to P can be reduced to the one we studied in the previous subsection. Otherwise we have to work out other necessary and sufficient conditions. First, we present a simple necessary condition without a proof, which is the same as the necessary condition for the case that S ∪ P is convex. Lemma 5: Problems 2 and 3 are unsolvable if F (vi ) ∩ (S ∪ P) = ∅ for some vi ∈ vert(S). Next, we present our main results. Theorem 5: Problem 2 is solvable if and only if there exist vectors wi ∈ F (vi ) (i = 1, . . . , n + 1) and ξ ∈ Rn such that (i) co(w1 , . . . , wn+1 ) ⊂ S ∪ P, (ii) ξ T (wi − vi ) > 0 for i = 1, . . . , n + 1. Proof: (⇒) If the problem is solvable, then the closed-loop system is x(k +1) = (A+BF )x(k)+a+Bg. Moreover, for the above autonomous affine system, we know that there is no fixed point in S and that for any x0 ∈ S, x(1, x0 ) ∈ S∪P. Denote X1 the set of points x(1, x0 ) for all x0 ∈ S. Clearly, X1 ⊂ S ∪ P. Now let
has a solution ξ. Proof: (⇐) Suppose that ξ is a solution of (7) for a set of vectors e1,k1 , . . . , en+1,kn+1 . For i = 1, . . . , n + 1, let ui be a vector such that ei,ki = (A − I)vi + a + Bui . Since ei,ki is selected in N (vi ), condition (5a) is satisfied for these ui ’s. In addition, from (7), it follows that condition (5b) is satisfied. Hence, the problem is solvable. (⇒) Since the problem is solvable, by Corollary 1, there exist vectors u1 , . . . , un+1 , and ξ such that (5) holds. As a result, (A − I)v1 + a + Bu1 ∈ co{N (vi )}. Then we claim that there must be a vector in N (vi ), say e1,k1 , such that ξ T e1,k1 < 0. (To see this, suppose by contradiction that ξ T e1,1 ≥ 0, . . . , ξ T e1,l1 ≥ 0, where e1,1 , . . . , e1,l1 are the elements of N (vi ). Then by convex argument, it follows that ξ T ((A − I)v1 + a + Bu1 ) ≥ 0, a contradiction.) Repeating this argument, we obtain that for every i, there is a vector ei,ki ∈ N (vi ) satisfying ξ T ei,ki < 0. Hence, (7) has a solution ξ. Next, we present a result for Problem 3. A necessary and sufficient condition for solving Problem 3 is similar to the one for Problem 2, but the unknown ξ in (5b) is replaced by the unit normal vector of F, which leads to find a solution easily. The solution to Problem 3 (if exists) is also a solution to Problem 2. In what follows, we use the following notations. Let h be the unit normal vector of F pointing towards P. For z ∈ Rn , we let Hz− denote the closed half space
wi = (A + BF )vi + a + Bg for i = 1, . . . , n + 1. By convexity argument, we know co(w1 , . . . , wn+1 ) = X1 . So co(w1 , . . . , wn+1 ) ⊂ S ∪ P. Also, from the fact that there is no fixed point in S for the autonomous affine system, it follows from Lemma 1 that condition (ii) holds. (⇐) Since there are wi ∈ F (vi ) and ξ satisfying conditions (i) and (ii), we can solve for ui (i = 1, . . . , n + 1) by letting Avi + a + Bui = wi and then construct an affine feedback u = F x + g. Since condition (ii) holds, Then by convexity argument and Lemma 1, all trajectories starting in S will leave S in finite time. Moreover, due to condition (i), the trajectories get into P after leaving S. Then the conclusion follows. Though it is assumed that S ∩ P is a common facet, the result above still holds for the following two special cases:
Hz− := {x ∈ Rn |hT (x − z) ≤ 0}. Theorem 4: Suppose that S ∪ P is convex. Problem 3 is solvable if and only if there are vectors u1 , . . . , un+1 such that (i) Avi + a + Bui ∈ S ∪ P, (ii) hT ((A − I)vi + a + Bui ) > 0, for all vi ∈ vert(S). Proof: (⇒) If (i) fails, it is clear that Problem 3 is unsolvable. Now suppose that (i) holds and (ii) fails at some vertex vi . Then we have hT ((A − I)vi + a + Bui ) = hT (x(1, vi ) − vi ) ≤ 0
(8)
66
WeA02.5 (a) S ∩ P = ∅ and (b) S ∩ P is a common face rather than connecting wn+1 and any wi is entirely in S ∪ P. Also, a facet. since w1 , . . . , wn are in P, due to convexity properties, we For Problem 3, the following theorem provides a solution obtain that condition (i) of Theorem 6 holds. when S ∪ P may not be convex. Again, we let h denote the V. ILLUSTRATIVE EXAMPLE unit normal vector of F pointing towards P. We illustrate our results by a temperature control system Theorem 6: Problem 3 is solvable if and only if there exist adopted from [16] with a few modifications. The temperature vectors wi ∈ F (vi ) (i = 1, . . . , n + 1) such that of room B is affected by an air conditioner in the next room (i) co(w1 , . . . , wn+1 ) ⊂ S ∪ P, T A (see Fig. 2). We define three discrete states: high, safe, (ii) h (wi − vi ) > 0 for i = 1, . . . , n + 1. This theorem can be proved similarly as Theorem 4 or 5. Air conditioner Next, we present a procedure to find a solution of Problem 3 based on Theorem 6. Similar idea can be applied to replacements find a solution of Problem 2 based on Theorem PSfrag 5. Room B Room A We denote by vn+1 the vertex of S not on F and by v1 , . . . , vn , the vertexes of S on F. For any vertex vi ∈ F, we define R(vi ) = {x ∈ F (vn+1 )∩S| ∃wi ∈ F (vi ), co(x, wi ) ⊂ S∪P} T and define R = ni=1 R(vi ). The set R(vi ) is the largest set of possible wn+1 in S such that the line segment connecting wn+1 and wi entirely lies in S ∪ P. Moreover, for a given wn+1 , we define E(vi , wn+1 ) = {x ∈ F (vi ) ∩ P| co(wn+1 , x) ⊂ S ∪ P}, which is the largest set of possible wi in P such that the line segment connecting wn+1 and wi entirely lies in S ∪ P for given wn+1 . Notice that both R(vi ) and E(vi , wn+1 ) are polytopes (possibly lower-dimensional), which can be computed. Finally, we present a procedure to find a solution of Problem 3. Procedure 1: (S1) If ∃vi such that F (vi ) ∩ (S ∪ P) = ∅, then no solution. (S2) Else if ∃vi ∈ F such that F (vi ) ∩ P = ∅, then no solution. (S3) Else if F (vn+1 )∩P 6= ∅, then select wn+1 in F (vn+1 )∩ P and select wi in F (vi ) ∩ P for i = 1, . . . , n. Then a solution is found. (S4) Else if R = ∅, then no solution. (S5) Else, select wn+1 ∈ R and select wi in E(vi , wn+1 ) for i = 1, . . . , n. Then a solution is found. Proof: (S1) It is straightforward from Lemma 5. (S2) For that vi ∈ F, the next state of vi is not in P, which violates the condition that for x0 ∈ S ∩ P, x(1, x0 ) ∈ P \ S. So there is no solution for Problem 3. (S3) In this and following stages, condition (ii) of Theorem 6 is satisfied. Moreover, at this stage w1 , . . . , wn+1 can be selected in P and therefore the convex hull co(w1 , . . . , wn+1 ) is entirely in P. Thus, condition (i) of Theorem 6 holds. (S4) Notice that at this stage, w1 , . . . , wn are in P and wn+1 are in S. R = ∅ means that there is no wn+1 such that the line segment connecting wn+1 and any wi is entirely in S ∪ P. So it violates condition (i) of Theorem 6. (S5) This stage implies that wn+1 can be found in S and w1 , . . . , wn can be found in P such that the line segment
Fig. 2.
A temperature control system
and low, to describe the temperature states in room B. The air conditioner has two operations, heating and cooling, to increase or decrease the temperature. Our control objective is to adjust the temperature at B to the safe region by controlling the air conditioner. Formulating the real control problem in our setting, we define the regions Ph := {x ∈ R2 |0 ≤ x1 ≤ b, a3 ≤ x2 ≤ a4 }, Ps := {x ∈ R2 |0 ≤ x1 ≤ b, a2 ≤ x2 ≤ a3 }, Pl := {x ∈ R2 |0 ≤ x1 ≤ b, a1 ≤ x2 ≤ a2 }, corresponding to discrete states: high, safe, and low (see Fig. 3), where x1 and x2 denote temperature of room A and B, and a1 = −5, a2 = 0, a3 = 5, a4 = 10, b = 20. The goal is to steer the sates in Ph and Pl to the safe region Ps . Difference equation x(k + 1) = A1 x(k) + B1 u(k) + a1 is for the mode that the air conditioner is on the heating operation (states in Pl ). The matrices in the above equation are given as 0.83 0.14 1.82 0.04 A1 = , B1 = , a1 = . 0.07 0.56 0.08 0.38 Similarly, when the air conditioner is on the cooling operation (states in Ph ), we have x(k + 1) = A2 x(k) + B2 u(k) + a2 , where A2 =
0.91 0.09 0.01 0.81
, B2 =
0.96 0.01
, a2 =
0.01 0.18
.
In order to apply our results, we partition Pl into two simplices: S1 and S2 ; and partition Ph into S3 and S4 as shown in Fig. 3. With this partition, the first reachability problem is to steer the sates in S1 to the target Ps . Select ξ = [0, 1]T in (5b) and then it is solved from (5) that u1 = 0.35, u2 = 0,
67
WeA02.5 Several extensions of the work are possible in future. One would be the extension from simplices to polytopes. Since polytopes can be triangulated into a collection of simplices, in this way, it is very important to find a good triangulation method such that the desired specifications on polytopes can be equivalently transformed to reachability specifications on the collection of simplices. Another direction is on how to approximate nonlinear systems using piecewise affine systems and how to apply reachability results of affine systems to some practical problems (e.g., maneuvering and/or regulation of nonlinear systems).
and u3 = 0 at vertices v1 = (0, a1 ), v2 = (0, a2 ), and v3 = (b, a2 ). Solving the equation Sfrag replacements F g v1 v2 v3 = u u u , 1 1 1 2 3 1 1 1 we obtain an affine feedback u = F1 x + g1 for the simplex S1 so that all the states in S1 reach Ps in finite time, where F1 = 0 −0.07 and g1 = 0. x2 a4
S4
a3
R EFERENCES Ph (high) S
[1] L. C. G. H. M. Habets and J. H. van Schuppen, “A control problem for affine dynamical systems on a full-dimensional polytope,” Automatica, vol. 40, pp. 21–35, 2004. [2] L. C. G. J. M. Habets and J. H. van Schuppen, “Control to facet problems for affine systems on simplices and polytope with applications to control of hybrid systems,” in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, 2005, pp. 4175–4180. [3] L. C. G. J. M. Habets, P. J. Collins, and J. H. van Schuppen, “Reachability and control synthesis for piecewise-affine hybrid systems on simplices,” IEEE Transactions on Automatic Control, vol. 51, no. 6, pp. 938–948, 2006. [4] B. Roszak and M. E. Broucke, “Reachability of a set of facets for linear affine systems with n - 1 inputs,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 359–364, 2007. [5] ——, “Necessary and sufficient conditions for reachability on a simplex,” Automatica, vol. 42, no. 11, pp. 1913–1918, 2006. [6] Z. Lin and M. E. Broucke, “Resolving control to facet problems for affine hypersurface systems,” in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 2006, pp. 2625–2630. [7] ——, “Reachability and control of affine hypersurface systems on polytopes,” in Proceedings of 46th IEEE Conference on Decision and Control, New Orleans, USA, 2007, pp. 12–14. [8] T. E. Hodrus, M. Buchholz, and V. Krebs, “A new local control strategy for control of discrete-time piecewise affine systems,” in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, 2005, pp. 4181– 4186. [9] T. E. Hodrus, F. Wolff, and V. Krebs, “A control strategy for control of discrete-time piecewise affine systems on polytopes using system inherent bounds,” in Proceedings of IEEE International Conference on Control Applications, Munich, Germany, 2006, pp. 1499 – 1504. [10] T. E. Hodrus, M. Buchholz, and V. Krebs, “Control of discrete-time piecewise affine systems,” in Proceedings of IFAC World Congress, Prague, 2005. [11] T. E. Hodrus, F. Wolff, M. Buchholz, and V. Krebs, “A local control strategy for the control of discrete-time piecewise affine systems on full-dimensional polytopes,” in Proceedings of 32nd IEEE Annual Conference on Industrial Electronics, Paris, France, 2006, pp. 4737 – 4742. [12] F. Blanchini, “Set invariance in control,” Automatica, vol. 35, no. 11, pp. 1747–1767, 1999. [13] F. Blanchini and S. Miani, Set-Theoretic Methods in Control. Birkhauser, 2008. [14] G. Bitsoris, “Positively invariant polyhedral sets of discrete-time linear systems,” International Journal of Control, vol. 47, no. 6, pp. 1713– 1726, 1988. [15] C. E. T. Dorea and J. C. Hennet, “(A,B)- invariant polyhedral sets of linear discrete-time systems,” Journal of Optimization Theory and Applications, vol. 103, no. 2, pp. 521–542, 1999. [16] X. D. Koutsoukos and P. J. Antsaklis, “Safety and reachability of piecewise linear hybrid dynamical systems based on discrete abstractions,” Discrete Event Dynamic Systems: Theory and Applications, vol. 13, pp. 203–243, 2003.
3
Ps (safe)
b
a2 S1 a1 Fig. 3.
Pl (low)
x1
S2
Simulated trajectories using the obtained control.
Next we consider the trajectories starting in S2 . It is expected that they enter S1 ∪ Ps to reach the safe region. Similarly it can be solved that the affine feedback u = F2 x + g2 defined on S2 achieves this goal, where F2 = −0.02 0 and g2 = 0.35.
Notice that (F1 , g1 ) and (F2 , g2 ) are different, so the closedloop systems are different in S1 and S2 . As long as a trajectory goes from S2 to S1 , its dynamics also switches to the one defined on S1 . Similarly, an affine state feedback u = F3 x + g3 defined on S3 with F3 = [0.05 0.22], g3 = −1.07 and an affine state feedback u = F4 x + g4 defined on S4 with F4 = [0.14 0], g4 = 0 can be found to drive all the sates in the high region to the safe region Ps . With the obtained control above, simulated trajectories are shown in Fig. 3. VI. CONCLUSION In the paper, we study reachability problems in discrete time affine systems on simplices. The analogous problems in a discrete time setting have several distinct features from the control-to-facet problem in a continuous time setting. We reformulate the problems in a discrete time setting. Necessary and sufficient conditions for solvability of the problems are investigated as well as feasible affine feedback controllers. As the reachability problems in a discrete time setting depend on not only the simplex itself but also the target polytope, we start from the simple case that the union of the simplex and the target polytope is convex and then generalize our results to a possibly non-convex case. Finally, a temperature control system is presented to illustrate our results.
68
