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Nonlinear optimal stabilization of unstable periodic orbits
4th IFAC Conference on Analysis and Control of Chaotic Systems 4th IFAC26-28, Conference on Analysis Control of Chaotic Systems August 2015. Tokyo, Japan and 4th IFAC IFAC Conference Conference on Analysis Analysis and Control Control of of Chaotic Chaotic Systems Systems 4th on August 26-28, 2015. Tokyo, Japan andAvailable online at www.sciencedirect.com August 26-28, 2015. Tokyo, Japan August 26-28, 2015. Tokyo, Japan
ScienceDirect IFAC-PapersOnLine 48-18 (2015) 215–220
Nonlinear optimal stabilization of unstable Nonlinear stabilization of of unstable Nonlinear optimal optimal stabilization unstable periodic orbits periodic orbits periodic orbits ∗∗∗ Yuuta Habaguchi ∗∗ Noboru Sakamoto ∗∗ ∗∗ Keisuke Nagata ∗∗∗ Yuuta Habaguchi ∗ Noboru Sakamoto ∗∗ ∗∗ Keisuke Nagata ∗∗∗ ∗∗∗ ∗ Yuuta Habaguchi Noboru Sakamoto Keisuke Nagata Yuuta Habaguchi Noboru Sakamoto Keisuke Nagata ∗ ∗ Department of Aerospace Engineering, Graduate School of of Aerospace Engineering, Graduate School of ∗ ∗ Department Department of Engineering, School of Engineering, Nagoya University, Furo-cho, Graduate Chikusa-ku, Nagoya, Department of Aerospace Aerospace Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Engineering, Nagoya University, Furo-cho, Furo-cho, Chikusa-ku, Chikusa-ku, Nagoya, Nagoya, Japan, (e-mail: [email protected]) Engineering, Nagoya University, (e-mail: [email protected]) ∗∗Japan, Japan, (e-mail: [email protected]) Department of Aerospace Engineering, Graduate School of (e-mail: [email protected]) ∗∗Japan, Department of Aerospace Engineering, Graduate School of ∗∗ ∗∗ Department of Aerospace Engineering, School of Engineering, Nagoya University, Furo-cho, Graduate Chikusa-ku, Nagoya, Department of Aerospace Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, (e-mail: [email protected]) Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, [email protected]) ∗∗∗ Japan, (e-mail: (e-mail: [email protected]) and Aerospace Engineering, Graduate School of Japan, (e-mail: [email protected]) ∗∗∗ Mechanical ∗∗∗ Mechanical and Aerospace Engineering, Graduate School of ∗∗∗ Mechanical and Aerospace Aerospace Engineering, Graduate School School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Mechanical and Engineering, Graduate of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, (e-mail: [email protected]) Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, (e-mail: [email protected]) Japan, Japan, (e-mail: (e-mail: [email protected]) [email protected])
Abstract: The framework for the optimal stabilization of unstable periodic orbits in nonlinear Abstract: The framework for the optimal stabilization of unstable periodic orbits in nonlinear Abstract: The optimal stabilization unstable periodic nonlinear systems is proposed. First, for thethe stable manifold methodof for approximately solvingin HamiltonAbstract: The framework framework for the optimal stabilization offor unstable periodic orbits orbits inHamiltonnonlinear systems is proposed. First, the stable manifold method approximately solving systems is proposed. proposed. First, the stable stable manifold method for approximately approximately solving HamiltonJacobi equations in the optimal control problemmethod is extended to nonlinear solving periodicHamiltonsystems. systems is First, the manifold for Jacobi equations in the optimal control problem is extended to nonlinear periodic systems. Jacobi equations optimal is extended periodic systems. Then, this methodin isthe applied for control optimalproblem stabilization problemto ofnonlinear error dynamics around an Jacobi equations in the optimal control problem is extended to nonlinear periodic systems. Then, this method is applied for optimal stabilization problem of error dynamics around an Then, this method is applied for optimal stabilization problem of error dynamics an unstable periodic orbit. The effectiveness of the proposed method is evaluated via around numerical Then, this method is applied for optimal stabilization problem of error dynamics around an unstable periodic orbit. The effectiveness of unstable periodic orbit. The effectiveness of the the proposed proposed method method is is evaluated evaluated via via numerical numerical simulations for theorbit. Rossler chaotic system. of unstable periodic The effectiveness the proposed method is evaluated via numerical simulations for the Rossler chaotic system. simulations for for the the Rossler Rossler chaotic chaotic system. system. simulations © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear optimal control, unstable periodic orbit, chaos control, Rossler chaotic Keywords: Nonlinear control, unstable periodic orbit, Keywords: Nonlinear optimal optimal control, unstable periodic orbit, chaos chaos control, control, Rossler Rossler chaotic chaotic system, Hamilton-Jacobi equation, stable manifold methods Keywords: Nonlinear optimal control, unstable periodic orbit, chaos control, Rossler chaotic system, Hamilton-Jacobi equation, stable manifold methods system, Hamilton-Jacobi Hamilton-Jacobi equation, equation, stable stable manifold manifold methods methods system, 1. INTRODUCTION tem has delay terms, analysis of the system controlled by 1. tem has terms, analysis system by 1. INTRODUCTION INTRODUCTION tem has delay delay terms, analysis of of the theHowever, system controlled controlled by the method is terms, not straightforward. the analysis 1. INTRODUCTION tem has delay analysis of the system controlled by the method is not straightforward. However, the analysis Chaos has been observed in various areas in physics, the the method is not not straightforward. However, the analysis method in the vicinity of the periodic orbits is proposed method is straightforward. However, the analysis Chaos has observed in areas physics, in the vicinity of periodic orbits is Chaos has been been observed in asvarious various areas in in celestial physics, method engineering and biology such fluid dynamics, method in al. the(1997), vicinityand of the the periodic orbits is proposed proposed in Just et the periodic optimal orbits gain tuning using Chaos has been observed in various areas in physics, method in the vicinity of the is proposed engineering and biology such as fluid dynamics, celestial in Just et al. (1997), and the optimal gain tuning using engineering and physiology, biology such such as fluid fluidthe dynamics, celestial mechanics, and therefore control of chaos in in Just et al. (1997), and the optimal gain tuning using linearization around the target orbits is reported in Biggs engineering and biology as dynamics, celestial Just et al. (1997), and the optimal gain tuning using mechanics, and therefore the control of around the mechanics, and physiology, physiology, therefore the controlthere of chaos chaos is an imperative subject. In chaotic the systems, are linearization linearization around the target target orbits orbits is is reported reported in in Biggs Biggs and R.McInnes (2009). mechanics, and physiology, therefore control of chaos linearization around the target orbits is reported in Biggs is an imperative subject. In chaotic systems, there are and R.McInnes (2009). is an imperative subject. In chaotic systems, there are interesting regionssubject. in which achaotic large number of there unstable and R.McInnes R.McInnes (2009). (2009). is an imperative In systems, are and interesting regions in which a large number of Although chaotic systems are nonlinear dynamics, there interesting regions in and which large numberthat of unstable unstable periodic orbits exist, it aais large well-known some of Although interesting regions in which number of unstable chaotic are dynamics, there periodic orbits exist, and it is well-known that some of Although chaotic systems systems are nonlinear nonlinear dynamics, there exists no method to explicitly consider dynamics, or make use of Although chaotic systems are nonlinear there periodic orbits exist,may andbeit itcapable is well-known well-known that some some of exists the unstable orbits of improving system periodic orbits exist, and is that of no method to explicitly consider or make use of the unstable orbits may be capable of improving system exists no method to explicitly consider or make use of the nonlinearity. Since the OGY method is based on exists no method to explicitly consider or make use of the unstable according orbits may maytobe besome capable of improving improving system performance criteria. Stabilizing such the nonlinearity. Since the OGY method is based on the unstable orbits capable of system performance according to some criteria. Stabilizing such the nonlinearity. Since the OGY method is based on linearized models around target orbits, the nonlinearity of the nonlinearity. Since the OGY method is based on performance according to some criteria. Stabilizing such orbits can be according beneficial to in many fields. From this respect, performance some fields. criteria. Stabilizing such linearized models around target orbits, the of orbits can be beneficial in many From this respect, linearized models around target orbits, the nonlinearity nonlinearity of the systemmodels is ignored in the control design. Furthermore, linearized around target orbits, the nonlinearity of orbits can be beneficial in many fields. From this respect, Ott et al. (1990) proposed a method to control chaotic orbits can be beneficial in many fields. From this respect, the is ignored in control design. Furthermore, Ott et (1990) proposed aa method to control chaotic the system system isgain ignored in the the control design. Furthermore, optimalis tuning method fordesign. TDFCFurthermore, do not take system ignored in the control Ott et al. al. (1990) the proposed method to to control chaotic systems utilizing sensitivity of chaos initialchaotic condi- the Ott et al. (1990) proposed a method to control the optimal gain tuning method for TDFC do not take systems utilizing the sensitivity of chaos to initial condithe optimal gain tuning method for TDFC do not take nonlinearity into account in that it solves thenot optimal the optimal gain tuning method for TDFC do take systems utilizing thechaos sensitivity ofhas chaos to initial initial condi- the nonlinearity into account in that it solves the tions andutilizing since then controlof attracted attention systems the sensitivity chaos to condioptimal tions and since then chaos control has attracted attention the nonlinearity into account in that it solves the optimal control problem for the linearized model. The chaotic the nonlinearity into account in that it solves the optimal tions and since then chaos control has attracted attention of many researchers. Theircontrol methodology is called OGY control problem for the linearized model. The chaotic tions and researchers. since then chaos has attracted attention of many Their methodology is OGY control problemsuch for as thethe linearized model. The chaotic chaotic characteristics, sensitivity on initial states, problem for the linearized model. The of manyand researchers. Their methodology isofcalled called OGY method became one of the major fieldsis ChaosOGY con- control of many researchers. Their methodology called characteristics, such as the sensitivity on initial states, method and became one of the major fields of Chaos concharacteristics, such as the sensitivity on initial states, which should be taken advantage of, as Ott et al. (1990) characteristics, such as the sensitivity on initial states, method and became one of oflittle the major major fields of Chaos Chaosabout con- which should be taken advantage of, as Ott et al. (1990) trol. Thisand method requires a priorifields information method became one the of control. This method requires little a priori information about which should be taken advantage of, as Ott et al. (1990) indicated, are created from its nonlinearity, and therefore, which should be taken advantage of, as Ott et al. (1990) trol. This method method requires little priori information information about controlled systemsrequires and haslittle beenaasuccessfully used in about many indicated, are created from its nonlinearity, and therefore, trol. This priori controlled systems and been successfully used in many indicated, are considering created from fromthe its nonlinearity nonlinearity, in andthe therefore, by explicitly control are created its nonlinearity, and therefore, controlled systems and has has been successfully used in many indicated, contexts (see Garfinkel et al. (1992); Schiff et al. in (1994)). controlled systems and has been successfully used many by explicitly considering the nonlinearity in the control contexts (see Garfinkel et al. (1992); Schiff et al. (1994)). by explicitly considering the nonlinearity in the control design, more effective chaos control than ever can be by explicitly considering the nonlinearity in the control contexts (see Garfinkel et al. (1992); Schiff et al. (1994)). The optimality of the method in convergence time is shown contexts (see Garfinkel et al. in (1992); Schiff et al. is (1994)). design, more effective chaos control than ever can be The optimality of the method convergence time shown design, more effective chaos control than ever can be achieved. design, more effective chaos control than ever can be The optimality ofal.the the method in convergence convergence time is is shown shown in Romeiras et of (1992) and, moreover, Epureanu and achieved. The optimality method in time in Romeiras et al. and, moreover, Epureanu and achieved. in Romeiras etproposes al. (1992) (1992) and, moreover, Epureanu and achieved. Dowell (2000)et an and, optimal control Epureanu scheme which in Romeiras al. (1992) moreover, and On the other hand, the research on control methods that Dowell (2000) proposes an optimal control scheme which the other hand, the on methods that Dowell (2000) proposes an optimal optimal control scheme which minimizes the proposes effort required in control. While the which OGY On On the other hand, the research research on control control methods that Dowell (2000) an control scheme make full use of nonlinearity has been attracting attention On the other hand, the research on control methods that minimizes the effort required in control. While the OGY full use of nonlinearity has been attracting attention minimizes the the effort required in control. While the to OGY method gives control law in only near the orbits be make make full use of nonlinearity has been attracting attention minimizes the effort required control. While the OGY in the areas of control theory/engineering. Among the make full use of nonlinearity has been attracting attention method gives the control only near orbits to the of control Among the method gives thestabilization control law law is only near the the orbits to be be in stabilized, global proposed using targeting in the areas areas of manifold control theory/engineering. theory/engineering. Among the method gives the control law only near the orbits to be methods, stable method developed byAmong Sakamoto in the areas of control theory/engineering. the stabilized, global stabilization is proposed using targeting methods, stable manifold method developed by Sakamoto stabilized, global stabilization is proposed proposed using targeting method in global Shinbrot et al. (1990); Kostelichusing et al.targeting (1993). methods, stable manifold method developed bydesigns Sakamoto stabilized, stabilization is and van der Schaft (2008) is thedeveloped one thatby the methods, stable manifold method Sakamoto method in Shinbrot et al. (1990); Kostelich et al. (1993). and van der Schaft (2008) is the one that designs the method in Shinbrot et al. (1990); Kostelich et al. (1993). and van der Schaft (2008) is the one that designs the method in Shinbrot et al. (1990); Kostelich et al. (1993). optimal control law taking into account of nonlinearities in and van der Schaft (2008) is the one that designs the On the other hand, Time Delayed Feedback Control optimal control law taking into account of nonlinearities in On the other hand, Time Delayed Feedback Control optimal control law taking into account of nonlinearities in the system, and have succeeded in the swing up control of optimal control law taking into account of nonlinearities in On the other hand, Time Delayed Feedback Control (TDFC) introduced by Pyragas (1992) constitutes another On the introduced other hand, Time Delayed Feedback another Control the system, and succeeded in swing up control of (TDFC) by Pyragas (1992) constitutes the system, and have have succeeded in the the swingof upthe control of inverted pendulum and the optimal control system the system, and have succeeded in the swing up control of (TDFC) introduced by Pyragas (1992) constitutes another major field in chaos control. It is able to stabilize a lot of (TDFC) introduced by Pyragas (1992) constitutes another inverted pendulum and optimal control of major field in control. It to stabilize aa lot inverted pendulum and the the optimal control (Yuasa of the the system system with input saturations andoptimal rate limiters et al. pendulum and the control of the system major field in chaos chaos control. It is is able able tosimple stabilize lot of of inverted unstable periodic orbits in spite of itsto algorithm major field in chaos control. It is able stabilize a lot of with input saturations and rate limiters (Yuasa et unstable periodic orbits in spite of its simple algorithm with input saturations and rate limiters (Yuasa et al. al. (2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). with input saturations and rate limiters (Yuasa et al. unstable periodicsignal orbitscomputed in spite spite from of its its the simple algorithm using a control difference be- (2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). unstable periodic orbits in of simple algorithm using a control signal computed from the difference be(2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). In the present paper, we propose a method designing (2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). using a control signal computed from the difference between athe current and computed delayed states (Postlethwaite and using control signal from(Postlethwaite the difference and be- In the paper, we propose a method tween the and delayed states In the present present paper, we law propose method designing designing nonlinear optimal control whichaa stabilizes unstable the present paper, we propose method designing tween the current current and delayed states (Postlethwaite and In Silber (2007); Ding and et al.delayed (2010)).states Since (Postlethwaite the closed loop systween the current and nonlinear optimal control law which stabilizes unstable Silber (2007); Ding et al. (2010)). Since the closed loop sysnonlinear optimal control law which stabilizes unstable nonlinear optimal control law which stabilizes unstable Silber (2007); (2007); Ding Ding et et al. al. (2010)). (2010)). Since Since the the closed closed loop loop syssysSilber Copyright 2015 IFAC 215 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 215 Copyright 2015 IFAC 215 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 215Control. 10.1016/j.ifacol.2015.11.039
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periodic orbits in chaotic systems using the stable manifold method. Here, our goal is to minimize input effort during the stabilization, which would lead an extension of the OGY method by introducing optimality with respect to a suitable cost function. The organization of the paper is as follows. In §2, we give the definition of the optimal control problem for nonlinear periodic systems and demonstrate that the problem attributes to obtaining the stable manifold of a time-periodic canonical equation. In §3, the method of calculating the stable manifold and important theories for the method is described. In §4, the proposed method of optimally stabilizing an unstable periodic orbit is applied to Rossler chaotic system. 2. OPTIMAL CONTROL OF PERIODIC NONLINEAR SYSTEMS Let us consider a nonlinear control system of the form Σ : x˙ = f (t, x) + g(t, x)u (1) n m n where x(t) ∈ R , u(t) ∈ R and f : R × R → Rn , g : R × Rn → Rn×m are C r (r ⩾ 2) functions. We also assume that f and g are periodic functions with respect to t with period T and that f (t, 0) = 0 for all t ∈ R. For system Σ, we seek a control law that minimizes a cost function ∫ ∞ J= xT Qx + uT Ru dt, (2) 0
where Q and R are symmetric with appropriate dimension and R is a positive definite matrix. More precisely, we define our optimal control problem as follows. Definition 2.1. (Stable regulator problem). Find a control law u(t) that minimizes the cost function J in (2) among all locally uniformly asymptotically stabilizing controls for Σ in (1) at x = 0.
An important application of the stable regulator problem for a nonlinear periodic system is the optimal stabilization of a periodic orbit in a nonlinear system, from which we would like to derive an optimal OGY control for chaotic systems. Let us consider a control system with a periodic solution system : q˙ = a(q) + b(q)u, (3) periodic solution: q˙c = a(qc ), qc (t + T ) = qc (t), (4) where q, qc (t) ∈ Rn , T is the period of the periodic solution qc and u(t) ∈ Rm is an input. It is also assumed that a(·) : Rn → Rn , b(·) : Rn → Rn×m are smooth vector fields. Definition 2.2. (Stabilization problem for a periodic orbit). A state feedback control u = u(q) is said to be an asymptotically stabilizing control for the periodic orbit qc in system (3) if u satisfies u(qc (t)) = 0 for all t and qc is an asymptotically stable periodic orbit of the closed loop system. We wish to find a stabilizing control of the periodic orbit qc in an optimal way. To define the optimal stabilization problem, first, introduce error x := q − qc and then derive the error system x˙ = q˙ − q˙c = a(x + qc (t)) − a(qc (t)) + b(x + qc (t))u. 216
If we set f (t, x) := a(x + qc (t)) − a(qc (t)), g(t, x) := b(x + qc (t)), then, the optimal stabilization problem of the periodic orbit amounts to the optimal regulator problem (1) and (2) with appropriate weighting matrices Q, R. Now, we will derive a Hamilton-Jacobi equation (HJE) for the stable regulator problem. Define the Hamiltonian for the optimal control problem for Σ and (2) by HD = pT {f (t, x) + g(t, x)u} + xT Qx + uT Ru. Since R > 0, 1 u ¯(t, x, p) := arg min HD (t, x, p, u) = − R−1 g(t, x)T p m 2 u∈R and let ¯(t, x, p)). (5) H(t, x, p) := HD (t, x, p, u Then, the Hamilton-Jacobi equation associated with the optimal control problem in Definition 2.1 is ( ) ∂V ∂V = H t, x, − ∂t ∂x ∂V T 1 ∂V ∂V f (t, x) − g(t, x)R−1 g(t, x)T = + xT Qx. ∂x 4 ∂x ∂x (6) It can be readily seen that if (6) has a periodic solution V (t, x) with period T defined for all t and for x in a neighborhood of x = 0 with V (t, 0) = 0 such that 1 ∂V T x˙ = f (t, x) − g(t, x)R−1 g(t, x)T 2 ∂x is locally uniformly asymptotically stable at x = 0, then, ∂V T 1 u∗ = − R−1 g(t, x)T 2 ∂x is the solution for the optimal control in Definition 2.1. The solution of (6) satisfying the above conditions is called periodic stabilizing solution of (6). Linearizing (6), one obtains a periodic Riccati differential equation −P˙ = P A(t) + A(t)T P − P B(t)R−1 B(t)T P + Q, (7) where ∂f (t, 0), B(t) = g(t, 0). A(t) = ∂x and these matrices are T -periodic. Also, related to HJE (6), the differential equation dxj ∂H ∂H dpj = =− (t, x, p), , j = 1, . . . , n, (8) dt ∂pj dt ∂xj with H in (5), is called the canonical equation. We can, now, state a fundamental theorem for the existence of a periodic stabilizing solution for (6). Theorem 2.1. Suppose that there exists a periodic stabilizing solution (see, Definition A.1) for the Riccati differential equation (7). Then, there exists an n-dimensional stable manifold L for (8) defined around the origin, which is T periodic with respect to t and canonically surjective to the n-dimensional base space for x. Furthermore, there exists a T -periodic function V (t, x) defined on R × U , where U is a neighborhood of the origin of the base space Rn , such that V (t, x) satisfies (6) and L ∩ π −1 (U ) = {(x, p) | pj = (∂V /∂xj )(t, x), j = 1, . . . , n, x ∈ U }.
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The detail of the stable manifold construction in the timeinvariant case can be found in Sakamoto (2013).
where π : (x, p) → x is the canonical projection. 3. CONSTRUCTION OF STABLE MANIFOLD This section is devoted to prove Theorem 2.1 in a constructive way. First, by direct computations, one can obtain the linear part of (8) as [ ] 1 A(t) − B(t)R−1 B(t)T , 2 −2Q −A(t)T which is T -periodic and will be denoted as F (t). The canonical equation (8) is, then, written [˙] [ ] [ ] x x Nx (t, x, p) =F (t) + , (9) p p Np (t, x, p) where Nx and Np are higher order nonlinear terms and T -periodic.
Then using Floquet-Lyapnov transformation (see Meyer et al. (2015)), one can transform this system into the system with a constant coefficient matrix by the following coordinate transformation [ ] [ ′] x x =S(t) ′ . (10) p p In the new coordinate, the linear part of the canonical equation (9) becomes time-invariant: [ ˙′ ] [ ′] [ ] x Nx′ (t, x′ , p′ ) ′ x , (11) ′ =F ′ + ′ ′ Np′ (t, x , p ) p p where the non-linear part satisfies [ ] ] [ Nx′ (t, x′ , p′ ) −1 Nx (t, x, p) . = S(t) Np (t, x, p) Np′ (t, x′ , p′ )
The assumption that (7) has a T -periodic stabilizing solution and Theorem A.1 assure that F ′ has no eigenvalues on the imaginary axis and therefore, there exists a 2n × 2n real constant V such that ] [ Λ 0 (12) V −1 F ′ V = 0 −ΛT where Λ is a real stable n × n matrix. Then, the canonical equation in (11) can be put into the form (b1) where the stable manifold computation (b2) proposed in Theorem B.1 in Appendix B can be applied.
Let (x′′k (t, ξ), p′′k (t, ξ)) be the sequence obtained by (b2) for the canonical equation with the block-diagonalized linear part with k (the number of iteration) sufficiently large. Then, the sequence [ ] [ ′′ ] xk (t, ξ) xk (t, ξ) = S(t)V ′′ pk (t, ξ) pk (t, ξ) are the solution on the stable manifold of (4).
Eliminating ξ from {(xk (t, ξ), pk (t, ξ)) | t ∈ R, |ξ| : sufficiently small} for sufficiently large k, one obtains an approximation of the stable manifold L of (9). Since L is locally canonically surjective to the x-space, it can be represented as p = p(t, x) (13) with some function p(t, x) which is T -periodic for t. The optimal control is then given by 1 u∗ = − R−1 g(t, x)T p(t, x). 2 217
4. STABILIZATION OF A PERIODIC ORBIT IN ROSSLER CHAOTIC SYSTEM 4.1 Rossler chaotic system and its periodic orbit Here, we demonstrate the effectiveness of the proposed method using the Rossler system, introduced by Rossler (1976). Let us consider the Rossler system with three inputs q˙ = A q + N + u, (14) where T T q = [q1 , q2 , q3 ] , u = [u1 , u2 , u3 ] ] ] [ [ 0 0 −1 −1 0 0 ,N= A= 1 a q1 q3 q30 0 q10 − c √ √ c − c2 − 4ab c − c2 − 4ab , q30 = . q10 = 2 2a In this system with some parameters (a, b, c), unstable periodic orbits can be observed. We choose the parameter set (a, b, c) = (0.2, 0.2, 3), for which it is known that there exists an unstable periodic orbit. Applying the method of delayed feedback to the system (14), we found an unstable periodic orbit. The characteristic multipliers of this periodic orbit are (−1.11, 1, −2.01× 10−7 ). Since the periodic orbit has a characteristic multiplier with modulus more than unity, the periodic orbit is unstable. We wish to design a control law stabilizing this periodic orbit in an optimal fashion. 4.2 Optimal control problem Let qc = [q1c (t), q2c (t), q3c (t)]T be the periodic solution given in the previous subsection and T be the period. Consider the error dynamics along qc (t) x˙ =Ac (t) x + Nc + u, where x = [x1 , x2 , x3 ]T = q − qc is the error state and ] ] [ [ 0 −1 −1 0 1 a 0 0 . Ac (t) = , Nc = q30 + q3c (t) 0 q10 + q1c (t) − c x 1 x3 For this system, we define the cost function as ∫ ∞ J= xT Qx + uT Ru dt. 0
To obtain a control law minimizing input, we select the following weight matrices; Q = 10−3 · I3×3 , R = I3×3 ∫∞ This is an approximation of the cost function 0 uT Ru dt, from which we expect to derive an extension of OGY control in the sense that the control input is kept as low as possible using the sensitivity of chaotic system. This cost function, however, does not lead a solution for the stable regulator problem in Definition 2.2. The inclusion of small Q > 0 is one of the easiest way for the stable regulator problem to have a solution. Let us define the Hamiltonian H(t, x, u, p) = pT (Ac x + Nc + u) + xT Qx + uT Ru.
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Setting u ¯(x, p) = −1/2R−1 p, the Hamilton-Jacobi equation for this problem is ∂V ∂V ∂V = H(t, x, u ¯(x, ), ) − ∂t ∂x ∂x T ∂V = (Ac x + Nc ) + xT Qx ∂x 1 ∂V T −1 ∂V , R − 4 ∂x ∂x and Hamilton’s canonical equations are { 1 x˙ = Ac x + Nc − R−1 p, (15) 2 p˙ = −2Q x − ATc p − dNc p, where ] [ 0 0 x3 dNc = 0 0 0 . 0 0 x1 Let p = p(t, x) be the stable manifold of the canonical equation, then the solution of this optimal control problem is given as u∗ = −1/2R−1 p(t, x).
Fig. 1. Set of optimal trajectories in Rossler system with control 7UDMHFWRU\ 3HULRGLFRUELW ,QLWLDOFRQGLWLRQ &RQWUROVWDUWLQJSRLQW
4.3 Derivation of Floquet factor
′ = S(t)−1 Nqp . Then the eigenvalues of F ′ is where Nqp obtained as λ = ±2.672, ±0.1640, ±0.0765 It shows that the assumption in §3 is satisfied. Thus, we can apply the stable manifold method.
4.4 Controller representation To obtain the approximate solution of Hamilton-Jacobi equation, we apply the stable manifold method in §3 to the system (17). When one employs the computational method of stable manifold to the optimal stabilization problem of an unstable periodic orbit qc (t), the periodic function p(t, x) in (13) can be written as p(t, x) = p˜(x + qc (t)) = p˜(q) and therefore, the control law u∗ = −1/2R−1 p˜(q) is in a feedback form. For the construction of the control law, we calculate qk (t, ξ) = qc (t) + xk (t, ξ) which correspond to the closed loop orbits converging to the periodic orbit qc (t) as t → ∞. In Fig.1, we show the set of the optimally controlled trajectories qk (t, ξ). Fig 1 shows projected data on q-space. From the corresponding data for p-space, we compute p˜(q) and the optimal control law u∗ (q) = −1/2R−1 p˜(q). 218
]
Extracting the linear part, (15) can be written as [ ] [ ] x˙ x =F + Nqp , (16) p˙ p where ] [ ] [ 1 Nc Ac − R−1 , Nqp = F = . 2 T −dNc p −2Q −Ac F is a T -periodic matrix, that is, F (t + T ) = F (t). Then, we can apply Floquet-Lyapnov transformation so as for the linear part to be time-invariant. Let S(t) and F ′ be the transformation and constant coefficient matrix each. Using the transformation (10), we have [ ′] [ ′] x˙ ′ x ′ + Nqp , F ′ : constant, (17) ′ =F p˙ p′
\
[
Fig. 2. Time responses for q(0) = [3, −8, 0]T (proposed method). The controller is activated from the beginning. 4.5 Simulation results To demonstrate the effectiveness of the proposed method, numerical simulations of the Rossler system under the control are performed. We set the initial condition q(0) = [3, −8, 0]T . The simulation results are shown in Fig. 2 and Fig. 3. Fig. 2 illustrates the behavior of the states in the Cartesian coordinate system, and Fig. 3 shows the error from the periodic orbit. From Fig. 2 or Fig. 3, it can be seen that the designed control law stabilizes the periodic orbit. We show the input time history during stabilization in Fig. 4. The original motivation of our research is to establish a framework for designing stabilizing control of unstable periodic orbits using as small input as possible by exploiting the initial condition sensitivity ofc chaos. To this end, we design a control law using optimal OGY method proposed by Epureanu and Dowell (2000), and compare the input behaviors by our control and optimal OGY control. The results of the simulation with the same initial condition by the optimal OGY method are presented in Fig.5 -
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219
HUURU
HUURU
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Fig. 3. Error from the periodic orbit (proposed method)
Fig. 6. Error from the periodic orbit (optimal OGY method)
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X X
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Fig. 7. Control input (optimal OGY method)
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Fig. 4. Control input (proposed method)
Table 1. The values of cost function Ju Method Proposed Optimal OGY
]
Ju 0.0074 0.023
The main advantage of the proposed method is the capability to take full nonlinearity into account. It can be seen that the control law works well even in the area far from the periodic orbit, whereas OGY method has to wait until the states come into the prescribed region to keep the input small enough.
\
[
5. CONCLUSIONS T
Fig. 5. Time responses for q(0) = [3, −8, 0] (optimal OGY method). The controller is activated t = 460.9s (shown as the red circle). Fig.7. Since the OGY method is based on linearization of the system around the target periodic orbit, the effective domain is limited. In our simulation, the optimal OGY control is activated when the error in the Euclidean norm becomes less than 0.5. If one activates the control outside of this domain, the input by this controller can be unrealistically large. Fig.5 shows the trajectory in the Cartesian coordinate system, and Fig.6 illustrates the error from the designated periodic orbit at each period of time. In Fig.7, we show the input behavior after the OGY control is activated at t = 460. Although the maximum input value in the optimal OGY method is similar to the one in the proposed method, the squared integral of the input over the control duration is significantly improved. The values of the cost function ∫∞ Ju = 0 uT u dt are listed in Table 1.
219
In this paper, we proposed an approach to design optimal control for unstable periodic orbits in chaotic systems using stable manifold method. First, we have developed a theory of optimal control for nonlinear periodic systems and apply it for the error dynamics of target periodic orbit in a chaotic system. The method is based on computing a stable manifold of canonical equation associated with the Hamilton-Jacobi equation that is equivalent to the optimal control problem for the unstable orbit. When one tries to apply the proposed method, more exact knowledge of the model is necessary unlike the OGY method or Time-Delayed Feedback Control. The method, however, yields a control law that optimally stabilizes the unstable orbit by taking the nonlinearity into account. Moreover, the effective domain of the proposed method is larger than the OGY method while control effort is kept even smaller than the OGY method. Simulation results for the Rossler chaotic system shows the effectiveness of the proposed method.
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REFERENCES Biggs, J. and R.McInnes, C. (2009). An optimal gains matrix for time-delay feedback control. In 2nd IFAC Conference on the Analysis and Control of Chaotic Systems, 22–24. Ding, Y., Jiang, W., and Wang, H. (2010). Delayed feedback control and bifurcation analysis of Rossler chaotic system. Nonlinear Dynamics, 61(4), 707–715. Epureanu, B.I. and Dowell, E.H. (2000). Optimal multi-dimensional OGY controller. Physica D, 139, 87–96. Fujimoto, R. and Sakamoto, N. (2011). The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input saturation. In the 18th IFAC World Congress, volume 18. 8046–8051. Garfinkel, A., Spano, M.L., Ditto, W.L., and Weiss, J.N. (1992). Controlling cardiac chaos. Science, 257, 1230–1235. Just, W., Bernard, T., Ostheimer, M., Reibold, E., and Benner, H. (1997). Mechanism of time-delayed feedback control. Physical Review Letters, 78(2), 203–206. Kostelich, E.J., Grebogi, C., Ott, E., and Yorke, J.A. (1993). Higherdimensional targeting. Physical Review E, 47(1), 305–310. Meyer, K.R., Hall, G.R., and Offin, D. (2015). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer, second edition. Ott, E., Grebogi, C., and Yorke, J.A. (1990). Controlling chaos. Physical Review Letters, 64(11), 1196–1199. Postlethwaite, C.M. and Silber, M. (2007). Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control. Physical Review E, 76(056214), 1–10. Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback. Physics Letters A, 170, 421–428. Romeiras, F.J., Grebogi, C., Ott, E., and Dayawansa, W. (1992). Controlling chaotic dynamical systems. Physica D, 58, 165–192. Rossler, O. (1976). An equation for continuous chaos. Physics Letters A, 57A(5), 397–398. Sakamoto, N. (2013). Case studies on the applications of the stable manifold approach for nonlinear optimal control design. Automatica, 49, 568–576. Sakamoto, N. and van der Schaft, A.J. (2008). Analytical approximation methods for the stabilizing solution of the Hamilton-Jacobi equation. IEEE Transactions on Automatic Control, 53(10), 2335–2350. Schiff, S.J., Jerger, K., Duong, D.H., Chang, T., Spano, M.L., and Ditto, W.L. (1994). Controlling chaos in the brain. Nature, 370(25), 615–620. Shinbrot, T., Ott, E., Grebogi, C., and Yorke, J.A. (1990). Using chaos to direct trajectories to targets. Physical Review Letters, 65(26), 3215–3218. Yuasa, Y., Sakamoto, N., and Umemura, Y. (2010). Optimal control designs for systems with input saturations and rate limiters. In SICE Annual Conference 2010, 2042–2045.
Appendix A. THEORY OF LINEAR PERIODIC SYSTEMS In this Appendix, we briefly review the theory of linear periodic systems that plays a major role throughout the paper. Linear periodic Riccati differential equation Suppose A(t) ∈ Rn×n and B(t) ∈ Rn×m are T -periodic matrices. Let us consider a T -periodic Riccati differential equation and the T -periodic Hamiltonian matrix associated with the Riccati equation −P˙ = P A(t) + A(t)T P − P B(t)R−1 B(t)T P + Q, (a1) ] [ 1 A(t) − B(t)R−1 B(t)T . (a2) 2 −2Q −A(t)T 220
Definition A.1. A T -periodic solution P (t) of (a1) is called a stabilizing solution when it satisfies (a1) and x˙ = (A(t)− R−1 B(t)T P (t))x is asymptotically stable. Theorem A.1. A necessary and sufficient condition for the Riccati equation (a1) to have a periodic stabilizing solution is that the Monodromy matrix of (a2) has no eigenvalues on the unit circle and the pair (A(·), B(·)) is stabilizable in the sense of periodic systems. Appendix B. STABLE MANIFOLD METHOD In this Appendix, we briefly present the stable manifold algorithm proposed in Sakamoto and van der Schaft (2008). The detail application procedure of the method, such as the convergence issue of the algorithm and the function representation of the stable manifold, can be found in Sakamoto (2013). Let us consider the equation of the form [ ] [ ][ ] [ ] x˙ Λ1 0 x Nx (t, x, p) = + , p˙ 0 Λ2 p Np (t, x, p)
(b1)
where all the eigenvalues of Λ1 , −Λ2 lie in the open left half plane i.e., Re λj (Λ1 ) < 0, Re λj (Λ2 ) > 0, j = 1, . . . , n. Assumption B.1. Nx , Np : R × Rn × Rn → Rn are continuous and satisfy the following. (i) For all t ∈ R, |u| + |v| < l and |u′ | + |v ′ | < l, |Nx (t, u, v)−Np (t, u′ , v ′ )| < δ1 (l)(|u − u′ | + |v − v ′ |). (ii) for all t ∈ R, |u| + |v| < l and |u′ | + |v ′ | < l, |Np (t, u, v)−Np (t, u′ , v ′ )| < δ2 (l)(|u − u′ | + |v − v ′ |), where δj : [0, ∞) → [0, ∞), j = 1, 2 are continuous and monotonically increasing on [0, Lj ] for some constants L1 , L2 > 0. Let us define the sequences {xk (t, ξ)} and {pk (t, ξ)} by ∫ t Λ1 t xk+1 = e ξ + eΛ1 (t−s) Nx (t, xk , pk ) ds ∫ ∞ 0 (b2) Λ2 (t−s) = − e N (t, x , p ) ds p k+1 p k k t
for k = 0, 1, 2, . . ., and {
x0 = eΛ1 t ξ p0 = 0
where ξ ∈ Rn is arbitrary as long as |ξ| is sufficiently small. Then, we have the following theorem. Theorem B.1. (Sakamoto and van der Schaft (2008)) Under Assumption B.1, the sequence xk (t, ξ) and pk (t, ξ) are convergent to zero for sufficiently small |ξ|, that is, xk (t, ξ), pk (t, ξ) → 0 as t → ∞ for all k = 0, 1, 2, . . .. Furthermore, xk (t, ξ) and pk (t, ξ) are uniformly convergent to a solution of (b1) on [0, ∞) as k → ∞. Let x(t, ξ) and p(t, ξ) be the limits of xk (t, ξ) and pk (t, ξ), respectively. Then, x(t, ξ) and p(t, ξ) are a solution on the stable manifold of (b1), that is, x(t, ξ), p(t, ξ) → 0 as t → ∞.
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Nonlinear optimal stabilization of unstable Nonlinear stabilization of of unstable Nonlinear optimal optimal stabilization unstable periodic orbits periodic orbits periodic orbits ∗∗∗ Yuuta Habaguchi ∗∗ Noboru Sakamoto ∗∗ ∗∗ Keisuke Nagata ∗∗∗ Yuuta Habaguchi ∗ Noboru Sakamoto ∗∗ ∗∗ Keisuke Nagata ∗∗∗ ∗∗∗ ∗ Yuuta Habaguchi Noboru Sakamoto Keisuke Nagata Yuuta Habaguchi Noboru Sakamoto Keisuke Nagata ∗ ∗ Department of Aerospace Engineering, Graduate School of of Aerospace Engineering, Graduate School of ∗ ∗ Department Department of Engineering, School of Engineering, Nagoya University, Furo-cho, Graduate Chikusa-ku, Nagoya, Department of Aerospace Aerospace Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Engineering, Nagoya University, Furo-cho, Furo-cho, Chikusa-ku, Chikusa-ku, Nagoya, Nagoya, Japan, (e-mail: [email protected]) Engineering, Nagoya University, (e-mail: [email protected]) ∗∗Japan, Japan, (e-mail: [email protected]) Department of Aerospace Engineering, Graduate School of (e-mail: [email protected]) ∗∗Japan, Department of Aerospace Engineering, Graduate School of ∗∗ ∗∗ Department of Aerospace Engineering, School of Engineering, Nagoya University, Furo-cho, Graduate Chikusa-ku, Nagoya, Department of Aerospace Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, (e-mail: [email protected]) Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, [email protected]) ∗∗∗ Japan, (e-mail: (e-mail: [email protected]) and Aerospace Engineering, Graduate School of Japan, (e-mail: [email protected]) ∗∗∗ Mechanical ∗∗∗ Mechanical and Aerospace Engineering, Graduate School of ∗∗∗ Mechanical and Aerospace Aerospace Engineering, Graduate School School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Mechanical and Engineering, Graduate of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, (e-mail: [email protected]) Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan, (e-mail: [email protected]) Japan, Japan, (e-mail: (e-mail: [email protected]) [email protected])
Abstract: The framework for the optimal stabilization of unstable periodic orbits in nonlinear Abstract: The framework for the optimal stabilization of unstable periodic orbits in nonlinear Abstract: The optimal stabilization unstable periodic nonlinear systems is proposed. First, for thethe stable manifold methodof for approximately solvingin HamiltonAbstract: The framework framework for the optimal stabilization offor unstable periodic orbits orbits inHamiltonnonlinear systems is proposed. First, the stable manifold method approximately solving systems is proposed. proposed. First, the stable stable manifold method for approximately approximately solving HamiltonJacobi equations in the optimal control problemmethod is extended to nonlinear solving periodicHamiltonsystems. systems is First, the manifold for Jacobi equations in the optimal control problem is extended to nonlinear periodic systems. Jacobi equations optimal is extended periodic systems. Then, this methodin isthe applied for control optimalproblem stabilization problemto ofnonlinear error dynamics around an Jacobi equations in the optimal control problem is extended to nonlinear periodic systems. Then, this method is applied for optimal stabilization problem of error dynamics around an Then, this method is applied for optimal stabilization problem of error dynamics an unstable periodic orbit. The effectiveness of the proposed method is evaluated via around numerical Then, this method is applied for optimal stabilization problem of error dynamics around an unstable periodic orbit. The effectiveness of unstable periodic orbit. The effectiveness of the the proposed proposed method method is is evaluated evaluated via via numerical numerical simulations for theorbit. Rossler chaotic system. of unstable periodic The effectiveness the proposed method is evaluated via numerical simulations for the Rossler chaotic system. simulations for for the the Rossler Rossler chaotic chaotic system. system. simulations © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Nonlinear optimal control, unstable periodic orbit, chaos control, Rossler chaotic Keywords: Nonlinear control, unstable periodic orbit, Keywords: Nonlinear optimal optimal control, unstable periodic orbit, chaos chaos control, control, Rossler Rossler chaotic chaotic system, Hamilton-Jacobi equation, stable manifold methods Keywords: Nonlinear optimal control, unstable periodic orbit, chaos control, Rossler chaotic system, Hamilton-Jacobi equation, stable manifold methods system, Hamilton-Jacobi Hamilton-Jacobi equation, equation, stable stable manifold manifold methods methods system, 1. INTRODUCTION tem has delay terms, analysis of the system controlled by 1. tem has terms, analysis system by 1. INTRODUCTION INTRODUCTION tem has delay delay terms, analysis of of the theHowever, system controlled controlled by the method is terms, not straightforward. the analysis 1. INTRODUCTION tem has delay analysis of the system controlled by the method is not straightforward. However, the analysis Chaos has been observed in various areas in physics, the the method is not not straightforward. However, the analysis method in the vicinity of the periodic orbits is proposed method is straightforward. However, the analysis Chaos has observed in areas physics, in the vicinity of periodic orbits is Chaos has been been observed in asvarious various areas in in celestial physics, method engineering and biology such fluid dynamics, method in al. the(1997), vicinityand of the the periodic orbits is proposed proposed in Just et the periodic optimal orbits gain tuning using Chaos has been observed in various areas in physics, method in the vicinity of the is proposed engineering and biology such as fluid dynamics, celestial in Just et al. (1997), and the optimal gain tuning using engineering and physiology, biology such such as fluid fluidthe dynamics, celestial mechanics, and therefore control of chaos in in Just et al. (1997), and the optimal gain tuning using linearization around the target orbits is reported in Biggs engineering and biology as dynamics, celestial Just et al. (1997), and the optimal gain tuning using mechanics, and therefore the control of around the mechanics, and physiology, physiology, therefore the controlthere of chaos chaos is an imperative subject. In chaotic the systems, are linearization linearization around the target target orbits orbits is is reported reported in in Biggs Biggs and R.McInnes (2009). mechanics, and physiology, therefore control of chaos linearization around the target orbits is reported in Biggs is an imperative subject. In chaotic systems, there are and R.McInnes (2009). is an imperative subject. In chaotic systems, there are interesting regionssubject. in which achaotic large number of there unstable and R.McInnes R.McInnes (2009). (2009). is an imperative In systems, are and interesting regions in which a large number of Although chaotic systems are nonlinear dynamics, there interesting regions in and which large numberthat of unstable unstable periodic orbits exist, it aais large well-known some of Although interesting regions in which number of unstable chaotic are dynamics, there periodic orbits exist, and it is well-known that some of Although chaotic systems systems are nonlinear nonlinear dynamics, there exists no method to explicitly consider dynamics, or make use of Although chaotic systems are nonlinear there periodic orbits exist,may andbeit itcapable is well-known well-known that some some of exists the unstable orbits of improving system periodic orbits exist, and is that of no method to explicitly consider or make use of the unstable orbits may be capable of improving system exists no method to explicitly consider or make use of the nonlinearity. Since the OGY method is based on exists no method to explicitly consider or make use of the unstable according orbits may maytobe besome capable of improving improving system performance criteria. Stabilizing such the nonlinearity. Since the OGY method is based on the unstable orbits capable of system performance according to some criteria. Stabilizing such the nonlinearity. Since the OGY method is based on linearized models around target orbits, the nonlinearity of the nonlinearity. Since the OGY method is based on performance according to some criteria. Stabilizing such orbits can be according beneficial to in many fields. From this respect, performance some fields. criteria. Stabilizing such linearized models around target orbits, the of orbits can be beneficial in many From this respect, linearized models around target orbits, the nonlinearity nonlinearity of the systemmodels is ignored in the control design. Furthermore, linearized around target orbits, the nonlinearity of orbits can be beneficial in many fields. From this respect, Ott et al. (1990) proposed a method to control chaotic orbits can be beneficial in many fields. From this respect, the is ignored in control design. Furthermore, Ott et (1990) proposed aa method to control chaotic the system system isgain ignored in the the control design. Furthermore, optimalis tuning method fordesign. TDFCFurthermore, do not take system ignored in the control Ott et al. al. (1990) the proposed method to to control chaotic systems utilizing sensitivity of chaos initialchaotic condi- the Ott et al. (1990) proposed a method to control the optimal gain tuning method for TDFC do not take systems utilizing the sensitivity of chaos to initial condithe optimal gain tuning method for TDFC do not take nonlinearity into account in that it solves thenot optimal the optimal gain tuning method for TDFC do take systems utilizing thechaos sensitivity ofhas chaos to initial initial condi- the nonlinearity into account in that it solves the tions andutilizing since then controlof attracted attention systems the sensitivity chaos to condioptimal tions and since then chaos control has attracted attention the nonlinearity into account in that it solves the optimal control problem for the linearized model. The chaotic the nonlinearity into account in that it solves the optimal tions and since then chaos control has attracted attention of many researchers. Theircontrol methodology is called OGY control problem for the linearized model. The chaotic tions and researchers. since then chaos has attracted attention of many Their methodology is OGY control problemsuch for as thethe linearized model. The chaotic chaotic characteristics, sensitivity on initial states, problem for the linearized model. The of manyand researchers. Their methodology isofcalled called OGY method became one of the major fieldsis ChaosOGY con- control of many researchers. Their methodology called characteristics, such as the sensitivity on initial states, method and became one of the major fields of Chaos concharacteristics, such as the sensitivity on initial states, which should be taken advantage of, as Ott et al. (1990) characteristics, such as the sensitivity on initial states, method and became one of oflittle the major major fields of Chaos Chaosabout con- which should be taken advantage of, as Ott et al. (1990) trol. Thisand method requires a priorifields information method became one the of control. This method requires little a priori information about which should be taken advantage of, as Ott et al. (1990) indicated, are created from its nonlinearity, and therefore, which should be taken advantage of, as Ott et al. (1990) trol. This method method requires little priori information information about controlled systemsrequires and haslittle beenaasuccessfully used in about many indicated, are created from its nonlinearity, and therefore, trol. This priori controlled systems and been successfully used in many indicated, are considering created from fromthe its nonlinearity nonlinearity, in andthe therefore, by explicitly control are created its nonlinearity, and therefore, controlled systems and has has been successfully used in many indicated, contexts (see Garfinkel et al. (1992); Schiff et al. in (1994)). controlled systems and has been successfully used many by explicitly considering the nonlinearity in the control contexts (see Garfinkel et al. (1992); Schiff et al. (1994)). by explicitly considering the nonlinearity in the control design, more effective chaos control than ever can be by explicitly considering the nonlinearity in the control contexts (see Garfinkel et al. (1992); Schiff et al. (1994)). The optimality of the method in convergence time is shown contexts (see Garfinkel et al. in (1992); Schiff et al. is (1994)). design, more effective chaos control than ever can be The optimality of the method convergence time shown design, more effective chaos control than ever can be achieved. design, more effective chaos control than ever can be The optimality ofal.the the method in convergence convergence time is is shown shown in Romeiras et of (1992) and, moreover, Epureanu and achieved. The optimality method in time in Romeiras et al. and, moreover, Epureanu and achieved. in Romeiras etproposes al. (1992) (1992) and, moreover, Epureanu and achieved. Dowell (2000)et an and, optimal control Epureanu scheme which in Romeiras al. (1992) moreover, and On the other hand, the research on control methods that Dowell (2000) proposes an optimal control scheme which the other hand, the on methods that Dowell (2000) proposes an optimal optimal control scheme which minimizes the proposes effort required in control. While the which OGY On On the other hand, the research research on control control methods that Dowell (2000) an control scheme make full use of nonlinearity has been attracting attention On the other hand, the research on control methods that minimizes the effort required in control. While the OGY full use of nonlinearity has been attracting attention minimizes the the effort required in control. While the to OGY method gives control law in only near the orbits be make make full use of nonlinearity has been attracting attention minimizes the effort required control. While the OGY in the areas of control theory/engineering. Among the make full use of nonlinearity has been attracting attention method gives the control only near orbits to the of control Among the method gives thestabilization control law law is only near the the orbits to be be in stabilized, global proposed using targeting in the areas areas of manifold control theory/engineering. theory/engineering. Among the method gives the control law only near the orbits to be methods, stable method developed byAmong Sakamoto in the areas of control theory/engineering. the stabilized, global stabilization is proposed using targeting methods, stable manifold method developed by Sakamoto stabilized, global stabilization is proposed proposed using targeting method in global Shinbrot et al. (1990); Kostelichusing et al.targeting (1993). methods, stable manifold method developed bydesigns Sakamoto stabilized, stabilization is and van der Schaft (2008) is thedeveloped one thatby the methods, stable manifold method Sakamoto method in Shinbrot et al. (1990); Kostelich et al. (1993). and van der Schaft (2008) is the one that designs the method in Shinbrot et al. (1990); Kostelich et al. (1993). and van der Schaft (2008) is the one that designs the method in Shinbrot et al. (1990); Kostelich et al. (1993). optimal control law taking into account of nonlinearities in and van der Schaft (2008) is the one that designs the On the other hand, Time Delayed Feedback Control optimal control law taking into account of nonlinearities in On the other hand, Time Delayed Feedback Control optimal control law taking into account of nonlinearities in the system, and have succeeded in the swing up control of optimal control law taking into account of nonlinearities in On the other hand, Time Delayed Feedback Control (TDFC) introduced by Pyragas (1992) constitutes another On the introduced other hand, Time Delayed Feedback another Control the system, and succeeded in swing up control of (TDFC) by Pyragas (1992) constitutes the system, and have have succeeded in the the swingof upthe control of inverted pendulum and the optimal control system the system, and have succeeded in the swing up control of (TDFC) introduced by Pyragas (1992) constitutes another major field in chaos control. It is able to stabilize a lot of (TDFC) introduced by Pyragas (1992) constitutes another inverted pendulum and optimal control of major field in control. It to stabilize aa lot inverted pendulum and the the optimal control (Yuasa of the the system system with input saturations andoptimal rate limiters et al. pendulum and the control of the system major field in chaos chaos control. It is is able able tosimple stabilize lot of of inverted unstable periodic orbits in spite of itsto algorithm major field in chaos control. It is able stabilize a lot of with input saturations and rate limiters (Yuasa et unstable periodic orbits in spite of its simple algorithm with input saturations and rate limiters (Yuasa et al. al. (2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). with input saturations and rate limiters (Yuasa et al. unstable periodicsignal orbitscomputed in spite spite from of its its the simple algorithm using a control difference be- (2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). unstable periodic orbits in of simple algorithm using a control signal computed from the difference be(2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). In the present paper, we propose a method designing (2010); Fujimoto and Sakamoto (2011); Sakamoto (2013)). using a control signal computed from the difference between athe current and computed delayed states (Postlethwaite and using control signal from(Postlethwaite the difference and be- In the paper, we propose a method tween the and delayed states In the present present paper, we law propose method designing designing nonlinear optimal control whichaa stabilizes unstable the present paper, we propose method designing tween the current current and delayed states (Postlethwaite and In Silber (2007); Ding and et al.delayed (2010)).states Since (Postlethwaite the closed loop systween the current and nonlinear optimal control law which stabilizes unstable Silber (2007); Ding et al. (2010)). Since the closed loop sysnonlinear optimal control law which stabilizes unstable nonlinear optimal control law which stabilizes unstable Silber (2007); (2007); Ding Ding et et al. al. (2010)). (2010)). Since Since the the closed closed loop loop syssysSilber Copyright 2015 IFAC 215 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 215 Copyright 2015 IFAC 215 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 215Control. 10.1016/j.ifacol.2015.11.039
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periodic orbits in chaotic systems using the stable manifold method. Here, our goal is to minimize input effort during the stabilization, which would lead an extension of the OGY method by introducing optimality with respect to a suitable cost function. The organization of the paper is as follows. In §2, we give the definition of the optimal control problem for nonlinear periodic systems and demonstrate that the problem attributes to obtaining the stable manifold of a time-periodic canonical equation. In §3, the method of calculating the stable manifold and important theories for the method is described. In §4, the proposed method of optimally stabilizing an unstable periodic orbit is applied to Rossler chaotic system. 2. OPTIMAL CONTROL OF PERIODIC NONLINEAR SYSTEMS Let us consider a nonlinear control system of the form Σ : x˙ = f (t, x) + g(t, x)u (1) n m n where x(t) ∈ R , u(t) ∈ R and f : R × R → Rn , g : R × Rn → Rn×m are C r (r ⩾ 2) functions. We also assume that f and g are periodic functions with respect to t with period T and that f (t, 0) = 0 for all t ∈ R. For system Σ, we seek a control law that minimizes a cost function ∫ ∞ J= xT Qx + uT Ru dt, (2) 0
where Q and R are symmetric with appropriate dimension and R is a positive definite matrix. More precisely, we define our optimal control problem as follows. Definition 2.1. (Stable regulator problem). Find a control law u(t) that minimizes the cost function J in (2) among all locally uniformly asymptotically stabilizing controls for Σ in (1) at x = 0.
An important application of the stable regulator problem for a nonlinear periodic system is the optimal stabilization of a periodic orbit in a nonlinear system, from which we would like to derive an optimal OGY control for chaotic systems. Let us consider a control system with a periodic solution system : q˙ = a(q) + b(q)u, (3) periodic solution: q˙c = a(qc ), qc (t + T ) = qc (t), (4) where q, qc (t) ∈ Rn , T is the period of the periodic solution qc and u(t) ∈ Rm is an input. It is also assumed that a(·) : Rn → Rn , b(·) : Rn → Rn×m are smooth vector fields. Definition 2.2. (Stabilization problem for a periodic orbit). A state feedback control u = u(q) is said to be an asymptotically stabilizing control for the periodic orbit qc in system (3) if u satisfies u(qc (t)) = 0 for all t and qc is an asymptotically stable periodic orbit of the closed loop system. We wish to find a stabilizing control of the periodic orbit qc in an optimal way. To define the optimal stabilization problem, first, introduce error x := q − qc and then derive the error system x˙ = q˙ − q˙c = a(x + qc (t)) − a(qc (t)) + b(x + qc (t))u. 216
If we set f (t, x) := a(x + qc (t)) − a(qc (t)), g(t, x) := b(x + qc (t)), then, the optimal stabilization problem of the periodic orbit amounts to the optimal regulator problem (1) and (2) with appropriate weighting matrices Q, R. Now, we will derive a Hamilton-Jacobi equation (HJE) for the stable regulator problem. Define the Hamiltonian for the optimal control problem for Σ and (2) by HD = pT {f (t, x) + g(t, x)u} + xT Qx + uT Ru. Since R > 0, 1 u ¯(t, x, p) := arg min HD (t, x, p, u) = − R−1 g(t, x)T p m 2 u∈R and let ¯(t, x, p)). (5) H(t, x, p) := HD (t, x, p, u Then, the Hamilton-Jacobi equation associated with the optimal control problem in Definition 2.1 is ( ) ∂V ∂V = H t, x, − ∂t ∂x ∂V T 1 ∂V ∂V f (t, x) − g(t, x)R−1 g(t, x)T = + xT Qx. ∂x 4 ∂x ∂x (6) It can be readily seen that if (6) has a periodic solution V (t, x) with period T defined for all t and for x in a neighborhood of x = 0 with V (t, 0) = 0 such that 1 ∂V T x˙ = f (t, x) − g(t, x)R−1 g(t, x)T 2 ∂x is locally uniformly asymptotically stable at x = 0, then, ∂V T 1 u∗ = − R−1 g(t, x)T 2 ∂x is the solution for the optimal control in Definition 2.1. The solution of (6) satisfying the above conditions is called periodic stabilizing solution of (6). Linearizing (6), one obtains a periodic Riccati differential equation −P˙ = P A(t) + A(t)T P − P B(t)R−1 B(t)T P + Q, (7) where ∂f (t, 0), B(t) = g(t, 0). A(t) = ∂x and these matrices are T -periodic. Also, related to HJE (6), the differential equation dxj ∂H ∂H dpj = =− (t, x, p), , j = 1, . . . , n, (8) dt ∂pj dt ∂xj with H in (5), is called the canonical equation. We can, now, state a fundamental theorem for the existence of a periodic stabilizing solution for (6). Theorem 2.1. Suppose that there exists a periodic stabilizing solution (see, Definition A.1) for the Riccati differential equation (7). Then, there exists an n-dimensional stable manifold L for (8) defined around the origin, which is T periodic with respect to t and canonically surjective to the n-dimensional base space for x. Furthermore, there exists a T -periodic function V (t, x) defined on R × U , where U is a neighborhood of the origin of the base space Rn , such that V (t, x) satisfies (6) and L ∩ π −1 (U ) = {(x, p) | pj = (∂V /∂xj )(t, x), j = 1, . . . , n, x ∈ U }.
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The detail of the stable manifold construction in the timeinvariant case can be found in Sakamoto (2013).
where π : (x, p) → x is the canonical projection. 3. CONSTRUCTION OF STABLE MANIFOLD This section is devoted to prove Theorem 2.1 in a constructive way. First, by direct computations, one can obtain the linear part of (8) as [ ] 1 A(t) − B(t)R−1 B(t)T , 2 −2Q −A(t)T which is T -periodic and will be denoted as F (t). The canonical equation (8) is, then, written [˙] [ ] [ ] x x Nx (t, x, p) =F (t) + , (9) p p Np (t, x, p) where Nx and Np are higher order nonlinear terms and T -periodic.
Then using Floquet-Lyapnov transformation (see Meyer et al. (2015)), one can transform this system into the system with a constant coefficient matrix by the following coordinate transformation [ ] [ ′] x x =S(t) ′ . (10) p p In the new coordinate, the linear part of the canonical equation (9) becomes time-invariant: [ ˙′ ] [ ′] [ ] x Nx′ (t, x′ , p′ ) ′ x , (11) ′ =F ′ + ′ ′ Np′ (t, x , p ) p p where the non-linear part satisfies [ ] ] [ Nx′ (t, x′ , p′ ) −1 Nx (t, x, p) . = S(t) Np (t, x, p) Np′ (t, x′ , p′ )
The assumption that (7) has a T -periodic stabilizing solution and Theorem A.1 assure that F ′ has no eigenvalues on the imaginary axis and therefore, there exists a 2n × 2n real constant V such that ] [ Λ 0 (12) V −1 F ′ V = 0 −ΛT where Λ is a real stable n × n matrix. Then, the canonical equation in (11) can be put into the form (b1) where the stable manifold computation (b2) proposed in Theorem B.1 in Appendix B can be applied.
Let (x′′k (t, ξ), p′′k (t, ξ)) be the sequence obtained by (b2) for the canonical equation with the block-diagonalized linear part with k (the number of iteration) sufficiently large. Then, the sequence [ ] [ ′′ ] xk (t, ξ) xk (t, ξ) = S(t)V ′′ pk (t, ξ) pk (t, ξ) are the solution on the stable manifold of (4).
Eliminating ξ from {(xk (t, ξ), pk (t, ξ)) | t ∈ R, |ξ| : sufficiently small} for sufficiently large k, one obtains an approximation of the stable manifold L of (9). Since L is locally canonically surjective to the x-space, it can be represented as p = p(t, x) (13) with some function p(t, x) which is T -periodic for t. The optimal control is then given by 1 u∗ = − R−1 g(t, x)T p(t, x). 2 217
4. STABILIZATION OF A PERIODIC ORBIT IN ROSSLER CHAOTIC SYSTEM 4.1 Rossler chaotic system and its periodic orbit Here, we demonstrate the effectiveness of the proposed method using the Rossler system, introduced by Rossler (1976). Let us consider the Rossler system with three inputs q˙ = A q + N + u, (14) where T T q = [q1 , q2 , q3 ] , u = [u1 , u2 , u3 ] ] ] [ [ 0 0 −1 −1 0 0 ,N= A= 1 a q1 q3 q30 0 q10 − c √ √ c − c2 − 4ab c − c2 − 4ab , q30 = . q10 = 2 2a In this system with some parameters (a, b, c), unstable periodic orbits can be observed. We choose the parameter set (a, b, c) = (0.2, 0.2, 3), for which it is known that there exists an unstable periodic orbit. Applying the method of delayed feedback to the system (14), we found an unstable periodic orbit. The characteristic multipliers of this periodic orbit are (−1.11, 1, −2.01× 10−7 ). Since the periodic orbit has a characteristic multiplier with modulus more than unity, the periodic orbit is unstable. We wish to design a control law stabilizing this periodic orbit in an optimal fashion. 4.2 Optimal control problem Let qc = [q1c (t), q2c (t), q3c (t)]T be the periodic solution given in the previous subsection and T be the period. Consider the error dynamics along qc (t) x˙ =Ac (t) x + Nc + u, where x = [x1 , x2 , x3 ]T = q − qc is the error state and ] ] [ [ 0 −1 −1 0 1 a 0 0 . Ac (t) = , Nc = q30 + q3c (t) 0 q10 + q1c (t) − c x 1 x3 For this system, we define the cost function as ∫ ∞ J= xT Qx + uT Ru dt. 0
To obtain a control law minimizing input, we select the following weight matrices; Q = 10−3 · I3×3 , R = I3×3 ∫∞ This is an approximation of the cost function 0 uT Ru dt, from which we expect to derive an extension of OGY control in the sense that the control input is kept as low as possible using the sensitivity of chaotic system. This cost function, however, does not lead a solution for the stable regulator problem in Definition 2.2. The inclusion of small Q > 0 is one of the easiest way for the stable regulator problem to have a solution. Let us define the Hamiltonian H(t, x, u, p) = pT (Ac x + Nc + u) + xT Qx + uT Ru.
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Setting u ¯(x, p) = −1/2R−1 p, the Hamilton-Jacobi equation for this problem is ∂V ∂V ∂V = H(t, x, u ¯(x, ), ) − ∂t ∂x ∂x T ∂V = (Ac x + Nc ) + xT Qx ∂x 1 ∂V T −1 ∂V , R − 4 ∂x ∂x and Hamilton’s canonical equations are { 1 x˙ = Ac x + Nc − R−1 p, (15) 2 p˙ = −2Q x − ATc p − dNc p, where ] [ 0 0 x3 dNc = 0 0 0 . 0 0 x1 Let p = p(t, x) be the stable manifold of the canonical equation, then the solution of this optimal control problem is given as u∗ = −1/2R−1 p(t, x).
Fig. 1. Set of optimal trajectories in Rossler system with control 7UDMHFWRU\ 3HULRGLFRUELW ,QLWLDOFRQGLWLRQ &RQWUROVWDUWLQJSRLQW
4.3 Derivation of Floquet factor
′ = S(t)−1 Nqp . Then the eigenvalues of F ′ is where Nqp obtained as λ = ±2.672, ±0.1640, ±0.0765 It shows that the assumption in §3 is satisfied. Thus, we can apply the stable manifold method.
4.4 Controller representation To obtain the approximate solution of Hamilton-Jacobi equation, we apply the stable manifold method in §3 to the system (17). When one employs the computational method of stable manifold to the optimal stabilization problem of an unstable periodic orbit qc (t), the periodic function p(t, x) in (13) can be written as p(t, x) = p˜(x + qc (t)) = p˜(q) and therefore, the control law u∗ = −1/2R−1 p˜(q) is in a feedback form. For the construction of the control law, we calculate qk (t, ξ) = qc (t) + xk (t, ξ) which correspond to the closed loop orbits converging to the periodic orbit qc (t) as t → ∞. In Fig.1, we show the set of the optimally controlled trajectories qk (t, ξ). Fig 1 shows projected data on q-space. From the corresponding data for p-space, we compute p˜(q) and the optimal control law u∗ (q) = −1/2R−1 p˜(q). 218
]
Extracting the linear part, (15) can be written as [ ] [ ] x˙ x =F + Nqp , (16) p˙ p where ] [ ] [ 1 Nc Ac − R−1 , Nqp = F = . 2 T −dNc p −2Q −Ac F is a T -periodic matrix, that is, F (t + T ) = F (t). Then, we can apply Floquet-Lyapnov transformation so as for the linear part to be time-invariant. Let S(t) and F ′ be the transformation and constant coefficient matrix each. Using the transformation (10), we have [ ′] [ ′] x˙ ′ x ′ + Nqp , F ′ : constant, (17) ′ =F p˙ p′
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Fig. 2. Time responses for q(0) = [3, −8, 0]T (proposed method). The controller is activated from the beginning. 4.5 Simulation results To demonstrate the effectiveness of the proposed method, numerical simulations of the Rossler system under the control are performed. We set the initial condition q(0) = [3, −8, 0]T . The simulation results are shown in Fig. 2 and Fig. 3. Fig. 2 illustrates the behavior of the states in the Cartesian coordinate system, and Fig. 3 shows the error from the periodic orbit. From Fig. 2 or Fig. 3, it can be seen that the designed control law stabilizes the periodic orbit. We show the input time history during stabilization in Fig. 4. The original motivation of our research is to establish a framework for designing stabilizing control of unstable periodic orbits using as small input as possible by exploiting the initial condition sensitivity ofc chaos. To this end, we design a control law using optimal OGY method proposed by Epureanu and Dowell (2000), and compare the input behaviors by our control and optimal OGY control. The results of the simulation with the same initial condition by the optimal OGY method are presented in Fig.5 -
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Fig. 6. Error from the periodic orbit (optimal OGY method)
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Table 1. The values of cost function Ju Method Proposed Optimal OGY
]
Ju 0.0074 0.023
The main advantage of the proposed method is the capability to take full nonlinearity into account. It can be seen that the control law works well even in the area far from the periodic orbit, whereas OGY method has to wait until the states come into the prescribed region to keep the input small enough.
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Fig. 5. Time responses for q(0) = [3, −8, 0] (optimal OGY method). The controller is activated t = 460.9s (shown as the red circle). Fig.7. Since the OGY method is based on linearization of the system around the target periodic orbit, the effective domain is limited. In our simulation, the optimal OGY control is activated when the error in the Euclidean norm becomes less than 0.5. If one activates the control outside of this domain, the input by this controller can be unrealistically large. Fig.5 shows the trajectory in the Cartesian coordinate system, and Fig.6 illustrates the error from the designated periodic orbit at each period of time. In Fig.7, we show the input behavior after the OGY control is activated at t = 460. Although the maximum input value in the optimal OGY method is similar to the one in the proposed method, the squared integral of the input over the control duration is significantly improved. The values of the cost function ∫∞ Ju = 0 uT u dt are listed in Table 1.
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In this paper, we proposed an approach to design optimal control for unstable periodic orbits in chaotic systems using stable manifold method. First, we have developed a theory of optimal control for nonlinear periodic systems and apply it for the error dynamics of target periodic orbit in a chaotic system. The method is based on computing a stable manifold of canonical equation associated with the Hamilton-Jacobi equation that is equivalent to the optimal control problem for the unstable orbit. When one tries to apply the proposed method, more exact knowledge of the model is necessary unlike the OGY method or Time-Delayed Feedback Control. The method, however, yields a control law that optimally stabilizes the unstable orbit by taking the nonlinearity into account. Moreover, the effective domain of the proposed method is larger than the OGY method while control effort is kept even smaller than the OGY method. Simulation results for the Rossler chaotic system shows the effectiveness of the proposed method.
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REFERENCES Biggs, J. and R.McInnes, C. (2009). An optimal gains matrix for time-delay feedback control. In 2nd IFAC Conference on the Analysis and Control of Chaotic Systems, 22–24. Ding, Y., Jiang, W., and Wang, H. (2010). Delayed feedback control and bifurcation analysis of Rossler chaotic system. Nonlinear Dynamics, 61(4), 707–715. Epureanu, B.I. and Dowell, E.H. (2000). Optimal multi-dimensional OGY controller. Physica D, 139, 87–96. Fujimoto, R. and Sakamoto, N. (2011). The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input saturation. In the 18th IFAC World Congress, volume 18. 8046–8051. Garfinkel, A., Spano, M.L., Ditto, W.L., and Weiss, J.N. (1992). Controlling cardiac chaos. Science, 257, 1230–1235. Just, W., Bernard, T., Ostheimer, M., Reibold, E., and Benner, H. (1997). Mechanism of time-delayed feedback control. Physical Review Letters, 78(2), 203–206. Kostelich, E.J., Grebogi, C., Ott, E., and Yorke, J.A. (1993). Higherdimensional targeting. Physical Review E, 47(1), 305–310. Meyer, K.R., Hall, G.R., and Offin, D. (2015). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer, second edition. Ott, E., Grebogi, C., and Yorke, J.A. (1990). Controlling chaos. Physical Review Letters, 64(11), 1196–1199. Postlethwaite, C.M. and Silber, M. (2007). Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control. Physical Review E, 76(056214), 1–10. Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback. Physics Letters A, 170, 421–428. Romeiras, F.J., Grebogi, C., Ott, E., and Dayawansa, W. (1992). Controlling chaotic dynamical systems. Physica D, 58, 165–192. Rossler, O. (1976). An equation for continuous chaos. Physics Letters A, 57A(5), 397–398. Sakamoto, N. (2013). Case studies on the applications of the stable manifold approach for nonlinear optimal control design. Automatica, 49, 568–576. Sakamoto, N. and van der Schaft, A.J. (2008). Analytical approximation methods for the stabilizing solution of the Hamilton-Jacobi equation. IEEE Transactions on Automatic Control, 53(10), 2335–2350. Schiff, S.J., Jerger, K., Duong, D.H., Chang, T., Spano, M.L., and Ditto, W.L. (1994). Controlling chaos in the brain. Nature, 370(25), 615–620. Shinbrot, T., Ott, E., Grebogi, C., and Yorke, J.A. (1990). Using chaos to direct trajectories to targets. Physical Review Letters, 65(26), 3215–3218. Yuasa, Y., Sakamoto, N., and Umemura, Y. (2010). Optimal control designs for systems with input saturations and rate limiters. In SICE Annual Conference 2010, 2042–2045.
Appendix A. THEORY OF LINEAR PERIODIC SYSTEMS In this Appendix, we briefly review the theory of linear periodic systems that plays a major role throughout the paper. Linear periodic Riccati differential equation Suppose A(t) ∈ Rn×n and B(t) ∈ Rn×m are T -periodic matrices. Let us consider a T -periodic Riccati differential equation and the T -periodic Hamiltonian matrix associated with the Riccati equation −P˙ = P A(t) + A(t)T P − P B(t)R−1 B(t)T P + Q, (a1) ] [ 1 A(t) − B(t)R−1 B(t)T . (a2) 2 −2Q −A(t)T 220
Definition A.1. A T -periodic solution P (t) of (a1) is called a stabilizing solution when it satisfies (a1) and x˙ = (A(t)− R−1 B(t)T P (t))x is asymptotically stable. Theorem A.1. A necessary and sufficient condition for the Riccati equation (a1) to have a periodic stabilizing solution is that the Monodromy matrix of (a2) has no eigenvalues on the unit circle and the pair (A(·), B(·)) is stabilizable in the sense of periodic systems. Appendix B. STABLE MANIFOLD METHOD In this Appendix, we briefly present the stable manifold algorithm proposed in Sakamoto and van der Schaft (2008). The detail application procedure of the method, such as the convergence issue of the algorithm and the function representation of the stable manifold, can be found in Sakamoto (2013). Let us consider the equation of the form [ ] [ ][ ] [ ] x˙ Λ1 0 x Nx (t, x, p) = + , p˙ 0 Λ2 p Np (t, x, p)
(b1)
where all the eigenvalues of Λ1 , −Λ2 lie in the open left half plane i.e., Re λj (Λ1 ) < 0, Re λj (Λ2 ) > 0, j = 1, . . . , n. Assumption B.1. Nx , Np : R × Rn × Rn → Rn are continuous and satisfy the following. (i) For all t ∈ R, |u| + |v| < l and |u′ | + |v ′ | < l, |Nx (t, u, v)−Np (t, u′ , v ′ )| < δ1 (l)(|u − u′ | + |v − v ′ |). (ii) for all t ∈ R, |u| + |v| < l and |u′ | + |v ′ | < l, |Np (t, u, v)−Np (t, u′ , v ′ )| < δ2 (l)(|u − u′ | + |v − v ′ |), where δj : [0, ∞) → [0, ∞), j = 1, 2 are continuous and monotonically increasing on [0, Lj ] for some constants L1 , L2 > 0. Let us define the sequences {xk (t, ξ)} and {pk (t, ξ)} by ∫ t Λ1 t xk+1 = e ξ + eΛ1 (t−s) Nx (t, xk , pk ) ds ∫ ∞ 0 (b2) Λ2 (t−s) = − e N (t, x , p ) ds p k+1 p k k t
for k = 0, 1, 2, . . ., and {
x0 = eΛ1 t ξ p0 = 0
where ξ ∈ Rn is arbitrary as long as |ξ| is sufficiently small. Then, we have the following theorem. Theorem B.1. (Sakamoto and van der Schaft (2008)) Under Assumption B.1, the sequence xk (t, ξ) and pk (t, ξ) are convergent to zero for sufficiently small |ξ|, that is, xk (t, ξ), pk (t, ξ) → 0 as t → ∞ for all k = 0, 1, 2, . . .. Furthermore, xk (t, ξ) and pk (t, ξ) are uniformly convergent to a solution of (b1) on [0, ∞) as k → ∞. Let x(t, ξ) and p(t, ξ) be the limits of xk (t, ξ) and pk (t, ξ), respectively. Then, x(t, ξ) and p(t, ξ) are a solution on the stable manifold of (b1), that is, x(t, ξ), p(t, ξ) → 0 as t → ∞.