- Email: [email protected]
Maximal compact positively invariant sets of discrete-time nonlinear systems
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Maximal compact positively invariant sets of discrete-time nonlinear systems ? Alexander P. Krishchenko ∗ Anatoly N. Kanatnikov ∗∗ ∗
Bauman Moscow State Technical University, Moscow, 105005 Russia (e-mail: [email protected]) ∗∗ Bauman Moscow State Technical University, Moscow, 105005 Russia (e-mail: [email protected]) Abstract: In this paper we examine the existence problem for maximal compact positively invariant sets of discrete-time nonlinear systems. To find these sets we propose the method based on the localization procedure of compact positively invariant sets. Analysis of a location of compact positively invariant sets and maximal compact positively invariant set of the Chatala system is realized for all values of its parameters. Keywords: Discrete-time system, maximal positively invariant set, localization 1. INTRODUCTION During the past years, the interest of many researchers has been attracted to the idea of finding some geometrical bounds for attractors, periodical orbits and chaotic dynamics of a nonlinear autonomous differentiable right-side system x˙ = f (x), f ∈ C 1 (Rn ). (1)
with the continuous right-side was proposed in Kanatnikov et al. [2010, 2011]. Using results obtained in these papers we suggest a method to find the maximal compact positively invariant sets of discrete-time nonlinear systems. In this paper, when we talk about a localization problem for the discrete-time system (2) we have in mind the same problem as in the case of continuous-time systems: find the localizing set.
Mainly, in the presented literature this problem was solved by Lyapunov-type functions, see e.g. articles of Doering & Gibbon [1995], Leonov et al. [1996], McMillen [1998], Swinnerton-Dyer [2001], Pogromsky et al. [2003], Li et al. [2005]. The method for finding families of semipermeable surfaces was proposed in Giacomini & Neukirch [1997] and then developed in other publications of these authors. The functional method was proposed in Krishchenko [1995, 1997] to the localization problem of periodical solutions of the system (1) and developed in Coria & Starkov [2009], Kanatnikov & Krishchenko [2009], Krishchenko [2005], Krishchenko & Starkov [2006a,b, 2008], Starkov [2007, 2009a,b,c, 2010] in the case of compact invariant sets.
The set V ⊂ Q ⊂ M is the localizing set for compact positively invariant sets of system (2) contained in Q if and only if V contains all compact positively invariant sets of system (2) contained in Q. The localizing set of a discrete-time system (2) is a localizing set for compact positively invariant sets of system (2) contained in M .
The theory of set invariance plays a fundamental role in the control of constrained discrete-time systems and has been a subject of research by many authors — see for instance Gilbert & Tan [1991], Blanchini [1999], Kolmanovsky & Gilbert [1998], Kerrigan & Maciejowski [2000], Rakovic et al. [2004, 2006] and the references therein. In this paper we consider the case of compact positively invariant sets.
In Sec. 2 and 4 we prove the main results describing the maximal compact positively invariant sets.
The localization method of compact invariant sets in the case of discrete-time nonlinear systems
In Sec. 5 we show the computation of localizing sets of the Chatala system and find outer approximations of the attractor of this system. Sec. 6 contains conclusions.
xk+1 = F (xk ),
F : M → M,
M ⊂ Rn
(2)
? This work was supported by the Russian Foundation for Basic Research, (projects 11-01-00733 and 09-07-00327) and the program of the President of the Russian Federation for state support of leading scientific schools (grant NSh-4144.2010.1)
978-3-902661-93-7/11/$20.00 © 2011 IFAC
This paper presents the methods for computation of a localizing set as well as the computation of decrescent outer approximations of the maximal compact positively invariant set. These approximations are achieved by computing a sequence of compact localizing sets. This paper is organized as follows.
Sec. 3 contains results of localization method in the case of discrete-time nonlinear systems. These results point out how to find the localizing sets of a discretetime nonlinear system and the localizing sets for compact positively invariant sets of a discrete-time nonlinear system contained in some subset of state space.
2. MAIN RESULTS A subset K ⊂ M is said to be positively invariant for the discrete-time nonlinear system (2) if F (K) ⊂ K.
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10.3182/20110828-6-IT-1002.00521
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
The positively invariant sets of a discrete-time system include equilibria (stationary points), periodic orbits, and attractors. Let G be any set in M , we introduce G0 = G,
Gi = Gi−1 ∩ F −1 (Gi−1 ),
and G∞ =
∞ \
i = 1, 2, . . . , (3)
Proof. Let K ⊂ Q be a compact positively invariant set. The continuous function ϕ(x) attains its maximum on K at some point x∗ ∈ K ⊂ Q since K is a compact set. Then F (x∗ ) ∈ K as K is a positively invariant set. As a result, ϕ(F (x∗ )) ≤ ϕ(x∗ ), i.e. ϕ(F (x∗ )) − ϕ(x∗ ) ≤ 0, ∗ and x∗ ∈ Σ− ϕ (Q) because x ∈ Q. Therefore for any point x ∈ K we get ϕ(x) ≤ ϕ(x∗ ) ≤
Gi .
sup x∈Σ− ϕ (Q)
i=0
ϕ(x) = ϕrsup (Q).
(7)
In the same way, for any point x ∈ K we get the inequalities
For these sets we get G = G0 ⊃ G1 ⊃ · · · ⊃ Gi ⊃ . . . ; x ∈ G∞ iff x ∈ Gi−1 and x ∈ F −1 (Gi−1 ), i = 1, 2, . . . Theorem 1. The set G∞ is a positively invariant set of the discrete-time system (2).
ϕ(x) ≥ ϕ(x∗ ) ≥
inf
x∈Σ+ ϕ (Q)
ϕ(x) = ϕrinf (Q),
(8)
where x∗ = argmin ϕ(x). Inequalities (7), (8) mean that x∈K
Proof. Let x ∈ G∞ . Then for any i > 0 we have x ∈ F −1 (Gi−1 ), i.e. F (x) ∈ Gi−1 . Therefore, F (x) ∈ ∞ T Gi−1 = G∞ , i.e. the set G∞ is a positively invariant i=1
set of the discrete-time system (2). The compact set V ⊂ G ⊂ M is the maximal compact positively invariant set contained in G if and only if V is positively invariant and contains all compact positively invariant sets contained in G. In the case G = M the compact set V is the maximal compact positively invariant set of system (2). Theorem 2. If the set G is a compact localizing set for all compact positively invariant sets of the discrete-time system (2), then G∞ is the maximal compact positively invariant set of this discrete-time system. Theorem 3. If the set G is a compact localizing set for compact positively invariant sets of the discrete-time system (2) contained in Q, then G∞ is the maximal compact positively invariant set contained in Q. Corollary 4. The discrete-time system (2) has the maximal compact positively invariant set iff this system has a compact localizing set for all compact positively invariant sets.
the compact positively invariant set K is contained in the set (6). Corollary 6. If the set G = Ωrϕ (Q) (6) is a compact set, then G∞ is the maximal compact positively invariant set contained in Q. − Corollary 7. If Σ+ ϕ (Q) = ∅ or Σϕ (Q) = ∅ then the discrete-time system (2) has no compact positively invariant sets contained in Q. Theorem 8. Let ϕα , α ∈ I, be a family of functions defined on M , ϕα ∈ C(Q), Q ⊂ M . Then each compact positively invariant set of the discrete-time T r system (2) contained in Q is contained in the set Ωϕα (Q). α∈I
Theorem 9. Let ϕ be a function defined on M and continuous on Q ⊂ M . If ψ(x) = h(ϕ(x)), where h : R → R is a strictly monotone function, then Ωrψ (Q) = Ωrϕ (Q). Particularly, this equality holds for h(t) = at + b, a 6= 0. Proof. Let us proof the equality Ωrψ (Q) = Ωrϕ (Q) in the case where h : R → R is a strictly increasing function. For strictly increasing function h the inequality ϕ(x1 ) ≤ ϕ(x2 ) is equivalent to ψ(x1 ) ≤ ψ(x2 ). Hence the set Σ− ψ (Q) is equal to the set Σ− (Q) and we get ϕ r ψsup =
To prove Theorems 2, 3 and to find G∞ we need some results of the localization method.
and define inf
ϕ(x),
ϕrsup (Q) =
sup
sup
h(ϕ(x))
x∈Σ− ϕ (Q)
sup x∈Σ− ϕ (Q)
Let Q be a subset of M . For a continuous function ϕ(x) defined on M we introduce the sets Σ+ ϕ (Q) = {x ∈ Q : ϕ(F (x)) − ϕ(x) ≥ 0} , (4) Σ− ϕ (Q) = {x ∈ Q : ϕ(F (x)) − ϕ(x) ≤ 0}
x∈Σ+ ϕ (Q)
ψ(x) =
=h
3. THE LOCALIZATION METHOD
ϕrinf (Q) =
sup x∈Σ− (Q) ψ
ϕ(x). (5)
x∈Σ− ϕ (Q)
Theorem 5. Each compact positively invariant set of the discrete-time system (2) contained in Q is contained in the set Ωrϕ (Q) = x ∈ Q : ϕrinf (Q) ≤ ϕ(x) ≤ ϕrsup (Q) , (6) i.e. the set Ωrϕ (Q) is a localizing set for compact positively invariant sets of system (2) contained in Q.
ϕ(x) = h ϕrsup .
r By the same way we obtain ψinf = h(ϕrinf ). Therefore the inequalities ψinf ≤ ψ(x) ≤ ψsup are equivalent to ϕinf ≤ ϕ(x) ≤ ϕsup , i.e. Ωrψ = Ωrϕ . Theorem 10. Suppose that a set G contains all compact positively invariant sets of the discrete-time system (2). Then the sets F −1 (G) and F −1 (G)∩G contain all compact positively invariant sets of this system as well.
Proof. Let K be a compact positively invariant set. Then K ⊂ G and F (K) ⊂ K. Hence F (K) ⊂ G; K ⊂ F −1 (G) and by Theorem 8 K ⊂ F −1 (G) ∩ G. In view of this theorem, given a localizing set for a discretetime system, we can obtain new localizing sets for compact positively invariant sets by applying shifts along the orbits of the system.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
4. THE PROOF OF THEOREM 2 The set G∞ is a closed set, because it is the intersection of closed sets. Therefore, the set G∞ is a compact set, because it is a closed subset of the compact set G. By Theorem 1 the set G∞ is a positively invariant set of the discrete-time system (2). Therefore, the set G∞ is a compact positively invariant set. If the set G is a localizing set for all positively invariant compact sets of the discrete-time system (2), then by Theorems 8, 10 the set Gk from (3) is a localizing set. ∞ T Gk = G∞ is a localizing set. Therefore, the set
5.1 Localization of compact positively invariant sets For the linear function ϕ(x, y) = Ax + By we get
k=0
As a result, we get that the set G∞ is a compact positively invariant set containing all other positively invariant compact sets of the system, i.e. the set G∞ is the maximal compact positively invariant set of the discrete-time system (2).
5. THE CHATALA SYSTEM The Chatala system (Mira et al. [1996]) is one of the discrete-time models of nonlinear dynamics exhibiting chaos. We consider this two-dimensional system in the form xn+1 = p1 xn + yn (9) yn+1 = p2 + x2n ,
p
(1 − p1 )2 − 4p2 , 2 p (1 − p1 )2 − (1 − p1 ) (1 − p1 )2 − 4p2 y1 = , 2 p 1 − p1 + (1 − p1 )2 − 4p2 x2 = , 2 p (1 − p1 )2 + (1 − p1 ) (1 − p1 )2 − 4p2 y2 = . 2 x1 =
1 − p1 −
ϕ(F (x, y)) − ϕ(x, y) = A(p1 x + y) + B(p2 + x2 ) − Ax − By = Bx2 + A(p1 − 1)x + (A − B)y + Bp2 . To find ϕrsup and ϕrinf , we have to solve two problems Ax + By → sup, Bx2 + A(p1 − 1)x + (A − B)y + Bp2 ≤ 0; (10) Ax + By → inf, Bx2 + A(p1 − 1)x + (A − B)y + Bp2 ≥ 0. Consider the case B > 0. Then we get ϕrinf = −∞. To find ϕrsup we can suppose that B = 1 (see Theorem 9), i.e. we have the problem Ax + y → sup, (11) x2 + A(p1 − 1)x + (A − 1)y + p2 ≤ 0.
where p1 , p2 ∈ R are parameters. Under p1 = 1, p2 = −0.5952 the system has an attractor (Fig. 1). This attractor is a compact positively invariant set.
Let us note the case A = 1 in which we get the problem x + y → sup, x2 + (p1 − 1)x + p2 ≤ 0. The inequality in this problem has no solutions in the case 2 (p1 −1)2 −4p2 < 0. Under this condition the set Σ− ϕ (R ) is empty and the Chatala system has no compact positively invariant sets. In the case A > 1 the problem (11) has solution A2 (A − p1 )2 − 4p2 . 4(A − 1) Hence we get the family GA of localizing sets defined by the inequality A2 (A − p1 )2 − 4p2 Ax + y ≤ , 4(A − 1) or A2 (A − p1 )2 − 4p2 y≤ − Ax, (12) 4(A − 1) where A > 1. ϕrsup =
Fig. 1. The attractor of the Chatala system In this section F : R2 → R2 is the Chatala mapping, F (x, y) = (p1 x + y, p2 + x2 )T . Note that F is not a reversible mapping. Under (p1 − 1)2 − 4p2 > 0 the Chatala system has two equilibria points (x1 , y1 ), (x2 , y2 ), where
The intersection of sets GA (Fig. 2) is described by the inequality 2 A (A − p1 )2 − 4p2 y ≤ inf − Ax , (13) A>1 4(A − 1) or, after some calculations, 2 A (A − p1 )2 − 4p2 y ≤ min − Ax . (14) A>1 4(A − 1)
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
or (p1 x + y)2 + Ax2 + Ap1 (p1 x + y) A2 (A − p1 )2 − 4p2 + (1 + A)p2 − ≤ 0. 4(A − 1) This set is bounded by the ellipse (p1 x + y)2 + Ax2 + Ap1 (p1 x + y) A2 (A − p1 )2 − 4p2 + (1 + A)p2 − = 0. 4(A − 1) Therefore we obtain a compact localization for all compact positively invariant sets of the Chatala system and this system has maximal compact positively invariant set containing in the set F −2 (GA ). Fig. 2. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the bound of the set (13)
The intersection G2 of the sets F −2 (GA ) is plotted on Fig. 4.
By Theorem 10 F −1 (GA ) is a localizing set. To find F −1 (GA ), we realize the substitution x → p1 x + y,
−4 In a similar T −4manner one can find F (GA ) and the set 4 G = F (GA ) (Fig. 5). A>1
y → p2 + x2
in (12): p2 + x2 ≤
A2 (A − p1 )2 − 4p2 − A(p1 x + y), 4(A − 1)
or in the equivalent form: y≤
A2 (A − p1 )2 − 4p2 x2 + p1 Ax + p2 − , 4A(A − 1) A
A > 1.
The intersection G1 of these sets (Fig. 3) is described by the inequality 2 A (A − p1 )2 − 4p2 x2 + p1 Ax + p2 y ≤ min − . A>1 4A(A − 1) A In a similar manner one can find the set F −2 (GA ) = F −1 (F −1 (GA )). As a result of the same substitution we obtain inequality defining the set F −2 (GA ) p2 + (p1 x + y)2 A2 (A − p1 )2 − 4p2 ≤ − Ap1 (p1 x + y) − A(p2 + x2 ), 4(A − 1)
Fig. 3. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the bound of the set G1
Fig. 4. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the localizing set G2
Fig. 5. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the localizing set G4 12524
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
By Theorems 8, 10 the sets \ F −k (GA ), Gk = F −k (GA ),
k = 2, 3, . . . ,
A>1
contain the maximal compact positively invariant set of the Chatala system. Now we can conclude that, due to Theorem 2, the maximal compact positively invariant set of the Chatala system is the limit of the sequence G0 = F −k (GA ), Gi = Gi−1 ∩ F −1 (Gi−1 ),
i = 1, 2, 3, . . . ,
where k > 1 can be chosen arbitrarily. 6. CONCLUSIONS In our paper we obtain the condition of existence of maximal compact positively invariant set of the discrete-time system. We indicate the method to construct the approximation of the maximal compact positively invariant set. This method is based on the localization of all compact positively invariant sets of the discrete-time system. It allows us to find the maximal compact positively invariant set as a limit of sequence of sets. As an example, the Chatala system is examined. REFERENCES F. Blanchini. Set invariance in control. Automatica, 35:1747–1767, 1999. L.N. Coria and K.E. Starkov. Bounding a domain containing all compact invariant sets of the permanentmagnet motor system. Commun. Nonlinear Sci. Numer. Simulat., 14:3879–3888, 2009. C. Doering and J. Gibbon. On the shape and dimension of the Lorenz attractor. Dynam. Stabil. Syst., 10:255–268, 1995. H. Giacomini and S. Neukirch. Integral of motion and the shape of the attractor for the Lorenz model. Phys. Lett. A, 240:157–150, 1997. E.G. Gilbert and K.T. Tan. Linear systems with state and control constraints: The theory and application of maximal output admissible sets. IEEE Transactions on Automatic Control, 36(9):1008–1020, 1991. A.N. Kanatnikov and A.P. Krishchenko. Localization of invariant compact sets of nonautonomous systems. Differential Equations, 45:46–52, 2009. A.N. Kanatnikov, S.K. Korovin, and A.P. Krishchenko. Localization of invariant compact sets of discrete systems. Dokl. Math., 81:326–328, 2010. A.N. Kanatnikov and A.P. Krishchenko, Localization of invariant compact sets of discrete-time systems. Int. J. Bifurcation and Chaos, submitted for publication, 2011. E.C. Kerrigan and J.M. Maciejowski. Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control Proceedings of the 39th IEEE Conference on Decision and Control, 5:4951–4956, 2000. I. Kolmanovsky and E.G. Gilbert. Theory and computation of disturbance invariance sets for discrete-time linear systems. Mathematical Problems in Engineering, 4:317–367, 1998.
A.P. Krishchenko. Localization of limit cycles. Differential Equations, 31:1826–1833, 1995. A.P. Krishchenko. Estimations of domain with cycles. Comp. Math. Appl., 34:325–332, 1997. A.P. Krishchenko. Localization of invariant compact sets of dynamical systems. Differential Equations, 41:1669– 1676, 2005. A.P. Krishchenko and K.E. Starkov. Localization of compact invariant sets of the Lorenz system. Phys. Lett. A, 353:383–388, 2006. A.P. Krishchenko and K.E. Starkov. Localization of compact invariant sets of nonlinear systems with applications to the Lanford system. Int. J. Bifurcation and Chaos, 16:3249–3256, 2006. A.P. Krishchenko and K.E. Starkov. Localization of compact invariant sets of nonlinear time-varying systems. Int. J. Bifurcation and Chaos, 18:1599–1604, 2008. G.A. Leonov, D.V. Ponomarenko, and V.B. Smirnova. Frequency-Domain Methods for Nonlinear Analysis, World Scientific, Singapore, 1996. D. Li, J. Lu, X. Wu, and G. Chen. Estimating the bounds for the Lorenz family of chaotic systems. Chaos, Solit. Fract., 23:529–534, 2005. Ch. Mira, L. Gardini, A. Barugola, and J.-C. Chatala. Chaotic dynamics in two-dimensional noninvertible maps. World Scientific, Singapore, 1996. T. McMillen. The shape and dynamics of the Rikitake attractor. The Nonlinear Journal, 1:1–10, 1998. A.Yu. Pogromsky, G. Santoboni, and H. Nijmeijer. An ultimate bound on the trajectories of the Lorenz system and its applications. Nonlinearity, 16:1597–1605, 2003. S.V. Rakovic, P. Grieder, M. Kvasnica, D.Q. Mayne, and M. Morari. Computation of Invariant Sets for Piecewise Affine Discrete-Time Systems subject to Bounded Disturbances. Proceedings of the 43rd IEEE Conference on Decision and Control, 1418–1423, 2004. S.V. Rakovic, E.C. Kerrigan, D.Q. Mayne, and J. Lygeros. Reachability Analysis of Discrete-Time Systems With Disturbances. IEEE Transactions on Automatic Control, 51(4):546–561, 2006. K.E. Starkov. Estimation of the domain containing all compact invariant sets of the optically injected laser system. Int. J. Bifurcation and Chaos, 17:4213–4217, 2007. K.E. Starkov. Bounds for compact invariant sets of the system describing dynamics of the nuclear spin generator. Commun. Nonlinear Sci. Numer. Simulat., 14:2565– 2570, 2009a. K.E. Starkov. Bounds for a domain containing all compact invariant sets of the system describing the laser-plasma interaction. Chaos, Solit. Fract., 39:1671–1676, 2009b. K.E. Starkov. Bounding a domain that contains all compact invariant sets of the Bloch system. Int. J. Bifurcation and Chaos, 19:1037–1042, 2009c. K.E. Starkov. Compact invariant sets of the static spherically symmetric Einstein–Yang–Mills equations. Phys. Lett. A, 374:1728–1731, 2010. P. Swinnerton-Dyer. Bounds for trajectories of the Lorenz system: an illlustration of how to choose Liapunov functions. Phys. Lett. A, 281:161–167, 2001.
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Maximal compact positively invariant sets of discrete-time nonlinear systems ? Alexander P. Krishchenko ∗ Anatoly N. Kanatnikov ∗∗ ∗
Bauman Moscow State Technical University, Moscow, 105005 Russia (e-mail: [email protected]) ∗∗ Bauman Moscow State Technical University, Moscow, 105005 Russia (e-mail: [email protected]) Abstract: In this paper we examine the existence problem for maximal compact positively invariant sets of discrete-time nonlinear systems. To find these sets we propose the method based on the localization procedure of compact positively invariant sets. Analysis of a location of compact positively invariant sets and maximal compact positively invariant set of the Chatala system is realized for all values of its parameters. Keywords: Discrete-time system, maximal positively invariant set, localization 1. INTRODUCTION During the past years, the interest of many researchers has been attracted to the idea of finding some geometrical bounds for attractors, periodical orbits and chaotic dynamics of a nonlinear autonomous differentiable right-side system x˙ = f (x), f ∈ C 1 (Rn ). (1)
with the continuous right-side was proposed in Kanatnikov et al. [2010, 2011]. Using results obtained in these papers we suggest a method to find the maximal compact positively invariant sets of discrete-time nonlinear systems. In this paper, when we talk about a localization problem for the discrete-time system (2) we have in mind the same problem as in the case of continuous-time systems: find the localizing set.
Mainly, in the presented literature this problem was solved by Lyapunov-type functions, see e.g. articles of Doering & Gibbon [1995], Leonov et al. [1996], McMillen [1998], Swinnerton-Dyer [2001], Pogromsky et al. [2003], Li et al. [2005]. The method for finding families of semipermeable surfaces was proposed in Giacomini & Neukirch [1997] and then developed in other publications of these authors. The functional method was proposed in Krishchenko [1995, 1997] to the localization problem of periodical solutions of the system (1) and developed in Coria & Starkov [2009], Kanatnikov & Krishchenko [2009], Krishchenko [2005], Krishchenko & Starkov [2006a,b, 2008], Starkov [2007, 2009a,b,c, 2010] in the case of compact invariant sets.
The set V ⊂ Q ⊂ M is the localizing set for compact positively invariant sets of system (2) contained in Q if and only if V contains all compact positively invariant sets of system (2) contained in Q. The localizing set of a discrete-time system (2) is a localizing set for compact positively invariant sets of system (2) contained in M .
The theory of set invariance plays a fundamental role in the control of constrained discrete-time systems and has been a subject of research by many authors — see for instance Gilbert & Tan [1991], Blanchini [1999], Kolmanovsky & Gilbert [1998], Kerrigan & Maciejowski [2000], Rakovic et al. [2004, 2006] and the references therein. In this paper we consider the case of compact positively invariant sets.
In Sec. 2 and 4 we prove the main results describing the maximal compact positively invariant sets.
The localization method of compact invariant sets in the case of discrete-time nonlinear systems
In Sec. 5 we show the computation of localizing sets of the Chatala system and find outer approximations of the attractor of this system. Sec. 6 contains conclusions.
xk+1 = F (xk ),
F : M → M,
M ⊂ Rn
(2)
? This work was supported by the Russian Foundation for Basic Research, (projects 11-01-00733 and 09-07-00327) and the program of the President of the Russian Federation for state support of leading scientific schools (grant NSh-4144.2010.1)
978-3-902661-93-7/11/$20.00 © 2011 IFAC
This paper presents the methods for computation of a localizing set as well as the computation of decrescent outer approximations of the maximal compact positively invariant set. These approximations are achieved by computing a sequence of compact localizing sets. This paper is organized as follows.
Sec. 3 contains results of localization method in the case of discrete-time nonlinear systems. These results point out how to find the localizing sets of a discretetime nonlinear system and the localizing sets for compact positively invariant sets of a discrete-time nonlinear system contained in some subset of state space.
2. MAIN RESULTS A subset K ⊂ M is said to be positively invariant for the discrete-time nonlinear system (2) if F (K) ⊂ K.
12521
10.3182/20110828-6-IT-1002.00521
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
The positively invariant sets of a discrete-time system include equilibria (stationary points), periodic orbits, and attractors. Let G be any set in M , we introduce G0 = G,
Gi = Gi−1 ∩ F −1 (Gi−1 ),
and G∞ =
∞ \
i = 1, 2, . . . , (3)
Proof. Let K ⊂ Q be a compact positively invariant set. The continuous function ϕ(x) attains its maximum on K at some point x∗ ∈ K ⊂ Q since K is a compact set. Then F (x∗ ) ∈ K as K is a positively invariant set. As a result, ϕ(F (x∗ )) ≤ ϕ(x∗ ), i.e. ϕ(F (x∗ )) − ϕ(x∗ ) ≤ 0, ∗ and x∗ ∈ Σ− ϕ (Q) because x ∈ Q. Therefore for any point x ∈ K we get ϕ(x) ≤ ϕ(x∗ ) ≤
Gi .
sup x∈Σ− ϕ (Q)
i=0
ϕ(x) = ϕrsup (Q).
(7)
In the same way, for any point x ∈ K we get the inequalities
For these sets we get G = G0 ⊃ G1 ⊃ · · · ⊃ Gi ⊃ . . . ; x ∈ G∞ iff x ∈ Gi−1 and x ∈ F −1 (Gi−1 ), i = 1, 2, . . . Theorem 1. The set G∞ is a positively invariant set of the discrete-time system (2).
ϕ(x) ≥ ϕ(x∗ ) ≥
inf
x∈Σ+ ϕ (Q)
ϕ(x) = ϕrinf (Q),
(8)
where x∗ = argmin ϕ(x). Inequalities (7), (8) mean that x∈K
Proof. Let x ∈ G∞ . Then for any i > 0 we have x ∈ F −1 (Gi−1 ), i.e. F (x) ∈ Gi−1 . Therefore, F (x) ∈ ∞ T Gi−1 = G∞ , i.e. the set G∞ is a positively invariant i=1
set of the discrete-time system (2). The compact set V ⊂ G ⊂ M is the maximal compact positively invariant set contained in G if and only if V is positively invariant and contains all compact positively invariant sets contained in G. In the case G = M the compact set V is the maximal compact positively invariant set of system (2). Theorem 2. If the set G is a compact localizing set for all compact positively invariant sets of the discrete-time system (2), then G∞ is the maximal compact positively invariant set of this discrete-time system. Theorem 3. If the set G is a compact localizing set for compact positively invariant sets of the discrete-time system (2) contained in Q, then G∞ is the maximal compact positively invariant set contained in Q. Corollary 4. The discrete-time system (2) has the maximal compact positively invariant set iff this system has a compact localizing set for all compact positively invariant sets.
the compact positively invariant set K is contained in the set (6). Corollary 6. If the set G = Ωrϕ (Q) (6) is a compact set, then G∞ is the maximal compact positively invariant set contained in Q. − Corollary 7. If Σ+ ϕ (Q) = ∅ or Σϕ (Q) = ∅ then the discrete-time system (2) has no compact positively invariant sets contained in Q. Theorem 8. Let ϕα , α ∈ I, be a family of functions defined on M , ϕα ∈ C(Q), Q ⊂ M . Then each compact positively invariant set of the discrete-time T r system (2) contained in Q is contained in the set Ωϕα (Q). α∈I
Theorem 9. Let ϕ be a function defined on M and continuous on Q ⊂ M . If ψ(x) = h(ϕ(x)), where h : R → R is a strictly monotone function, then Ωrψ (Q) = Ωrϕ (Q). Particularly, this equality holds for h(t) = at + b, a 6= 0. Proof. Let us proof the equality Ωrψ (Q) = Ωrϕ (Q) in the case where h : R → R is a strictly increasing function. For strictly increasing function h the inequality ϕ(x1 ) ≤ ϕ(x2 ) is equivalent to ψ(x1 ) ≤ ψ(x2 ). Hence the set Σ− ψ (Q) is equal to the set Σ− (Q) and we get ϕ r ψsup =
To prove Theorems 2, 3 and to find G∞ we need some results of the localization method.
and define inf
ϕ(x),
ϕrsup (Q) =
sup
sup
h(ϕ(x))
x∈Σ− ϕ (Q)
sup x∈Σ− ϕ (Q)
Let Q be a subset of M . For a continuous function ϕ(x) defined on M we introduce the sets Σ+ ϕ (Q) = {x ∈ Q : ϕ(F (x)) − ϕ(x) ≥ 0} , (4) Σ− ϕ (Q) = {x ∈ Q : ϕ(F (x)) − ϕ(x) ≤ 0}
x∈Σ+ ϕ (Q)
ψ(x) =
=h
3. THE LOCALIZATION METHOD
ϕrinf (Q) =
sup x∈Σ− (Q) ψ
ϕ(x). (5)
x∈Σ− ϕ (Q)
Theorem 5. Each compact positively invariant set of the discrete-time system (2) contained in Q is contained in the set Ωrϕ (Q) = x ∈ Q : ϕrinf (Q) ≤ ϕ(x) ≤ ϕrsup (Q) , (6) i.e. the set Ωrϕ (Q) is a localizing set for compact positively invariant sets of system (2) contained in Q.
ϕ(x) = h ϕrsup .
r By the same way we obtain ψinf = h(ϕrinf ). Therefore the inequalities ψinf ≤ ψ(x) ≤ ψsup are equivalent to ϕinf ≤ ϕ(x) ≤ ϕsup , i.e. Ωrψ = Ωrϕ . Theorem 10. Suppose that a set G contains all compact positively invariant sets of the discrete-time system (2). Then the sets F −1 (G) and F −1 (G)∩G contain all compact positively invariant sets of this system as well.
Proof. Let K be a compact positively invariant set. Then K ⊂ G and F (K) ⊂ K. Hence F (K) ⊂ G; K ⊂ F −1 (G) and by Theorem 8 K ⊂ F −1 (G) ∩ G. In view of this theorem, given a localizing set for a discretetime system, we can obtain new localizing sets for compact positively invariant sets by applying shifts along the orbits of the system.
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4. THE PROOF OF THEOREM 2 The set G∞ is a closed set, because it is the intersection of closed sets. Therefore, the set G∞ is a compact set, because it is a closed subset of the compact set G. By Theorem 1 the set G∞ is a positively invariant set of the discrete-time system (2). Therefore, the set G∞ is a compact positively invariant set. If the set G is a localizing set for all positively invariant compact sets of the discrete-time system (2), then by Theorems 8, 10 the set Gk from (3) is a localizing set. ∞ T Gk = G∞ is a localizing set. Therefore, the set
5.1 Localization of compact positively invariant sets For the linear function ϕ(x, y) = Ax + By we get
k=0
As a result, we get that the set G∞ is a compact positively invariant set containing all other positively invariant compact sets of the system, i.e. the set G∞ is the maximal compact positively invariant set of the discrete-time system (2).
5. THE CHATALA SYSTEM The Chatala system (Mira et al. [1996]) is one of the discrete-time models of nonlinear dynamics exhibiting chaos. We consider this two-dimensional system in the form xn+1 = p1 xn + yn (9) yn+1 = p2 + x2n ,
p
(1 − p1 )2 − 4p2 , 2 p (1 − p1 )2 − (1 − p1 ) (1 − p1 )2 − 4p2 y1 = , 2 p 1 − p1 + (1 − p1 )2 − 4p2 x2 = , 2 p (1 − p1 )2 + (1 − p1 ) (1 − p1 )2 − 4p2 y2 = . 2 x1 =
1 − p1 −
ϕ(F (x, y)) − ϕ(x, y) = A(p1 x + y) + B(p2 + x2 ) − Ax − By = Bx2 + A(p1 − 1)x + (A − B)y + Bp2 . To find ϕrsup and ϕrinf , we have to solve two problems Ax + By → sup, Bx2 + A(p1 − 1)x + (A − B)y + Bp2 ≤ 0; (10) Ax + By → inf, Bx2 + A(p1 − 1)x + (A − B)y + Bp2 ≥ 0. Consider the case B > 0. Then we get ϕrinf = −∞. To find ϕrsup we can suppose that B = 1 (see Theorem 9), i.e. we have the problem Ax + y → sup, (11) x2 + A(p1 − 1)x + (A − 1)y + p2 ≤ 0.
where p1 , p2 ∈ R are parameters. Under p1 = 1, p2 = −0.5952 the system has an attractor (Fig. 1). This attractor is a compact positively invariant set.
Let us note the case A = 1 in which we get the problem x + y → sup, x2 + (p1 − 1)x + p2 ≤ 0. The inequality in this problem has no solutions in the case 2 (p1 −1)2 −4p2 < 0. Under this condition the set Σ− ϕ (R ) is empty and the Chatala system has no compact positively invariant sets. In the case A > 1 the problem (11) has solution A2 (A − p1 )2 − 4p2 . 4(A − 1) Hence we get the family GA of localizing sets defined by the inequality A2 (A − p1 )2 − 4p2 Ax + y ≤ , 4(A − 1) or A2 (A − p1 )2 − 4p2 y≤ − Ax, (12) 4(A − 1) where A > 1. ϕrsup =
Fig. 1. The attractor of the Chatala system In this section F : R2 → R2 is the Chatala mapping, F (x, y) = (p1 x + y, p2 + x2 )T . Note that F is not a reversible mapping. Under (p1 − 1)2 − 4p2 > 0 the Chatala system has two equilibria points (x1 , y1 ), (x2 , y2 ), where
The intersection of sets GA (Fig. 2) is described by the inequality 2 A (A − p1 )2 − 4p2 y ≤ inf − Ax , (13) A>1 4(A − 1) or, after some calculations, 2 A (A − p1 )2 − 4p2 y ≤ min − Ax . (14) A>1 4(A − 1)
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
or (p1 x + y)2 + Ax2 + Ap1 (p1 x + y) A2 (A − p1 )2 − 4p2 + (1 + A)p2 − ≤ 0. 4(A − 1) This set is bounded by the ellipse (p1 x + y)2 + Ax2 + Ap1 (p1 x + y) A2 (A − p1 )2 − 4p2 + (1 + A)p2 − = 0. 4(A − 1) Therefore we obtain a compact localization for all compact positively invariant sets of the Chatala system and this system has maximal compact positively invariant set containing in the set F −2 (GA ). Fig. 2. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the bound of the set (13)
The intersection G2 of the sets F −2 (GA ) is plotted on Fig. 4.
By Theorem 10 F −1 (GA ) is a localizing set. To find F −1 (GA ), we realize the substitution x → p1 x + y,
−4 In a similar T −4manner one can find F (GA ) and the set 4 G = F (GA ) (Fig. 5). A>1
y → p2 + x2
in (12): p2 + x2 ≤
A2 (A − p1 )2 − 4p2 − A(p1 x + y), 4(A − 1)
or in the equivalent form: y≤
A2 (A − p1 )2 − 4p2 x2 + p1 Ax + p2 − , 4A(A − 1) A
A > 1.
The intersection G1 of these sets (Fig. 3) is described by the inequality 2 A (A − p1 )2 − 4p2 x2 + p1 Ax + p2 y ≤ min − . A>1 4A(A − 1) A In a similar manner one can find the set F −2 (GA ) = F −1 (F −1 (GA )). As a result of the same substitution we obtain inequality defining the set F −2 (GA ) p2 + (p1 x + y)2 A2 (A − p1 )2 − 4p2 ≤ − Ap1 (p1 x + y) − A(p2 + x2 ), 4(A − 1)
Fig. 3. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the bound of the set G1
Fig. 4. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the localizing set G2
Fig. 5. The attractor of the Chatala system with p1 = 1, p2 = −0,5952 and the localizing set G4 12524
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
By Theorems 8, 10 the sets \ F −k (GA ), Gk = F −k (GA ),
k = 2, 3, . . . ,
A>1
contain the maximal compact positively invariant set of the Chatala system. Now we can conclude that, due to Theorem 2, the maximal compact positively invariant set of the Chatala system is the limit of the sequence G0 = F −k (GA ), Gi = Gi−1 ∩ F −1 (Gi−1 ),
i = 1, 2, 3, . . . ,
where k > 1 can be chosen arbitrarily. 6. CONCLUSIONS In our paper we obtain the condition of existence of maximal compact positively invariant set of the discrete-time system. We indicate the method to construct the approximation of the maximal compact positively invariant set. This method is based on the localization of all compact positively invariant sets of the discrete-time system. It allows us to find the maximal compact positively invariant set as a limit of sequence of sets. As an example, the Chatala system is examined. REFERENCES F. Blanchini. Set invariance in control. Automatica, 35:1747–1767, 1999. L.N. Coria and K.E. Starkov. Bounding a domain containing all compact invariant sets of the permanentmagnet motor system. Commun. Nonlinear Sci. Numer. Simulat., 14:3879–3888, 2009. C. Doering and J. Gibbon. On the shape and dimension of the Lorenz attractor. Dynam. Stabil. Syst., 10:255–268, 1995. H. Giacomini and S. Neukirch. Integral of motion and the shape of the attractor for the Lorenz model. Phys. Lett. A, 240:157–150, 1997. E.G. Gilbert and K.T. Tan. Linear systems with state and control constraints: The theory and application of maximal output admissible sets. IEEE Transactions on Automatic Control, 36(9):1008–1020, 1991. A.N. Kanatnikov and A.P. Krishchenko. Localization of invariant compact sets of nonautonomous systems. Differential Equations, 45:46–52, 2009. A.N. Kanatnikov, S.K. Korovin, and A.P. Krishchenko. Localization of invariant compact sets of discrete systems. Dokl. Math., 81:326–328, 2010. A.N. Kanatnikov and A.P. Krishchenko, Localization of invariant compact sets of discrete-time systems. Int. J. Bifurcation and Chaos, submitted for publication, 2011. E.C. Kerrigan and J.M. Maciejowski. Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control Proceedings of the 39th IEEE Conference on Decision and Control, 5:4951–4956, 2000. I. Kolmanovsky and E.G. Gilbert. Theory and computation of disturbance invariance sets for discrete-time linear systems. Mathematical Problems in Engineering, 4:317–367, 1998.
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