Stability of Hybrid Systems? - Semantic Scholar
Stability of Hybrid Systems? Mikhail Kourjanski and Pravin Varaiya Department of Electrical Engineering and Computer Science University of California at Berkeley Berkeley, California 94720 fmichaelk, [email protected]
Abstract. Hybrid systems combine discrete and continuous behavior.
We study properties of trajectories of a rectangular hybrid system in which the discrete state goes through a loop. This system is viable if there exists an in nite trajectory starting from some state. We show that the system is viable if and only if it has a limit cycle or xed point. The set of xed points is a polyhedron. The viability kernel may not be a polyhedron. However, under a \controllability" condition, the viability kernel is a polyhedron.
1 Introduction Hybrid systems (HS) couple discrete and continuous dynamics. The state of the system is a pair|the discrete state and the continuous state. Within each discrete state, the continuous state obeys a dynamical law until an event occurs. Then the discrete state changes instantaneously; in addition there may be a reset of the continuous state. The continuous dynamics are represented by constant di erential inclusions. Each discrete state can be identi ed with an \enabling zone," the inclusion, and the reset map. When the continuous state trajectory enters the enabling zone, the condition for a transition of the discrete state is satis ed. (Thus the enabling set is a \guard.") The reset map is given by an ane map. The control of the system is achieved via the choice of an admissible velocity, a switching point within the enabling zone, and an admissible reset value. In this paper we study properties of systems with a prede ned order of enabling zones. In other words, the discrete states form a nite chain or a loop. The study of a nite chain leads to a problem of convex programming. When the system permits a loop, there arises the question of the existence of a xed point (limit cycle) and its stability. Most results presented here use a general assumption of convexity and compactness of the controls; but special attention is paid to \rectangular" systems where the controls belong to the sets described by linear inequalities, especially rectangles ([1], [4], [7], [8]). ?
Research supported by NSF Grant ECS9417370. The authors are grateful to Mireille Broucke for posing the questions addressed here, and to Anuj Puri for discussion.
2 Fixed Chain Hybrid System (FCHS) 2.1 FCHS Setting Discrete states are triples of the form [EZ, RS, DYN]. EZ is a zone which enables an abrupt change in dynamics and reset. RS is a reset map which speci es the change of state. DYN is the new inclusion to be used after the discrete change. The control of the system is implemented via the choice of the switching point within the EZs, admissible velocity, and admissible resets. The order in which the trajectory visits the EZs is given; and the reset is applied at a switching point of our choice followed by the change of the continuous dynamics. Each EZ is a convex compact set of IRN . RS is a linear map applied to the continuous state. The continuous dynamics is described by the di erential inclusion x_ 2 Ui ; (1) N where i is the discrete state and Ui is a convex compact set in IR . We choose a starting point within Z1 |the rst EZ of the chain, apply the reset, and immediately start to move with the dynamics given by (1). If we can reach points in Z2 then we choose a switching point within Z2 ; when we are in Z2 we apply the reset map and follow the next dynamics towards Z3 ; and so on. Our main interest is to evaluate the reach set of the system. The reach sets are the subsets of the Zi that can be reached from Z1 in nite time. We distinguish two situations: { Finite chain of EZs: the question is whether the nal EZ of the chain Z1 ; :::; ZM is reachable? { Looping phenomenon, Z1 ZM : the question is the existence of a \ xed point" (which is a point or a limit cycle as we will see later) of the system, and its stability.
2.2 Reduction to Multistage Discrete Dynamic System We show that FCHS is a special type of discrete dynamic system. First we describe the one{step evolution of the FCHS. The set reachable from 0 using (1) is the convex cone Ci = [ 0 Ui : (2) N The set of points reachable from X IR using (1) is Fic(X) = X + Ci: (3) Departing from x 2 Zi we apply the reset which is speci ed in the form Firs (x) = Ai x + Ri; (4) where Ai is a linear \diagonal" operator, and Ri are the admissible \resets." The operator Ai may be not invertible. The set reachable from Z1 is given by the composition Y1 (Z1 ) = F1c F1rs(Z1 ): (5)
The reach set for the chain fZ1; Z2 g is the set{valued map T1+ (Z1 ) = Y1 \ Z2 . The backward map T1? (Z2 ) gives all points in Z1 from which there starts a feasible trajectory for the chain fZ1 ; Z2 g: T1? (Z2 ) = A?1 1 [Z2 ? C1 ? R1] \ Z1 : (6) Following these de nitions we note that T1+ (Z1 ) = T1+ (T1? (Z2 )); T1? (Z2 ) = T1? (T1+ (Z1 )): (7) We extend this approach with the following maps: 8 (i < j) Reach set in Z starting from Z < j i Ti;j = : (i > j) Backward mapping from Zi to Zj (8) (i = j) Identity mapping on Zi Using these de nitions we can represent the FCHS as a discrete dynamic system: x(i + 1) = Ai x(i) + u(i)ti + r(i); i = 1; :::; M: (9) Here Ai is a N N diagonal matrix. The elements on the diagonal represent the scaling factor and r(i) is a vector of resets. M is the length of the chain. The elements of the system should satisfy the following constraints: x(i) 2 Zi ; u(i) 2 Ui ; r(i) 2 Ri ; ti 0; (10) where Zi ; Ui ; Ri are speci ed convex compact sets. The scalar form of (9) is xj (i + 1) = aj (i)xj (i) + uN +1 (i)uj (i) + rj (i); j = 1; :::; N; (11) xN +1 (i + 1) = xN +1 (i) + uN +1 (i): The coordinate xN +1 represents time. A sequence x(i), i = 1; :::M,which satis es (9){(10) is called a trajectory of the FCHS.
2.3 Rectangular Hybrid Systems and Linear Programming We pay special attention to the case of the rectangular FCHS (RFCHS). In this case the sets in (10) are rectangles. Each nite{length RFCHS is represented by a nite number of linear inequalities. Denote: ti ? instants of switching xij ? switching points, i = 1; :::; M; j = 1; :::; N; Zijl ; Ziju ? enabling zones Uijl ; Uiju ? velocity limitations Rlij ; Ruij ? reset additions Starting from the rst EZ of the given chain Z1l j x1j Z1uj ; j = 1; :::; N; (12)
we proceed further through the chain of EZs (m = 2; :::; M) subject to the following: l xmj Z u Zmj (13) mj l tm xmj ? r(m?1)j a(m?1)j x(m?1)j + U u tm (14) a(m?1)j x(m?1)j + Umj mj l u R(m?1)j r(m?1)j R(m?1)j (15) j = 1; :::; N The feasibility of the nite{length RFCHS can be checked by Linear Programming techniques. The resulting LP system is a \staircase-structured" problem; the relevant methods can be found in [6]. Remark. For each xed set of fti g; i = 1; :::; M, the reachability set within each of the Zi ; i = 1; :::; M, is a rectangle.
3 Loops in FCHS Consider the case Z1 ZM .
De nition1. A loop is a trajectory which starts and ends in the same EZ. A
simple loop is a loop in which each EZ is visited exactly once during the loop. A xed loop is a loop which ends precisely at its own starting point of a given EZ: x(M) = x(1). A looping trajectory is a nite or in nite number of trajectories
fx(i)gj ; fx(i)gj ; ::::; fx(i)gjL; 1
2
(L may be equal to in nity), with x(M)j1 = x(1)j2; ::::; x(M)jl?1 = x(1)jl ; ::::; x(M)jL?1 = x(1)jL: The looping trajectory is viable i it is in nite (L = 1 in the de nition of a looping trajectory). A point is viable if there exists a viable trajectory starting from that point. The phase portrait of the FCHS is in the Euclidean space IRN . For each starting point x(1) within Z1 , the rst EZ of the chain, there is a reach set F(x(1)) Z1 . Here F is a point-to-set mapping. The reach set for each starting point is a convex compact set since it is an intersection of a closed convex cone with a convex compact EZ. We need the following lemma.
Lemma 2. The set of viable points is convex and compact. Proof. Follows from de nition of compactness and convexity.
De nition3. The set K Z of all viable points is called the viability kernel
[2], [3].
Theorem4. The FCHS has a viable point i it has a xed point. Proof. Suppose there is a non{empty viability kernel K. Then K is a convex compact set, K Z1 . There exists a set{valued function FK which is closed and maps each point x 2 K to the nonempty convex compact subset of K:
FK (x) = F(x) \ K: This set is not empty since x is viable. By the Kakutani xed point theorem there is a point x such that x 2 FK (x). But this immediately means that there is a set of velocities, switching points, times and reset additions such that: x(2) = A(1)x (1) + u (1)t (1) + r (1); .. . x (1) = A(M ? 1)x (M ? 1) + u (M ? 1)t (M ? 1) + r (M ? 1) and all these equalities satisfy (10). We can repeat this simple xed loop in nitely. This is a xed point of the system. Now suppose that there is a xed point in the system. It is obvious that this is a viable trajectory.
De nition5. An FP-set is a set of all xed points of the system within one Zi. Corollary6. If there is no xed point in the FCHS then each looping trajectory can be extended through only a nite number of simple loops (L is nite).
Corollary7. The FP-set in the RFCHS can be determined by solving a nite
number of LP problems as in subsection 2.3 by adding the condition x1j = xMj ; j = 1; :::; N to the conditions (12){(15).
Note that an FP-set is always a polyhedron.
4 Stability 4.1 Single{valued FCHS First we investigate stability properties of a single{valued subclass of FCHS. No control is available in this case, i.e., the Ui are singletons.
De nition8. The FCHS is single-valued i for each starting point x0 its image T1;j (x0) is either a single point or an empty set for every j > 1.
We assume the EZs are convex polyhedra contained in (N ? 1)-dimensional hyperplanes; the velocities are constants; the velocity vectors make a non-zero angle with the relevant EZs and the reset additions are constants. This design provides us with a single{valued FCHS: there is a unique single{valued trajectory for each starting point. The xed points of a single{valued FCHS are then the solutions to a linear system x = Hx + D; (16) where H is a square matrix and D is a vector. This system may have one, many or no solution at all; but for a xed point to exist, the solution of (16) should also satisfy all state constraints of FCHS. We will now try to investigate and characterize the stability properties of the system.
Theorem 9. Suppose there is a xed point in the FCHS. The stability properties of a single{valued FCHS are then characterized by a (N ? 1){dimensional linear return map.
Proof. Following the discrete dynamic representation (9) we construct a com-
position of the linear steps which is also linear. Without loss of generality we may consider one of the EZs to be an oriented hyperplane (parallel to (N ? 1) coordinate axes). Taking the composition of transformations of a perturbation from a xed point through the loop we have a system x(i + 1) = G x(i) (17) where G is a (N ? 1) (N ? 1) matrix.
Corollary 10. Consider all EZs to be oriented. The transformation of the perturbation x(i) through one step of single{valued RFCHS is
0 xi 1 0 ai xi ? aim uuimi xim CC BBB ... BB ... BB xi = 0 CC BB ?ai uik xi BB . k CC BB k uim m . x(i) = B BB .. i CCC ! BBB .. i i m CC BB am xk = 0 BB x A B@ ... @ ... i i 1
xN
1
1
1
aiN xiN ? aim uuimN xim
1 CC CC CC CC = x(i + 1) CC CC CA
(18)
when k 6= m | that is, when the coordinates of degeneracy of EZs are di erent. The perturbation x(i) is multiplied by the scaling factor when k = m.
The eigenvalues i ; i = 1; :::; N ? 1 of G characterize the stability of the singlevalued FCHS. The system is hyperbolic if none of the eigenvalues has magnitude 1 [5].
4.2 An Example Consider the3following example. There are six discrete states and the continuous state is in IR . The EZs are two{dimensional rectangles with sides parallel to the coordinate axis. The reset map is identity. DIMENSION 3 STATE 1 EZONE (x1: 7..10; x2: 5.5..9; x3: 0.0) DYNAMICS (u1: -6; u2: 6; u3: 4) STATE 2 EZONE (x1: 0; x2: 11..19; x3: 1..9) DYNAMICS (u1: 4; u2: 6; u3: 2) STATE 3 EZONE (x1: 50..65; x2: 100; x3: 30..39) DYNAMICS (u1: -2; u2: -4; u3: 4) STATE 4 EZONE (x1: 30..39; x2: 50..59; x3: 80) DYNAMICS (u1: 2; u2: 2; u3: 3) STATE 5 EZONE (x1: 40; x2: 53..68; x3: 80..99) DYNAMICS (u1: -23; u2: -12; u3: -40) STATE 6 EZONE (x1: 16..25; x2: 50; x3: 50..63) DYNAMICS (u1: -9; u2: -36.1..-36; u3: -46..-45.9)
enabling zone of STATE 1 x2 FP-set map
7.60 7.50 7.40 7.30 7.20 7.10 7.00
x1 8.40
8.60
8.80
Fig.1. Example: Poincare map of a stable trajectory; diamond-shaped FP-set
enabling zone of STATE 1 x2 enab. zone map
9.00 8.50 8.00 7.50 7.00 6.50 6.00 5.50
x1 8.00
10.00
Fig. 2. Example: ellipsoidal viability kernel There is a choice of admissible velocities only in state 6. There is a unique set of parameters (u2: -36; u3: -46), which gives stable (but not asymptotically stable!) eigenvalues. The characteristic matrix for this case is 1:2005 ?0:8768 G = ?1:587 0:3261 The eigenvalues are 1;2 = ?0:4372 0:8994i; their magnitude is 1. The return map from the EZ of state 1 into itself for this case is shown in gure 1. The FP-set for the whole (set-valued) case is also shown in this picture. It is the thin diamond{shaped region. Let us investigate the shape of invariant curves of this example. Note that in fact it is a 2-dimensional linear recurrence: xk+1 = axk + byk ; yk+1 = cxk + dyk The linear transformation x = u + v; y = p1 u + p2 v; (19) converts the initial recurrence into diagonal form uk+1 = 1 uk ; vk+1 = 2 vk ; (20)
where ; are constants, and p1;2 = (1;2 ? a)=b; b 6= 0; p1;2 = c=(1;2 ? d); b = 0: (21) The eigenvalues are complex: 1;2 = i = ei . Here we apply one more transformation = u + iv; = ?i(u ? v) (22) which converts the complex{valued recurrence (20) into the real{valued canonical form: k+1 = (k cos ? k sin ); k+1 = (k sin + k cos ): (23) Consider the case = 1. The general form of the solution of the recurrence is k = a cos(n + 0 ); k = a sin(n + 0); 0 = a cos 0; 0 = a sin 0: (24) This yields 2 + 2 = a2 = 02 + 02. The invariant curves of (24) are the concentric circles and of the initial recurrence are the concentric ellipses. When the rotation angle is commensurable with 2, the trajectories are closed l?periodic orbits (this is an exceptional case, or resonance). Otherwise the subsequent iterations cover the whole ellipse. Returning back to the single{valued FCHS sample system we note that each variation of the velocity parameters yields a variation of the xed point. However the impact of those perturbations on the stability properties is somewhat di erent. Each transition in the system is from one oriented hyperplane to another. The variation (not shown in the example) of the velocity component whose coordinate is not degenerated during this transition, for example, STATE 6; DYNAMICS f u1: 9 g, changes the rotation angle only2. This immediately follows from (18). Indeed the characteristic equation is 2 + + 1 + = 0 (25) | the variation above a ects only ; = 0. Variations of 2 (?2; 2) in uence the rotation angle only. Variations of other velocity components a ect as well; this leads to abrupt change to asymptotically stable or unstable behavior. In our example every xed set of parameters gives an unstable system except for the unique extremal case STATE 6; DYNAMICS (u1: -9; u2: -36; u3: -46). Suppose the rotation angle is not commensurable with 2. The viability kernel for this extremal stable case is an ellipse|which implies that it can not be represented by a nite number of set-valued operations over the initial data| such as union, intersection, sum and di erence. The viability kernel is shown in gure 2. We note that there exists a sequence of non-viable points which converges to the viable point located on the boundary of the viability kernel, and the time for which a trajectory starting at each of the elements of the sequence is de ned, grows to in nity. The viability kernel for all other xed sets of parameters is a single ( xed!) point. 2
Note that the viability kernel changes discontinuously with the variation of this parameter.
Returning to the original set-valued case, we note that its viability kernel can not be represented as a union of viability kernels of all single{valued cases. For example, the point (7:5; 7:6; 0) is not viable in any of the single{valued cases, but if the whole set of velocities is available then the dynamics STATE 6;
DYNAMICS (u1: -9; u2: -36; u3: -45.9)
delivers it after 2 simple loops to the point (8:837483; 8:325626; 0) which is viable under the xed stable set of velocities. This shows that the viability kernel for the whole set{valued case is larger than the union of viability kernels of all possible single{valued cases. Next we provide sucient conditions guaranteeing a nite procedure for evaluating of the viability kernel.
4.3 Set-valued Procedure for Viability Kernel We rst specify an iterative set-valued process to nd the viability kernel. In fact, the viability kernel is a maximal invariant set|that is, for every point from this set there exists a trajectory which returns to this set. To x notation, we choose the \ rst" enabling zone; the choice is irrelevant for the procedure. For each EZ we build a chain of subsets Zi = Zi;0 Zi;1 ::: We de ne the forward and backward maps through one discrete state (see (7)) as T~1+ (Z1;k ) = Y1 (Z1;k ) \ Z2;k ; T~1? (Z2;k ) = A?1 1[Z2;k ? C1 ? R1] \ Z1;k ; (26) and through the chain of states as 8 (i < j) Reach set in Z starting from Z < j;k i;k k = (i > j) Backward mapping from Zi;k to Zj;k T~i;j (27) : (i = j) Identity mapping on Zi;k k (ZM;k ) we construct the next iteration Taking Ei;k = T~1k;i(Z1;k ) and Si;k = T~M;i of the procedure as Zi;k+1 = Si;k \ Ei;k . Suppose that the viability kernel is non{empty. Then it is clear that Zi;1 = \k Zi;k is the viability kernel. However Zi;1 may not be a polyhedron even though the Zi;k is a polyhedron. The only situations which may occur over each iteration of the procedure are the following: a S1;k = E1;k b E1;k S1;k c S1;k E1;k d S1;k \ E1;k 6= ; Conditions [a],[b] mean the viability kernel is found (it is TM;1 (S1;k )); in these cases the procedure stops after a nite numbers of steps. Conditions [c],[d] require further iterations. The procedure builds an exterior approximation of a viability kernel with polyhedra. The example in section 4.2 shows that an in nite number of iterations may occur. Note: if there exist a number K such that no nonviable point is feasible after K loops then the procedure will stop after a nite number of iterations.
Proposition11. Suppose there is a transition (Zi ! Zi+1 ) (that is a discrete state i ) such that:
h
9 8x 2 Zi ; 8y 2 fA?i [x ? Ci ? Ri ] \ Zi g +1
1
i
Yi (y) \ A?i1 [x ? Ci ? Ri ] contains a union of -balls :
Then the procedure will stop after a nite number of iterations.
The proof of this proposition can be derived from Theorem 5 of [3].
5 Conclusion The study of rectangular hybrid systems has largely been concerned with the questions of veri cation. Those questions often are decidable because the hybrid system can be shown to be equivalent (in an appropriate sense) to a nite automaton. In this paper we consider the question of existence of xed points (limit cycles) and their stability. We have shown that the set of xed points is given by a nite set of linear inequalities. That set is non-empty if and only if there is a viable state, i.e., a state from which a trajectory can be continued for in nite time. These results bring together the question of hybrid system viability with that of the existence of a xed point or limit cycle and the latter with stabilizability.
References 1. R. Alur, L. Fix and T.A. Henzinger. A Determinizable Class of Timed Automata. Proc. 6th Workshop Computer-Aided Veri cation 1994, LNCS 818, SpringerVerlag, 1994. pp. 1-13. 2. J.-P. Aubin. Viability Theory , Birkhauser, 1991. 3. A. Deshpande and P. Varaiya. Viable Control of Hybrid Systems. to appear. 4. T.A. Henzinger, P. Kopka, A. Puri and P. Varaiya. What's decidable about hybrid automata?. STOCS, 1995. 5. A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1995. 6. J.L. Nazareth. Computer Solution of Linear Programs. Oxford University Press, 1987. 7. A. Olivero, J. Sifakis and S. Yovine. Using Abstractions for the Veri cation of Linear Hybrid Systems. Proc. 6th Workshop Computer-Aided Veri cation 1994, LNCS 818, Springer-Verlag, 1994. pp. 81-94. 8. A. Puri and P. Varaiya. Decidability of Hybrid Systems with Rectangular Di erential Inclusions. Proc. 6th Workshop Computer-Aided Veri cation 1994, LNCS 818, Springer-Verlag, 1994. pp. 95-104. This article was processed using the LaTEX macro package with LLNCS style
Abstract. Hybrid systems combine discrete and continuous behavior.
We study properties of trajectories of a rectangular hybrid system in which the discrete state goes through a loop. This system is viable if there exists an in nite trajectory starting from some state. We show that the system is viable if and only if it has a limit cycle or xed point. The set of xed points is a polyhedron. The viability kernel may not be a polyhedron. However, under a \controllability" condition, the viability kernel is a polyhedron.
1 Introduction Hybrid systems (HS) couple discrete and continuous dynamics. The state of the system is a pair|the discrete state and the continuous state. Within each discrete state, the continuous state obeys a dynamical law until an event occurs. Then the discrete state changes instantaneously; in addition there may be a reset of the continuous state. The continuous dynamics are represented by constant di erential inclusions. Each discrete state can be identi ed with an \enabling zone," the inclusion, and the reset map. When the continuous state trajectory enters the enabling zone, the condition for a transition of the discrete state is satis ed. (Thus the enabling set is a \guard.") The reset map is given by an ane map. The control of the system is achieved via the choice of an admissible velocity, a switching point within the enabling zone, and an admissible reset value. In this paper we study properties of systems with a prede ned order of enabling zones. In other words, the discrete states form a nite chain or a loop. The study of a nite chain leads to a problem of convex programming. When the system permits a loop, there arises the question of the existence of a xed point (limit cycle) and its stability. Most results presented here use a general assumption of convexity and compactness of the controls; but special attention is paid to \rectangular" systems where the controls belong to the sets described by linear inequalities, especially rectangles ([1], [4], [7], [8]). ?
Research supported by NSF Grant ECS9417370. The authors are grateful to Mireille Broucke for posing the questions addressed here, and to Anuj Puri for discussion.
2 Fixed Chain Hybrid System (FCHS) 2.1 FCHS Setting Discrete states are triples of the form [EZ, RS, DYN]. EZ is a zone which enables an abrupt change in dynamics and reset. RS is a reset map which speci es the change of state. DYN is the new inclusion to be used after the discrete change. The control of the system is implemented via the choice of the switching point within the EZs, admissible velocity, and admissible resets. The order in which the trajectory visits the EZs is given; and the reset is applied at a switching point of our choice followed by the change of the continuous dynamics. Each EZ is a convex compact set of IRN . RS is a linear map applied to the continuous state. The continuous dynamics is described by the di erential inclusion x_ 2 Ui ; (1) N where i is the discrete state and Ui is a convex compact set in IR . We choose a starting point within Z1 |the rst EZ of the chain, apply the reset, and immediately start to move with the dynamics given by (1). If we can reach points in Z2 then we choose a switching point within Z2 ; when we are in Z2 we apply the reset map and follow the next dynamics towards Z3 ; and so on. Our main interest is to evaluate the reach set of the system. The reach sets are the subsets of the Zi that can be reached from Z1 in nite time. We distinguish two situations: { Finite chain of EZs: the question is whether the nal EZ of the chain Z1 ; :::; ZM is reachable? { Looping phenomenon, Z1 ZM : the question is the existence of a \ xed point" (which is a point or a limit cycle as we will see later) of the system, and its stability.
2.2 Reduction to Multistage Discrete Dynamic System We show that FCHS is a special type of discrete dynamic system. First we describe the one{step evolution of the FCHS. The set reachable from 0 using (1) is the convex cone Ci = [ 0 Ui : (2) N The set of points reachable from X IR using (1) is Fic(X) = X + Ci: (3) Departing from x 2 Zi we apply the reset which is speci ed in the form Firs (x) = Ai x + Ri; (4) where Ai is a linear \diagonal" operator, and Ri are the admissible \resets." The operator Ai may be not invertible. The set reachable from Z1 is given by the composition Y1 (Z1 ) = F1c F1rs(Z1 ): (5)
The reach set for the chain fZ1; Z2 g is the set{valued map T1+ (Z1 ) = Y1 \ Z2 . The backward map T1? (Z2 ) gives all points in Z1 from which there starts a feasible trajectory for the chain fZ1 ; Z2 g: T1? (Z2 ) = A?1 1 [Z2 ? C1 ? R1] \ Z1 : (6) Following these de nitions we note that T1+ (Z1 ) = T1+ (T1? (Z2 )); T1? (Z2 ) = T1? (T1+ (Z1 )): (7) We extend this approach with the following maps: 8 (i < j) Reach set in Z starting from Z < j i Ti;j = : (i > j) Backward mapping from Zi to Zj (8) (i = j) Identity mapping on Zi Using these de nitions we can represent the FCHS as a discrete dynamic system: x(i + 1) = Ai x(i) + u(i)ti + r(i); i = 1; :::; M: (9) Here Ai is a N N diagonal matrix. The elements on the diagonal represent the scaling factor and r(i) is a vector of resets. M is the length of the chain. The elements of the system should satisfy the following constraints: x(i) 2 Zi ; u(i) 2 Ui ; r(i) 2 Ri ; ti 0; (10) where Zi ; Ui ; Ri are speci ed convex compact sets. The scalar form of (9) is xj (i + 1) = aj (i)xj (i) + uN +1 (i)uj (i) + rj (i); j = 1; :::; N; (11) xN +1 (i + 1) = xN +1 (i) + uN +1 (i): The coordinate xN +1 represents time. A sequence x(i), i = 1; :::M,which satis es (9){(10) is called a trajectory of the FCHS.
2.3 Rectangular Hybrid Systems and Linear Programming We pay special attention to the case of the rectangular FCHS (RFCHS). In this case the sets in (10) are rectangles. Each nite{length RFCHS is represented by a nite number of linear inequalities. Denote: ti ? instants of switching xij ? switching points, i = 1; :::; M; j = 1; :::; N; Zijl ; Ziju ? enabling zones Uijl ; Uiju ? velocity limitations Rlij ; Ruij ? reset additions Starting from the rst EZ of the given chain Z1l j x1j Z1uj ; j = 1; :::; N; (12)
we proceed further through the chain of EZs (m = 2; :::; M) subject to the following: l xmj Z u Zmj (13) mj l tm xmj ? r(m?1)j a(m?1)j x(m?1)j + U u tm (14) a(m?1)j x(m?1)j + Umj mj l u R(m?1)j r(m?1)j R(m?1)j (15) j = 1; :::; N The feasibility of the nite{length RFCHS can be checked by Linear Programming techniques. The resulting LP system is a \staircase-structured" problem; the relevant methods can be found in [6]. Remark. For each xed set of fti g; i = 1; :::; M, the reachability set within each of the Zi ; i = 1; :::; M, is a rectangle.
3 Loops in FCHS Consider the case Z1 ZM .
De nition1. A loop is a trajectory which starts and ends in the same EZ. A
simple loop is a loop in which each EZ is visited exactly once during the loop. A xed loop is a loop which ends precisely at its own starting point of a given EZ: x(M) = x(1). A looping trajectory is a nite or in nite number of trajectories
fx(i)gj ; fx(i)gj ; ::::; fx(i)gjL; 1
2
(L may be equal to in nity), with x(M)j1 = x(1)j2; ::::; x(M)jl?1 = x(1)jl ; ::::; x(M)jL?1 = x(1)jL: The looping trajectory is viable i it is in nite (L = 1 in the de nition of a looping trajectory). A point is viable if there exists a viable trajectory starting from that point. The phase portrait of the FCHS is in the Euclidean space IRN . For each starting point x(1) within Z1 , the rst EZ of the chain, there is a reach set F(x(1)) Z1 . Here F is a point-to-set mapping. The reach set for each starting point is a convex compact set since it is an intersection of a closed convex cone with a convex compact EZ. We need the following lemma.
Lemma 2. The set of viable points is convex and compact. Proof. Follows from de nition of compactness and convexity.
De nition3. The set K Z of all viable points is called the viability kernel
[2], [3].
Theorem4. The FCHS has a viable point i it has a xed point. Proof. Suppose there is a non{empty viability kernel K. Then K is a convex compact set, K Z1 . There exists a set{valued function FK which is closed and maps each point x 2 K to the nonempty convex compact subset of K:
FK (x) = F(x) \ K: This set is not empty since x is viable. By the Kakutani xed point theorem there is a point x such that x 2 FK (x). But this immediately means that there is a set of velocities, switching points, times and reset additions such that: x(2) = A(1)x (1) + u (1)t (1) + r (1); .. . x (1) = A(M ? 1)x (M ? 1) + u (M ? 1)t (M ? 1) + r (M ? 1) and all these equalities satisfy (10). We can repeat this simple xed loop in nitely. This is a xed point of the system. Now suppose that there is a xed point in the system. It is obvious that this is a viable trajectory.
De nition5. An FP-set is a set of all xed points of the system within one Zi. Corollary6. If there is no xed point in the FCHS then each looping trajectory can be extended through only a nite number of simple loops (L is nite).
Corollary7. The FP-set in the RFCHS can be determined by solving a nite
number of LP problems as in subsection 2.3 by adding the condition x1j = xMj ; j = 1; :::; N to the conditions (12){(15).
Note that an FP-set is always a polyhedron.
4 Stability 4.1 Single{valued FCHS First we investigate stability properties of a single{valued subclass of FCHS. No control is available in this case, i.e., the Ui are singletons.
De nition8. The FCHS is single-valued i for each starting point x0 its image T1;j (x0) is either a single point or an empty set for every j > 1.
We assume the EZs are convex polyhedra contained in (N ? 1)-dimensional hyperplanes; the velocities are constants; the velocity vectors make a non-zero angle with the relevant EZs and the reset additions are constants. This design provides us with a single{valued FCHS: there is a unique single{valued trajectory for each starting point. The xed points of a single{valued FCHS are then the solutions to a linear system x = Hx + D; (16) where H is a square matrix and D is a vector. This system may have one, many or no solution at all; but for a xed point to exist, the solution of (16) should also satisfy all state constraints of FCHS. We will now try to investigate and characterize the stability properties of the system.
Theorem 9. Suppose there is a xed point in the FCHS. The stability properties of a single{valued FCHS are then characterized by a (N ? 1){dimensional linear return map.
Proof. Following the discrete dynamic representation (9) we construct a com-
position of the linear steps which is also linear. Without loss of generality we may consider one of the EZs to be an oriented hyperplane (parallel to (N ? 1) coordinate axes). Taking the composition of transformations of a perturbation from a xed point through the loop we have a system x(i + 1) = G x(i) (17) where G is a (N ? 1) (N ? 1) matrix.
Corollary 10. Consider all EZs to be oriented. The transformation of the perturbation x(i) through one step of single{valued RFCHS is
0 xi 1 0 ai xi ? aim uuimi xim CC BBB ... BB ... BB xi = 0 CC BB ?ai uik xi BB . k CC BB k uim m . x(i) = B BB .. i CCC ! BBB .. i i m CC BB am xk = 0 BB x A B@ ... @ ... i i 1
xN
1
1
1
aiN xiN ? aim uuimN xim
1 CC CC CC CC = x(i + 1) CC CC CA
(18)
when k 6= m | that is, when the coordinates of degeneracy of EZs are di erent. The perturbation x(i) is multiplied by the scaling factor when k = m.
The eigenvalues i ; i = 1; :::; N ? 1 of G characterize the stability of the singlevalued FCHS. The system is hyperbolic if none of the eigenvalues has magnitude 1 [5].
4.2 An Example Consider the3following example. There are six discrete states and the continuous state is in IR . The EZs are two{dimensional rectangles with sides parallel to the coordinate axis. The reset map is identity. DIMENSION 3 STATE 1 EZONE (x1: 7..10; x2: 5.5..9; x3: 0.0) DYNAMICS (u1: -6; u2: 6; u3: 4) STATE 2 EZONE (x1: 0; x2: 11..19; x3: 1..9) DYNAMICS (u1: 4; u2: 6; u3: 2) STATE 3 EZONE (x1: 50..65; x2: 100; x3: 30..39) DYNAMICS (u1: -2; u2: -4; u3: 4) STATE 4 EZONE (x1: 30..39; x2: 50..59; x3: 80) DYNAMICS (u1: 2; u2: 2; u3: 3) STATE 5 EZONE (x1: 40; x2: 53..68; x3: 80..99) DYNAMICS (u1: -23; u2: -12; u3: -40) STATE 6 EZONE (x1: 16..25; x2: 50; x3: 50..63) DYNAMICS (u1: -9; u2: -36.1..-36; u3: -46..-45.9)
enabling zone of STATE 1 x2 FP-set map
7.60 7.50 7.40 7.30 7.20 7.10 7.00
x1 8.40
8.60
8.80
Fig.1. Example: Poincare map of a stable trajectory; diamond-shaped FP-set
enabling zone of STATE 1 x2 enab. zone map
9.00 8.50 8.00 7.50 7.00 6.50 6.00 5.50
x1 8.00
10.00
Fig. 2. Example: ellipsoidal viability kernel There is a choice of admissible velocities only in state 6. There is a unique set of parameters (u2: -36; u3: -46), which gives stable (but not asymptotically stable!) eigenvalues. The characteristic matrix for this case is 1:2005 ?0:8768 G = ?1:587 0:3261 The eigenvalues are 1;2 = ?0:4372 0:8994i; their magnitude is 1. The return map from the EZ of state 1 into itself for this case is shown in gure 1. The FP-set for the whole (set-valued) case is also shown in this picture. It is the thin diamond{shaped region. Let us investigate the shape of invariant curves of this example. Note that in fact it is a 2-dimensional linear recurrence: xk+1 = axk + byk ; yk+1 = cxk + dyk The linear transformation x = u + v; y = p1 u + p2 v; (19) converts the initial recurrence into diagonal form uk+1 = 1 uk ; vk+1 = 2 vk ; (20)
where ; are constants, and p1;2 = (1;2 ? a)=b; b 6= 0; p1;2 = c=(1;2 ? d); b = 0: (21) The eigenvalues are complex: 1;2 = i = ei . Here we apply one more transformation = u + iv; = ?i(u ? v) (22) which converts the complex{valued recurrence (20) into the real{valued canonical form: k+1 = (k cos ? k sin ); k+1 = (k sin + k cos ): (23) Consider the case = 1. The general form of the solution of the recurrence is k = a cos(n + 0 ); k = a sin(n + 0); 0 = a cos 0; 0 = a sin 0: (24) This yields 2 + 2 = a2 = 02 + 02. The invariant curves of (24) are the concentric circles and of the initial recurrence are the concentric ellipses. When the rotation angle is commensurable with 2, the trajectories are closed l?periodic orbits (this is an exceptional case, or resonance). Otherwise the subsequent iterations cover the whole ellipse. Returning back to the single{valued FCHS sample system we note that each variation of the velocity parameters yields a variation of the xed point. However the impact of those perturbations on the stability properties is somewhat di erent. Each transition in the system is from one oriented hyperplane to another. The variation (not shown in the example) of the velocity component whose coordinate is not degenerated during this transition, for example, STATE 6; DYNAMICS f u1: 9 g, changes the rotation angle only2. This immediately follows from (18). Indeed the characteristic equation is 2 + + 1 + = 0 (25) | the variation above a ects only ; = 0. Variations of 2 (?2; 2) in uence the rotation angle only. Variations of other velocity components a ect as well; this leads to abrupt change to asymptotically stable or unstable behavior. In our example every xed set of parameters gives an unstable system except for the unique extremal case STATE 6; DYNAMICS (u1: -9; u2: -36; u3: -46). Suppose the rotation angle is not commensurable with 2. The viability kernel for this extremal stable case is an ellipse|which implies that it can not be represented by a nite number of set-valued operations over the initial data| such as union, intersection, sum and di erence. The viability kernel is shown in gure 2. We note that there exists a sequence of non-viable points which converges to the viable point located on the boundary of the viability kernel, and the time for which a trajectory starting at each of the elements of the sequence is de ned, grows to in nity. The viability kernel for all other xed sets of parameters is a single ( xed!) point. 2
Note that the viability kernel changes discontinuously with the variation of this parameter.
Returning to the original set-valued case, we note that its viability kernel can not be represented as a union of viability kernels of all single{valued cases. For example, the point (7:5; 7:6; 0) is not viable in any of the single{valued cases, but if the whole set of velocities is available then the dynamics STATE 6;
DYNAMICS (u1: -9; u2: -36; u3: -45.9)
delivers it after 2 simple loops to the point (8:837483; 8:325626; 0) which is viable under the xed stable set of velocities. This shows that the viability kernel for the whole set{valued case is larger than the union of viability kernels of all possible single{valued cases. Next we provide sucient conditions guaranteeing a nite procedure for evaluating of the viability kernel.
4.3 Set-valued Procedure for Viability Kernel We rst specify an iterative set-valued process to nd the viability kernel. In fact, the viability kernel is a maximal invariant set|that is, for every point from this set there exists a trajectory which returns to this set. To x notation, we choose the \ rst" enabling zone; the choice is irrelevant for the procedure. For each EZ we build a chain of subsets Zi = Zi;0 Zi;1 ::: We de ne the forward and backward maps through one discrete state (see (7)) as T~1+ (Z1;k ) = Y1 (Z1;k ) \ Z2;k ; T~1? (Z2;k ) = A?1 1[Z2;k ? C1 ? R1] \ Z1;k ; (26) and through the chain of states as 8 (i < j) Reach set in Z starting from Z < j;k i;k k = (i > j) Backward mapping from Zi;k to Zj;k T~i;j (27) : (i = j) Identity mapping on Zi;k k (ZM;k ) we construct the next iteration Taking Ei;k = T~1k;i(Z1;k ) and Si;k = T~M;i of the procedure as Zi;k+1 = Si;k \ Ei;k . Suppose that the viability kernel is non{empty. Then it is clear that Zi;1 = \k Zi;k is the viability kernel. However Zi;1 may not be a polyhedron even though the Zi;k is a polyhedron. The only situations which may occur over each iteration of the procedure are the following: a S1;k = E1;k b E1;k S1;k c S1;k E1;k d S1;k \ E1;k 6= ; Conditions [a],[b] mean the viability kernel is found (it is TM;1 (S1;k )); in these cases the procedure stops after a nite numbers of steps. Conditions [c],[d] require further iterations. The procedure builds an exterior approximation of a viability kernel with polyhedra. The example in section 4.2 shows that an in nite number of iterations may occur. Note: if there exist a number K such that no nonviable point is feasible after K loops then the procedure will stop after a nite number of iterations.
Proposition11. Suppose there is a transition (Zi ! Zi+1 ) (that is a discrete state i ) such that:
h
9 8x 2 Zi ; 8y 2 fA?i [x ? Ci ? Ri ] \ Zi g +1
1
i
Yi (y) \ A?i1 [x ? Ci ? Ri ] contains a union of -balls :
Then the procedure will stop after a nite number of iterations.
The proof of this proposition can be derived from Theorem 5 of [3].
5 Conclusion The study of rectangular hybrid systems has largely been concerned with the questions of veri cation. Those questions often are decidable because the hybrid system can be shown to be equivalent (in an appropriate sense) to a nite automaton. In this paper we consider the question of existence of xed points (limit cycles) and their stability. We have shown that the set of xed points is given by a nite set of linear inequalities. That set is non-empty if and only if there is a viable state, i.e., a state from which a trajectory can be continued for in nite time. These results bring together the question of hybrid system viability with that of the existence of a xed point or limit cycle and the latter with stabilizability.
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