Stabilizing Unstable Periodic Orbits with Delayed Feedback Control in Act-and-Wait Fashion
Ahmet Cetinkaya, Tomohisa Hayakawa, Mohd Amir Fikri bin Mohd Taib
aa r X i v : . [ c s . S Y ] F e b Stabilizing Unstable Periodic Orbits with Delayed Feedback Controlin Act-and-Wait Fashion
Ahmet Cetinkaya a , Tomohisa Hayakawa b , Mohd Amir Fikri bin Mohd Taib b a Department of Computer Science,Tokyo Institute of Technology, Yokohama 226-8502, Japan b Department of Systems and Control Engineering,Tokyo Institute of Technology, Tokyo 152-8552, Japan [email protected], [email protected], [email protected]
Abstract
A delayed feedback control framework for stabilizing unstable periodic orbits of linear periodic time-varying systems is proposed.In this framework, act-and-wait approach is utilized for switching a delayed feedback controller on and off alternately at everyinteger multiples of the period of the system. By analyzing the monodromy matrix of the closed-loop system, we obtain conditionsunder which the closed-loop system’s state converges towards a periodic solution under our proposed control law. We discussthe application of our results in stabilization of unstable periodic orbits of nonlinear systems and present numerical examples toillustrate the efficacy of our approach.
Keywords:
Stabilization of periodic orbits, periodic systems, time-varying systems, delayed feedback stabilization
1. Introduction
Stabilization of unstable periodic orbits of nonlinear sys-tems using delayed feedback control was first explored in [1].In the delayed feedback control scheme, the difference betweenthe current state and the delayed state is utilized as a controlinput to stabilize an unstable orbit. The delay time is set to cor-respond to the period of the orbit to be stabilized so that thecontrol input vanishes when the stabilization is achieved.Delayed feedback controllers have been used in many stud-ies for stabilization of the periodic orbits of both continuous-and discrete-time nonlinear systems (see, e.g., [2, 3, 4], andthe references therein). More recently, [5] investigated delayedfeedback control of nonlinear systems that are subject to noise,[6] explored delayed feedback control of a delay differentialequation, and [7] utilized delayed feedback control for stabiliz-ing quasi periodic orbits. The work [8] studied the relation be-tween the delayed feedback control approach and the harmonicoscillator-based control methods for stabilizing periodic orbitsin chaotic systems [9]. Furthermore, [10] and [11] exploredthe situation where the period of the orbit and the delay timein the delayed feedback controller do not match due to imper-fect information about the periodic orbit or inaccuracies in theimplementation of the controller.The physical structure of delayed feedback control schemeis simple. However, the analysis of the closed-loop system isdifficult. This is due to the fact that to investigate the systemunder delayed feedback control, one has to deal with delay-differential equations, the state space of which is infinite dimen-sional. To deal with the difficulties in the analysis of delay dif-ferential equations, an approach is to use approximation tech- niques (see, for instance, [12] and [13]). Another approach wastaken in [14]. There, stabilization of a linear time-invariant sys-tem with a time-delay controller was considered, and “act-and-wait” concept was introduced. This concept is characterized byalternately applying and cutting off the controller in finite in-tervals. It is shown in [14] that by utilizing the act-and-waitconcept, one may be able to derive a finite-sized monodromymatrix for the closed-loop system, which can then be used forstability analysis. Act-and-wait concept has been extended todiscrete-time systems in [15], and tested through experimentsin [16]. Furthermore, act-and-wait approach has been used to-gether with delayed feedback control in [17] for stabilizing un-stable fixed points of nonlinear systems, and more recently in[18] for stabilizing unstable periodic orbits of nonautonomousnonlinear systems.In this paper, we explore the stabilization of periodic solu-tions to linear periodic systems with an act-and-wait-fashioneddelayed feedback control framework. In this framework, aswitching mechanism is utilized to turn the delayed feedbackcontroller on and off alternately at every integer multiple of theperiod of a given linear periodic system. Act-and-wait schemeallows us to obtain the monodromy matrix associated with theclosed-loop system under our proposed controller. We then usethe obtained monodromy matrix for obtaining conditions underwhich the closed-loop system’s state converges to a periodicsolution. Our main motivation for studying a delayed feedbackcontrol problem for periodic systems stems from our desire toanalyze the stability of a periodic orbit of a nonlinear systemunder delayed feedback control. In this paper we apply ourresults for linear periodic systems in analyzing periodic linearvariational equations obtained after linearizing nonlinear sys-
Preprint submitted to Elsevier August 22, 2018 ems (under delayed feedback control) around periodic trajec-tories corresponding to periodic orbits. The uncontrolled non-linear systems that we consider are autonomous and as a resulttheir stability assessment under the act-and-wait-fashioned de-layed feedback controller differs from the nonautonomous casediscussed in [18]. We also note that our delayed feedback con-trol approach and therefore our analysis techniques differ fromthose in earlier works on stabilization of linear periodic systemswhere researchers have employed Gramian-based controllers[19], periodic Lyapunov functions [20], and linear matrix in-equalities [21].The paper is organized as follows. In Section 2, we in-troduce our act-and-wait-fashioned delayed feedback controlframework for stabilizing periodic solutions of linear periodicsystems; we present a method for assessing the asymptotic sta-bility of a periodic solution of the closed-loop system under ourproposed framework. Furthermore, in Section 3 we discuss anapplication of our results in stabilizing unstable periodic orbitsof nonlinear systems. We present illustrative numerical exam-ples in Section 4. Finally, we conclude our paper in Section 5.We note that a preliminary version of this work was pre-sented in [22]. In this paper, we provide additional discussionsand examples.
2. Delayed Feedback Stabilization of Periodic Orbits
In this section, we provide the mathematical model for a lin-ear periodic time-varying system and introduce a new delayedfeedback control framework based on act-and-wait approach.We then characterize a method for evaluating convergence ofstate trajectories of a closed-loop linear time-varying periodicsystem towards a periodic solution.
Consider the linear periodic time-varying system ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) , x ( t ) = x , t ≥ t , (1)where x ( t ) ∈ R n is the state vector, u ( t ) ∈ R m is the con-trol input, and A ( t ) ∈ R n × n and B ( t ) ∈ R n × m are peri-odic matrices with period T > , that is, A ( t + T ) = A ( t ) and B ( t + T ) = B ( t ) , t ≥ t . For simplicity of exposition,we assume t = 0 for the rest of the discussion because thecase where t = 0 can be similarly handled. Furthermore,we assume that the uncontrolled ( u ( t ) ≡ ) dynamics pos-sess a periodic solution x ( t ) ≡ x ∗ ( t ) with period T satisfying x ∗ ( t + T ) = x ∗ ( t ) , t ≥ . It follows from Floquet’s theoremthat there exists such a periodic solution to the uncontrolledsystem (1) of period T if and only if there exists a nonsingu-lar matrix C ∈ R n × n possessing in its spectrum such that V ( t + T ) = V ( t ) C , where V ( t ) denotes a fundamental ma-trix of the uncontrolled system (1). Moreover, note that since x ( t ) ≡ x ∗ ( t ) is a T -periodic solution of the uncontrolled sys-tem (1), x ( t ) ≡ αx ∗ ( t ) is also a T -periodic solution for all α ∈ R , that is, x ( t ) ≡ αx ∗ ( t ) satisfies (1) with u ( t ) ≡ . We investigate the asymptotic stability of periodic solutionsof the closed-loop system (1) under the delayed feedback con-trol input u ( t ) = − g ( t ) F ( x ( t ) − x ( t − T )) , (2)where F ∈ R m × n is a constant gain matrix and g ( t ) , (cid:26) , kT ≤ t < (2 k + 1) T, , (2 k + 1) T ≤ t < k + 1) T, k ∈ N . (3)is a time-varying function that switches the controller on andoff alternately at every integer multiples of the period T .Note that the feedback term characterized in (2) vanishesafter the periodic solution is stabilized. Specifically, for x ( t ) ≡ x ∗ ( t ) , we have u ( t ) = 0 , t ≥ , since x ( t ) = x ( t − T ) .We remark that our control approach is a specific case of theact-and-wait approach introduced in [14]. In particular, in ourcontrol law (2), both the acting and the waiting durations havelength T . Specifically, in every T period, the controller firstwaits for a duration of length T , and then acts for a duration oflength T . Note that the controllers in [14] are more general inthe sense that acting and waiting times need not be equal. InSection 4, we also consider different switching functions g ( t ) that lead to different acting and waiting times.The reason why g ( t ) is set to be a time-varying functioncan be understood if we compare it to the case where g ( t ) isconstant. For instance, if g ( t ) ≡ in (2), then (1) becomes ˙ x ( t ) = ( A ( t ) − B ( t ) F ) x ( t ) + B ( t ) F x ( t − T ) , (4)which is a delay-differential equation. Analysis of the solu-tion of (4) is difficult, as the state space associated with (4) isinfinite-dimensional.On the other hand, for the linear periodic system ˙ x ( t ) = A ( t ) x ( t ) , A ( t ) = A ( t + T ) , (5)where there are no delay terms, stability of an equilibrium so-lution can be assessed by analyzing the corresponding mon-odromy matrix. Let Φ( · , · ) denote the state-transition matrixof (5). The monodromy matrix associated with the T -periodicsystem (5) is given by Φ( T, ∈ R n × n . The eigenvalues ofthe monodromy matrix, known as the Floquet multipliers, areessential in the analysis of the long-term behavior of the state-transition matrix of (5), because Φ( t + kT,
0) = Φ( t, k ( T, , k ∈ N . (6)Moreover, the state of the periodic system (5) satisfies x (( k + 1) T ) = Φ x ( kT ) , k ∈ N . Observe that if g ( t ) ≡ in (2), we would not be able to finda homogeneous expression in the form of (5), let alone find acorresponding “monodromy matrix”, because of the existenceof the delay term.However, in our case, following the act-and-wait approach,we define g ( t ) as in (3) as a switching function. Consequently,2e are able to construct a monodromy matrix Λ ∈ R n × n forthe closed-loop system (1), (2) with the doubled period T suchthat x (2( k + 1) T ) = Λ x (2 kT ) , k ∈ N . (7)Note that the spectrum of the monodromy matrix Λ character-izes long-term behavior of the state trajectory.In the following sections, we first derive the monodromymatrix, and then we present conditions for the convergence ofthe state trajectory towards a periodic solution of the closed-loop system (1), (2). In this section, we obtain the monodromy matrix associatedwith the closed-loop system given by (1), (2). In our deriva-tions, we use Φ( · , · ) to denote the state-transition matrix associ-ated with (5). Furthermore, let Υ( · , · ) denote the state-transitionmatrix for the linear T -periodic system ˙ x ( t ) = ( A ( t ) − B ( t ) F ) x ( t ) . (8)Now, let T ( k ) , [2 kT, (2 k + 1) T ) , T ( k ) , [(2 k +1) T, k + 1) T ) , k ∈ N . Note that when t ∈ T ( k ) , thecontroller is on, that is, g ( t ) = 1 . Hence, it follows from (1)and (2) that for t ∈ T ( k ) , ˙ x ( t ) = ( A ( t ) − B ( t ) F ) x ( t ) + B ( t ) F x ( t − T ) . (9)Observe that for t ∈ T ( k ) , we have t − T ∈ T ( k ) . Since thecontroller is turned off during the interval T ( k ) , the evolutionof the state in this interval is described by (5) correspondingto the uncontrolled dynamics. Therefore, x ( t − T ) can be ex-pressed by x ( t − T ) = Φ( t − T, kT ) x (2 kT ) , t ∈ T ( k ) . (10)Now, by using (9) and (10), we obtain ˙ x ( t ) = ( A ( t ) − B ( t ) F ) x ( t )+ B ( t ) F Φ( t − T, kT ) x (2 kT ) , t ∈ T ( k ) . (11)By multiplying both sides of (11) from left with the matrix Υ − ( t, (2 k + 1) T ) , we obtain Υ − ( t, (2 k + 1) T ) ˙ x ( t )= Υ − ( t, (2 k + 1) T )( A ( t ) − B ( t ) F ) x ( t )+ Υ − ( t, (2 k + 1) T ) B ( t ) F Φ( t − T, kT ) x (2 kT ) . (12)Since Υ − ( t, (2 k + 1) T )( A ( t ) − B ( t ) F ) = − dd t Υ − ( t, (2 k +1) T ) , we have Υ − ( t, (2 k + 1) T ) ˙ x ( t ) − Υ − ( t, (2 k + 1) T )( A ( t ) − B ( t ) F ) x ( t )= Υ − ( t, (2 k + 1) T ) ˙ x ( t ) + dd t Υ − ( t, (2 k + 1) T ) x ( t )= dd t (cid:0) Υ − ( t, (2 k + 1) T ) x ( t ) (cid:1) . (13) It follows from (12) and (13) that dd t (cid:0) Υ − ( t, (2 k + 1) T ) x ( t ) (cid:1) = Υ − ( t, (2 k + 1) T ) B ( t ) F Φ( t − T, kT ) x (2 kT ) . (14)Next, we integrate both sides of (14) over the interval [(2 k +1) T, t ) to obtain Υ − ( t, (2 k + 1) T ) x ( t )= Υ − ((2 k + 1) T, (2 k + 1) T ) x ((2 k + 1) T )+ (cid:16) Z t (2 k +1) T Υ − ( s, (2 k + 1) T ) B ( s ) F Φ( s − T, kT )d s (cid:17) · x (2 kT ) . (15)Noting that Υ − ((2 k + 1) T, (2 k + 1) T ) = I n and x ((2 k +1) T ) = Φ((2 k + 1) T, kT ) x (2 kT ) , we obtain x ( t ) = Υ( t, (2 k + 1) T )Φ((2 k + 1) T, kT ) x (2 kT )+ Υ( t, (2 k + 1) T ) · (cid:16) Z t (2 k +1) T Υ − ( s, (2 k + 1) T ) B ( s ) F Φ( s − T, kT )d s (cid:17) · x (2 kT ) . (16)Since both (5) and (8) are T -periodic, we have Υ( t, (2 k +1) T ) =Υ( t − (2 k + 1) T, and Φ((2 k + 1) T, kT ) = Φ( T, . Fur-thermore, since Υ( t, (2 k +1) T ) = Υ( t, s )Υ( s, (2 k +1) T ) , s ∈ [(2 k + 1) T, t ] , we have Υ( t, (2 k + 1) T )Υ − ( s, (2 k + 1) T ) =Υ( t, s ) . Consequently, it follows from (16) that x ( t ) = (cid:16) Υ( t − (2 k + 1) T, T, Z t (2 k +1) T Υ( t, s ) B ( s ) F Φ( s − T, kT )d s (cid:17) x (2 kT ) , (17)for t ∈ T ( k ) . We now change the variable of the integral termin (17) by setting ¯ s = s − kT . As a result, we obtain Z t (2 k +1) T Υ( t, s ) B ( s ) F Φ( s − T, kT )d s = Z t − kTT Υ( t, ¯ s + 2 kT ) B (¯ s + 2 kT ) · F Φ(¯ s + 2 kT − T, kT )d¯ s, t ∈ T ( k ) . (18)By T -periodicity of (5) and (8), we have Υ( t, ¯ s +2 kT ) = Υ( t − kT, ¯ s ) and Φ(¯ s + 2 kT − T, kT ) = Φ(¯ s − T, . Moreover, B (¯ s +2 kT ) = B (¯ s ) , since B ( · ) is a T -periodic matrix function.It then follows from (18) that Z t (2 k +1) T Υ( t, s ) B ( s ) F Φ( s − T, kT )d s = Z t − kTT Υ( t − kT, ¯ s ) B (¯ s ) F Φ(¯ s − T, s. (19)3ow, from (17) and (19), we obtain x ( t ) = (cid:16) Υ( t − (2 k + 1) T, T, Z t − kTT Υ( t − kT, ¯ s ) B (¯ s ) F Φ(¯ s − T, s (cid:17) · x (2 kT ) , t ∈ T ( k ) . (20)By continuity of the state, we can compute x (2( k + 1) T ) byusing (20). Specifically, we set t = 2( k + 1) T in (20) andobtain (7), where the monodromy matrix Λ is given by Λ = (cid:16) Υ( T, T, Z TT Υ(2 T, ¯ s ) B (¯ s ) F Φ(¯ s − T, s (cid:17) . (21)Notice that (7) characterizes the evolution of the state at times t = 2 kT , k ∈ N . Consequently, stability of the equilibriumsolutions of the closed-loop system (1), (2) can be deducedthrough the eigenvalues of the monodromy matrix.Moreover, note that x ∗ ( t ) satisfies x ∗ (2 T ) = x ∗ (0) . Inaddition, from (7) we have x ∗ (2 T ) = Λ x ∗ (0) . (22)It follows that x ∗ (0) = Λ x ∗ (0) , and hence, the monodromymatrix Λ associated with the closed-loop system (1), (2) pos-sesses as an eigenvalue with the eigenvector x ∗ (0) . Note thatboth the algebraic and the geometric multiplicity of the eigen-value may be greater than .Let κ ∈ { , , . . . , n } denote the algebraic multiplicity ofthe eigenvalue . We represent generalized eigenvectors of themonodromy matrix Λ by vectors v , v , . . . , v n ∈ C n out ofwhich v , v . . . , v κ ∈ R n denote the generalized eigenvec-tors associated with the eigenvalue . The generalized eigen-vectors v , v , . . . , v n are linearly independent [23], and henceform a basis for C n , that is, for any y ∈ C n , there exist α , α , . . . , α n − , α n ∈ C such that y = P ni =1 α i v i . Note that α i v i characterizes the component of y along v i (the i th ele-ment of the basis). Note also that linear independence of vec-tors v , v , . . . , v n guarantees that the constants α , α , . . . , α n are uniquely determined.The long-term behavior of the state trajectory is determinedby the spectrum of the monodromy matrix Λ . In Theorem 2.1below, we present the condition for the convergence of the statetrajectory towards a periodic solution. Theorem 2.1.
Consider the linear time-varying periodic sys-tem (1) with the periodic solution x ∗ ( t ) . Let the initial conditionbe given by x (0) = x , P ni =1 α i v i , where v , v , . . . , v n ∈ C n are the generalized eigenvectors of the monodromy matrix Λ ∈ R n × n , and α , α , . . . , α n ∈ C . Suppose that is asemisimple eigenvalue of Λ . Let κ ∈ { , , . . . , n } denote thealgebraic multiplicity of the eigenvalue associated with theeigenspace spanned by the eigenvectors v , v . . . , v κ ∈ R n .If all the eigenvalues, other than the eigenvalue of the mon-odromy matrix Λ are strictly inside the unit circle of the complexplane, then lim k →∞ x (2 kT ) = P κi =1 α i v i . Proof.
First, we define P , [ v , v , . . . , v n ] . Note thatsince the columns of the matrix P given by v , v , . . . , v n arelinearly independent, it follows that P is nonsingular.Furthermore, note that with the similarity transformation J , P − Λ P , we obtain the Jordan form of the monodromymatrix Λ such that J = diag[ J , J , . . . , J r ] , where r ∈ N denotes the number of Jordan blocks, which is also equal tothe sum of the geometric multiplicities of the eigenvalues ofthe monodromy matrix Λ . The Jordan blocks J i ∈ C n i × n i , i ∈ { , , . . . , r } , have the form J i = λ i I n i + N i , where λ i ∈ C is an eigenvalue of the monodromy matrix Λ and N i ∈ R n i × n i is a nilpotent matrix of degree n i .We use (7), the definition for the Jordan form J of the mon-odromy matrix Λ , and [ α , α , . . . , α n ] T = P − x to obtain x (2 kT ) = Λ k x = P J k P − x = P diag[ J k , J k , . . . , J kr ][ α , α , . . . , α n ] T . (23)Note that the eigenvalue , which has algebraic multiplicity κ ∈ { , , . . . , n } , is a semisimple eigenvalue associated withthe eigenvectors v , v . . . , v κ ∈ R n . As a result, J i = 1 , i ∈{ , , . . . , κ } , and hence lim k →∞ J ki = 1 , i ∈ { , , . . . , κ } .On the other hand, for each i ∈ { κ + 1 , κ + 2 , . . . , r } the eigen-value λ i is strictly inside the unit circle; therefore, lim k →∞ J ki =0 , i ∈ { κ + 1 , κ + 2 , . . . , r } . Thus, lim k →∞ J k takes the formof a diagonal matrix with as the first κ diagonal entries and elsewhere. It follows from (23) that lim k →∞ x (2 kT ) = P diag[ κ terms z }| { , . . . , , n − κ terms z }| { , . . . , · [ α , α , . . . , α n ] T = P [ α , . . . , α κ , , . . . , T = κ X i =1 α i v i , (24)which completes the proof. (cid:3) Under the condition in Theorem 2.1 that all the eigenval-ues, other than the semisimple eigenvalue , of the monodromymatrix Λ are strictly inside the unit circle, the state evaluated atinteger multiples of the doubled period T converges to a pointon a periodic solution of the uncontrolled ( u ( t ) ≡ ) system(1). Hence, the state trajectory converges towards a periodicsolution. The location of the limiting periodic solution dependson the initial condition x . Specifically, the state evaluated at in-teger multiples of the doubled period T converges to the pointgiven by P κi =1 α i v i , where α i v i characterizes the componentof the initial state x along the eigenvector v i associated withthe eigenvalue of the monodromy matrix Λ . Note that if the Here without loss of generality we are considering the case where in theconstruction of P , the generalized eigenvectors associated with each eigenvalueare next to each other and they are in the same order that they appear in the Jor-dan chain (see [23]) associated with that eigenvalue. Generalized eigenvectorscan always be reordered in this way to obtain a Jordan form. is κ = 1 , then thelimiting periodic solution is given by α x ∗ ( t ) , where α x ∗ (0) is the component of x along the eigenvector v = x ∗ (0) .
3. Stabilization of Unstable Periodic Orbits of NonlinearSystems
We now employ the results obtained for linear periodic sys-tems for stabilizing an unstable periodic orbit of a nonlinearsystem.Consider the nonlinear system given by ˙ x ( t ) = f ( x ( t )) + u ( t ) , x (0) = x , t ≥ , (25)where x ( t ) ∈ R n is the state vector, u ( t ) ∈ R n is the controlinput, and f : R n → R n is a nonlinear function. Suppose thatthe uncontrolled ( u ( t ) ≡ ) system (25) possesses a periodicsolution x ( t ) ≡ x ∗ ( t ) with a known period T > such that x ∗ ( t + T ) = x ∗ ( t ) , (26) ˙ x ∗ ( t ) = f ( x ∗ ( t )) , t ∈ [0 , ∞ ) . (27)The periodic orbit associated with the periodic solution x ∗ ( · ) is given by O , { x ∗ ( t ) : t ∈ [0 , T ) } . The stability of theperiodic orbit O ⊂ R n is characterized through the stability of afixed point of a Poincaré map defined on an ( n − -dimensionalhypersurface that is transversal to the periodic orbit (see [24,25]). In this paper, we consider the case where O is an unstableperiodic orbit (UPO) and discuss its stabilization.There are several methods known for stabilizing the UPO.One of them is the Pyragas-type delayed feedback control frame-work (see [1, 3, 4]). In this framework the control input is givenby u ( t ) = − F ( x ( t ) − x ( t − T )) , (28)where F ∈ R n × n is the gain matrix of the controller. Thecontrol input is computed based on the difference between thecurrent state and the delayed state. The delay time is set tocorrespond to the period T of the desired UPO so that the con-trol input vanishes after the UPO is stabilized. In the delayedfeedback control method, the controller uses the delayed stateinstead of the UPO as a reference signal to which the currentstate is desired to be stabilized. Therefore this method doesnot require a preliminary calculation of the UPO if its period isgiven.The analysis of the closed-loop system under delayed feed-back controller is difficult, because the closed-loop dynamics isdescribed by a delay-differential equation (25), (28), the statespace of which is infinite-dimensional. This fact is our moti-vation for employing the act-and-wait-fashioned delayed feed-back control law (2), since the closed-loop system system (2),(25) can be analyzed by utilizing the methods that we developedin Section 2. Remark 3.1.
Act-and-wait-fashioned delayed-feedback controllaws were previously used in [17] and [18] for different prob-lem settings. In [17], the stabilization of a fixed-point is con-sidered. Furthermore, in [18], periodic orbit stabilization is considered for a nonautonomous system. Specifically, the un-controlled system in [18] is affected by an external periodicforce, which induces the periodic orbit. The controller in [18]is designed so that all Floquet multipliers of the linearized sys-tem are strictly inside the unit circle of the complex plane. Inour case, the uncontrolled system is not driven by a periodicforce and it is autonomous. The periodic orbit in our case isembedded in the dynamics. The stability assessment methodin this paper differs from that in [18] due to the difference inthe analysis of autonomous and nonautonomous systems (seeSection 7.1.3 of [26]). In particular, as we discuss below, thelinearized system in our case always possesses as a Floquetmultiplier regardless of the choice of the feedback gain matrix,and moreover, the stability of the periodic orbit under the act-and-wait-fashioned controller can be analyzed by assessing theFloquet multipliers that are not . We analyze the stability of the periodic orbit
O ⊂ R n un-der the act-and-wait-fashioned control input by assessing themonodromy matrix for the linear variational equation associ-ated with the closed-loop dynamics (2), (25). Specifically, welinearize the closed-loop system (2), (25) around the periodictrajectory x ∗ ( t ) . First, we write the solution of (2), (25) as x ( t ) = x ∗ ( t ) + δx ( t ) , (29)where δx ( t ) is the state deviation from the periodic solution x ∗ ( t ) at time t . It then follows from (25) and (29) that ˙ x ∗ ( t ) + ˙ δx ( t )= f ( x ∗ ( t ) + δx ( t )) − g ( t ) F ( x ∗ ( t ) + δx ( t ) − x ∗ ( t − T ) − δx ( t − T ))= f ( x ∗ ( t )) + ∂f ( x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x ∗ ( t ) δx ( t ) + H.O.T. − g ( t ) F ( δx ( t ) − δx ( t − T )) , (30)where H.O.T. denotes the higher-order terms in δx .By using (27) and neglecting the higher-order terms for in-finitely small deviations in (30), we obtain the linear variationalequation ˙ δx ( t ) = A ( t ) δx ( t ) − g ( t ) F ( δx ( t ) − δx ( t − T )) , (31)where A ( t ) , ∂f ( x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x ∗ ( t ) . (32)Since x ∗ ( t ) is a T -periodic trajectory, it follows from (32) that A ( t ) is also T -periodic, that is, A ( t ) = A ( t + T ) . Hence, thelinear variational dynamics (31) is in fact characterized by thelinear periodic system (1) (with B ( t ) = I n ) under the act-and-wait-fashioned delayed feedback controller (2). Consequently,the monodromy matrix Λ associated with the linear variationaldynamics (31) can be obtained by using (21).Next, we show that ˙ x ∗ ( t ) is a periodic solution of the linearvariational dynamics. First, by using (25) with u ( t ) ≡ , weobtain d ˙ x ( t )d t = ∂f ( x ( t )) ∂x ( t ) d x ( t )d t . (33)5ence, for x ( t ) ≡ x ∗ ( t ) , it follows from (32) and (33) that d ˙ x ∗ ( t )d t = A ( t ) ˙ x ∗ ( t ) . (34)Thus, the linear variational equation characterized by (31) has ˙ x ∗ ( t ) as a solution, that is, δx ( t ) ≡ ˙ x ∗ ( t ) satisfies (31). Tosee that ˙ x ∗ ( t ) is T -periodic, note that for x ( t ) ≡ x ∗ ( t ) , wehave u ( t ) ≡ , and hence, (26) and (27) imply ˙ x ∗ ( t + T ) = f ( x ∗ ( t + T )) = f ( x ∗ ( t )) = ˙ x ∗ ( t ) , t ≥ .Note that ˙ x ∗ ( T ) = Φ( T,
0) ˙ x ∗ (0) , where Φ( T, is themonodromy matrix for the uncontrolled system. Since, ˙ x ∗ ( T ) =˙ x ∗ (0) , it follows that ˙ x ∗ (0) is an eigenvector of the monodromymatrix Φ( T, corresponding to the eigenvalue . Note alsothat ˙ x ∗ (0) = ˙ x ∗ (2 T ) = Λ ˙ x ∗ (0) , (35)where Λ is the monodromy matrix associated with the closed-loop linear variational dynamics (31). It follows from (35)that Λ possesses as an eigenvalue with the correspondingeigenspace { λ ˙ x ∗ (0) : λ ∈ C } regardless of the choice of feed-back gain matrix F in the control law (2).Note that the eigenvalues of the monodromy matrix Λ thatare not associated with the eigenspace { λ ˙ x ∗ (0) : λ ∈ C } cor-respond to the eigenvalues of a linearized Poincaré map, whichcharacterize the local asymptotic stability of the periodic orbit O of the nonlinear system (25) (see [24]). Remark 3.2.
Following the approach presented in [25] and[27], we characterize the asymptotic stability of the periodicorbit x ∗ ( · ) of the closed-loop nonlinear system (2) , (25) , by as-sessing the spectrum of the monodromy matrix associated withthe linear variational dynamics (31) . In particular, the act-and-wait-fashioned Pyragas-type delayed feedback control law (2) asymptotically stabilizes the periodic orbit O of (25) if all theeigenvalues, other than the eigenvalue associated with theeigenspace { λ ˙ x ∗ (0) : λ ∈ C } , of the monodromy matrix Λ ofthe linear variational equation (31) are strictly inside the unitcircle of the complex plane. Note that the monodromy matrix of the linear variationalequation (31) can be calculated using (21) with B ( t ) = I n . Forthe periodic matrix A ( t ) ∈ R n × n and a feedback gain matrix F ∈ R m × n , numerical methods can be employed to calculatethe eigenvalues of the monodromy matrix Λ , which determinethe asymptotic stability. Note also that the conditions on themonodromy matrix of the linear variational equation is onlyenough to guarantee local stability of the periodic orbit of thenonlinear system (2), (25) (see Chapter 7 of [26]). For obtain-ing global stability results, the higher-order terms in (30) haveto be taken into consideration.
4. Illustrative Numerical Examples
In this section, we provide two numerical examples to demon-strate our main results. − − − − − PSfrag replacements x ∗ ( t ) x ∗ ( t )Time [ t ] Figure 1: Trajectory of the T -periodic solution x ∗ ( t ) given by (36) Example 4.1
Consider the linear time-varying periodic sys-tem (1) described by periodic matrices A ( t ) , " . . π sin(2 πt )2 − cos(2 πt ) , B ( t ) , (cid:20)
01 + sin (2 πt ) (cid:21) . The period of the time-varying system is T = 1 , that is, A ( t +1) = A ( t ) and B ( t + 1) = B ( t ) . The uncontrolled ( u ( t ) ≡ )dynamics possess a T -periodic solution x ( t ) ≡ x ∗ ( t ) given by x ∗ ( t ) , (cid:20) π ) (cos(2 πt ) − π sin(2 πt )) − − cos(2 πt ) (cid:21) . (36)Note that x ( t ) ≡ αx ∗ ( t ) , where α ∈ R , is also a T -periodicsolution of the uncontrolled system, that is, x ( t ) ≡ αx ∗ ( t ) sat-isfies (1) with u ( t ) ≡ . Fig. 1 shows the trajectory of theperiodic solution x ( t ) ≡ x ∗ ( t ) .The monodromy matrix associated with the uncontrolledsystem is given by Φ( T,
0) = (cid:20) . . (cid:21) , (37)which has the eigenvalues . and . Note that the eigen-value . of the monodromy matrix Φ( T, lies outside theunit circle. The periodic system without control input, hence,shows unstable behavior (see Figure 2 for state trajectories ob-tained for the initial condition x = [ − . , . T ).We are interested in finding a feedback gain matrix F ∈ R × such that the delayed-feedback control characterized in(2) guarantees convergence of the state trajectory towards a pe-riodic solution. In order to evaluate the asymptotic behaviorof solutions under the control law (2) we need to examine theeigenvalues of the monodromy matrix Λ associated with the6 T =10 2 4 6 8 1011.522.53 PSfrag replacements x ( t ) x ( t ) Time [ t ]Time [ t ] Figure 2: State trajectories of the uncontrolled ( u ( t ) ≡ ) system (1) for theinitial condition x = [ − . , . T closed-loop system. It is difficult to find an analytical expres-sion for the monodromy matrix Λ . For that reason, we nu-merically calculate the value of Λ for a certain range of feed-back gain parameters and search for a feedback gain matrix F ∈ R × that satisfies the condition in Theorem 2.1. Note thatfor the feedback gain matrix F = [4 . , . , the correspondingmonodromy matrix is given by Λ = (cid:20) . . − . − . (cid:21) , (38)which has the eigenvalues λ = 1 and λ = 0 . associ-ated with the eigenvectors v = x ∗ (0) = [ − . , T and v = [ − . , . T , respectively. Note that the eigen-value λ is inside the unit circle of the complex plane. There-fore, it follows from Theorem 2.1 that under the control law(2), the state trajectory evaluated at integer multiples of thedoubled period T converges to α v , where α v representsthe component of a given initial condition x along the eigen-vector v = x ∗ (0) . Figures 3 and 4 respectively show thephase portrait and state trajectories of the closed-loop system(1), (2) for the initial condition x = [ − . , . T , which canbe represented as x = α v + α v where α = 0 . and α = − . . Note that lim k →∞ x (2 kT ) = α v = α x ∗ (0) and hence the state trajectory converges to the periodic solution α x ∗ ( t ) . Note that the convergence is achieved with the helpof the proposed delayed feedback controller, which is turnedon and off alternately at every integer multiples of the period T = 1 of the uncontrolled system, and hence the control input(shown in Figure 5) is discontinuous at time instants T , T , T , . . . . Note also that the control input converges to as the stateconverges to the periodic solution. Example 4.2
In this example we demonstrate the utilityof our proposed control framework for the stabilization of anunstable periodic orbit of a nonlinear system. Specifically, we − − − − − Figure 3: Phase portrait of the closed-loop system (1), (2) obtained with theinitial condition x = [ − . , . T PSfrag replacements x ( t ) x ( t ) x ( t ) x ( t ) α x ∗ ( t ) α x ∗ ( t ) Time [ t ]Time [ t ] Figure 4: State trajectories of the closed-loop system (1), (2) obtained with theinitial condition x = [ − . , . T T =1 PSfrag replacements u ( t ) Time [ t ] Figure 5: Control input versus time − − − − − Figure 6: Phase portrait of the uncontrolled ( u ( t ) ≡ ) nonlinear system (25) consider the nonlinear dynamical system (25) with f ( x ) , (cid:20) − x (cid:0) ϕ ( x ) − ϕ ( x ) (cid:1) + 2 πx − x (cid:0) ϕ ( x ) − ϕ ( x ) (cid:1) − πx (cid:21) , (39)where ϕ ( x ) , x + x . This system is a modified version ofan example nonlinear dynamical system considered in Section2.7 of [28]. The phase portrait of the uncontrolled ( u ( t ) ≡ )nonlinear system (25) is shown in Figure 6. The system hasclockwise-revolving unstable periodic orbit O , { x ∗ ( t ) : t ∈ [0 , T ) } , where x ∗ ( t ) = [cos 2 πt, − sin 2 πt ] T is a -periodic so-lution of the uncontrolled system. The linear variational equa-tion associated with the closed-loop system (25) under our pro-posed controller (2) (with T = 1 ) is given by (31), where A ( t ) = (cid:20) a , ( t ) a , ( t ) a , ( t ) a , ( t ) (cid:21) (40)with a , ( t ) = 2 cos (2 πt ) , (41) a , ( t ) = − πt ) sin(2 πt ) + 2 π, (42) a , ( t ) = − πt ) sin(2 πt ) − π, (43) a , ( t ) = 2 sin (2 πt ) . (44)Note that it is difficult to find an analytical expression for themonodromy matrix Λ associated with the linear variational equa-tion (31). For this reason, we numerically calculate the value of Λ for a certain range of the elements of the gain matrix. Inparticular, for the case F = (cid:20) . . − . . (cid:21) , (45)the corresponding monodromy matrix is given by Λ = (cid:20) − . − . (cid:21) , (46) − − − − − − − − Figure 7: Phase portrait of the closed-loop system (2), (25). Inset figure showsthe initial condition x = [1 , − . T as well as the state evaluated at integermultiples of T . which has the eigenvalues λ = − . , λ = 1 , and eigen-vectors v = [ − . , − . T , v = [0 , − T . Note thatthe eigenvalue λ (eigenvalue that is not ) is inside the unitcircle. Therefore, the feedback control law (2) with the feed-back gain matrix (45) asymptotically stabilizes the periodic or-bit x ∗ ( · ) of the nonlinear system (25) (see Remark 3.2).Figures 7 and 8 respectively show the phase portrait and thestate magnitude of the closed-loop system (2), (25) obtainedfor the initial condition x = [1 , − . T . The state trajectoryconverges to the periodic orbit and the state magnitude k x ( t ) k converges to the desired value .Note that in this paper, we investigate local stability of theperiodic orbit of the nonlinear system through a linearizationapproach. The initial condition x in the simulation is selectedclose to x ∗ (0) , so that δx ( t ) (the state deviation from the pe-riodic solution) remains small and the effect of higher-orderterms in the variational equation (30) is negligible. Note that,if the initial state is far from the orbit, then the control law (2)with the feedback gain F may no longer achieve stabilization,since the higher order terms in the variational equation mayhave strong effects on the state trajectory. For achieving globalstabilization, the higher-order terms in (30) have to be takeninto consideration for control design.The control input trajectory (shown in Figure 9) is discon-tinuous at time instants T, T, T, . . . . At these time instants,the delayed-feedback controller is turned on and off alternatelyaccording to the switching function g ( t ) defined in (3). For the T -periodic switching function g ( t ) , the closed-loop system is T -periodic; hence, the stabilization of the UPO could be char-acterized through the monodromy matrix Λ of the T -periodicclosed-loop linear variational equation (31).Note that stabilization of the UPO could also be achieved8 T =1 PSfrag replacements k x ( t ) k Time [ t ] Figure 8: State magnitude versus time T =10 1 2 3 4 5 6 7 8 9 10−0.0500.05 Integer multiples of period T =1 PSfrag replacements u ( t ) u ( t ) Time [ t ]Time [ t ] Figure 9: Control input versus time through utilization of alternative switching sequences that aredifferent from the one induced by g ( t ) . For example, one canconsider the switching function ˆ g ( t ) , (cid:26) , kT ≤ t < (3 k + 1) T, , (3 k + 1) T ≤ t < k + 1) T, k ∈ N . (47)In this case the act-and-wait fashioned control law is given by u ( t ) , − ˆ g ( t ) F ( x ( t ) − x ( t − T )) . (48)Note that the closed-loop system under the controller (48) is T -periodic, thus by analyzing the spectrum of the monodromymatrix associated with the T -periodic closed-loop linear vari-ational equation, we can assess whether the control law (48)guarantees local asymptotic stabilization of the UPO or not.According to the switching sequence induced by ˆ g ( t ) , thecontroller is active two-thirds of the time in average. On theother hand, in the case of g ( t ) defined in (3), the controlleris active only half of the time. Hence, one may think that afeedback gain matrix that guarantees stabilization of the UPOfor the case of g ( t ) guarantees stabilization also for the case of ˆ g ( t ) . However, this is not true. A feedback gain that guaranteesstabilization for one switching sequence does not necessarilyguarantee stabilization for another. For instance, the control in-put (2) with the gain matrix F given by (45) achieves stabiliza-tion (as illustrated in Figures 7 and 8), whereas the control input(48) with the same gain matrix F does not stabilize the UPO. In fact, the monodromy matrix of the T -periodic closed-looplinear variational equation under the control law (48) possessesthe eigenvalue − . , which is outside the unit circle.In addition to utilizing a different switching sequence forthe act-and-wait controller, we may also employ a different de-lay term for the feedback for stabilizing the UPO. For example,consider the act-and-wait fashioned control law u ( t ) , − ¯ g ( t ) F ( x ( t ) − x ( t − T )) , (49)with the switching function ¯ g ( t ) , (cid:26) , kT ≤ t < (4 k + 2) T, , (4 k + 2) T ≤ t < k + 1) T, k ∈ N . (50)Note that in this case the control law (49) harnesses the differ-ence between the state at time t (current state) and the state attime t − T . Furthermore, with the switching function ¯ g ( t ) ,the controller is turned on and off alternately at time instants T, T, T, . . . . The closed-loop system under the control law(49) is T -periodic. It is important to note here that the con-trol input (49) with a feedback gain F does not necessarilyachieve stabilization of the UPO, even if the control law (2)with the same feedback gain guarantees stabilization. For ex-ample, with the feedback gain (45), the control law (2) achievesstabilization of the UPO, whereas the control law (49) does not.In fact, the monodromy matrix of the linear variational equa-tion associated with the T -periodic closed-loop system underthe control law (49) with the feedback gain (45) has the eigen-value . , which is outside the unit circle. The discussionabove illustrates that it is possible to obtain different controllaws by changing the act-and-wait sequence and/or changingthe delayed-feedback term; furthermore, each case with a dif-ferent control law requires independent analysis for assessingasymptotic stabilization of the UPO.In practical implementations of our act-and-wait-fashionedcontrol laws, the timing of the switching may not always be ex-act. Furthermore, it may also be the case that the delay timeand the period of the orbit do not exactly match. The effectsof such practical issues require further analysis. We note thatfor the standard delayed feedback control, the effect of mis-matches between the delay time and the period of the orbit wasanalyzed in [11]. We also note that although the trajectories ofthe act-and-wait-fashioned control law has discontinuities, thisdoes not cause a practical problem in the form of chattering,because the switching in the control law happens only period-ically with a period that is an integer multiple of the period ofthe orbit to be stabilized.
5. Conclusion
We explored stabilization of the periodic orbits of linear pe-riodic time-varying systems through an act-and-wait-fashioneddelayed feedback control framework. Our proposed frameworkemploys a switching mechanism to turn the control input onand off alternately at every integer multiples of the period ofthe desired orbit. The use of this mechanism allows us to derive9he monodromy matrix associated with the closed-loop dynam-ics. By analyzing the eigenvalues of the monodromy matrix, weobtained conditions under which the state trajectory convergestowards a periodic solution. We then applied our results in sta-bilization of unstable periodic orbits of nonlinear systems. Wediscussed alternative switching sequences for turning the con-troller on and off in our control framework. We observe that afeedback gain that guarantees stabilization for one switching se-quence does not necessarily guarantee stabilization for another.It is also interesting to observe that increasing the amount oftime the control input is turned on does not necessarily increasethe performance; in fact it may result in instability.In the act-and-wait-fashioned control laws that we consid-ered, the waiting duration where the control input is turned offis larger than or equal to the delay amount. It is shown in [29]that the act-and-wait approach can also be useful in obtainingfinite-dimensional monodromy matrices even if the waiting du-ration is smaller than the delay. One of our future research di-rections is to investigate the utility of small waiting and actingdurations in the delayed feedback control of periodic orbits.
Acknowledgements
This research was supported by the Aihara Project, the FIRSTprogram from JSPS, initiated by CSTP.
References [1] K. Pyragas, Continuous control of chaos by self-controlling feedback,Phys. Lett. A 170 (1992) 421–428.[2] I. Harrington, J. E. S. Socolar, Design and robustness of delayed feedbackcontrollers for discrete systems, Phys. Rev. E 69 (2004) 056207.[3] Y. Tian, J. Zhu, G. Chen, A survey on delayed feedback control of chaos,J. Control Theo. Appl. 3 (4) (2005) 311–319.[4] P. Hövel, Control of Complex Nonlinear Systems with Delay, Springer-Verlag: Berlin, 2010.[5] V. Pyragas, K. Pyragas, Adaptive search for the optimal feedback gain oftime-delayed feedback controlled systems in the presence of noise, Eur.Phys. J. B 86 (7) (2013).[6] B. Fiedler, S. M. Oliva, Delayed feedback control of a delay equation atHopf bifurcation, J. Dyn. Differ. Equ. 28 (3-4) (2016) 1357–1391.[7] N. Ichinose, M. Komuro, Delayed feedback control and phase reductionof unstable quasi-periodic orbits, Chaos 24 (3) (2014) 033137.[8] V. Pyragas, K. Pyragas, Relation between the extended time-delayed feed-back control algorithm and the method of harmonic oscillators, Phys. Rev.E 92 (2) (2015) 022925. [9] A. A. Olyaei, C. Wu, Controlling chaos using a system of harmonic os-cillators, Phys. Rev. E 91 (1) (2015) 012920.[10] V. Noviˇcenko, K. Pyragas, Phase-reduction-theory-based treatment of ex-tended delayed feedback control algorithm in the presence of a small timedelay mismatch, Phys. Rev. E 86 (2) (2012) 026204.[11] A. S. Purewal, C. M. Postlethwaite, B. Krauskopf, Effect of delay mis-match in Pyragas feedback control, Phys. Rev. E 90 (5) (2014) 052905.[12] H. Ma, V. Deshmukh, E. Butcher, V. Averina, Controller design for lineartime-periodic delay systems via a symbolic approach, in: Proc. Amer.Contr. Conf., 2003, pp. 2126–2131.[13] E. A. Butcher, H. Ma, E. Bueler, V. Averina, Z. Szabo, Stability of lin-ear time-periodic delay-differential equations via Chebyshev polynomi-als, Int. J. Numer. Meth. Engng. 59 (2004) 895–922.[14] T. Insperger, Act-and-wait concept for continuous-time control systemswith feedback delay, IEEE Trans. Contr. Sys. Tech. 14 (5) (2006) 974–977.[15] T. Insperger, G. Stepan, Act-and-wait control concept for discrete-timesystems with feedback delay, IET Cont. Theory Applications 1 (3) (2007)553–557.[16] T. Insperger, L. Kovacs, P. Galambos, G. Stepan, Increasing the ac-curacy of digital force control process using the act-and-wait concept,IEEE/ASME Trans. Mechatronics 15 (2) (2010) 291–298.[17] K. Konishi, H. Kokame, N. Hara, Delayed feedback control based on theact-and-wait concept, Nonlinear Dynamics 63 (3) (2011) 513–519.[18] V. Pyragas, K. Pyragas, Act-and-wait time-delayed feedback control ofnonautonomous systems, Phys. Rev. E 94 (1) (2016) 012201.[19] P. Montagnier, R. J. Spiteri, A Gramian-based controller for linear peri-odic systems, IEEE Trans. Autom. Contr. 49 (8) (2004) 1380–1385.[20] B. Zhou, G.-R. Duan, Periodic Lyapunov equation based approaches tothe stabilization of continuous-time periodic linear systems, IEEE Trans.Autom. Contr. 57 (8) (2012) 2139–2146.[21] C. E. De Souza, A. Trofino, An LMI approach to stabilization of lineardiscrete-time periodic systems, Int. J. Contr. 73 (8) (2000) 696–703.[22] M. A. F. Mohd Taib, T. Hayakawa, A. Cetinkaya, Delayed feedback con-trol for linear time-varying periodic systems in act-and-wait fashion, in:Proc. 5th IFAC Int. Workshop on Periodic Control Systems, 2013, pp.11–16.[23] D. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas, Prince-ton University Press: Princeton, 2011.[24] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields, Springer: New York, 2002.[25] C. Chicone, Ordinary Differential Equations with Applications, Springer:New York, 2006.[26] R. Leine, H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Me-chanical Systems, Springer-Verlag: Berlin, 2013.[27] L. Meirovitch, Methods of Analytical Dynamics, Dover: New York,2003.[28] H. K. Khalil, Nonlinear Systems, Prentice Hall: New Jersey, 2002.[29] T. Insperger, G. Stepan, On the dimension reduction of systems with feed-back delay by act-and-wait control, IMA J. Math Contr. Inform. 27 (2011)457–473.[1] K. Pyragas, Continuous control of chaos by self-controlling feedback,Phys. Lett. A 170 (1992) 421–428.[2] I. Harrington, J. E. S. Socolar, Design and robustness of delayed feedbackcontrollers for discrete systems, Phys. Rev. E 69 (2004) 056207.[3] Y. Tian, J. Zhu, G. Chen, A survey on delayed feedback control of chaos,J. Control Theo. Appl. 3 (4) (2005) 311–319.[4] P. Hövel, Control of Complex Nonlinear Systems with Delay, Springer-Verlag: Berlin, 2010.[5] V. Pyragas, K. Pyragas, Adaptive search for the optimal feedback gain oftime-delayed feedback controlled systems in the presence of noise, Eur.Phys. J. B 86 (7) (2013).[6] B. Fiedler, S. M. Oliva, Delayed feedback control of a delay equation atHopf bifurcation, J. Dyn. Differ. Equ. 28 (3-4) (2016) 1357–1391.[7] N. Ichinose, M. Komuro, Delayed feedback control and phase reductionof unstable quasi-periodic orbits, Chaos 24 (3) (2014) 033137.[8] V. Pyragas, K. Pyragas, Relation between the extended time-delayed feed-back control algorithm and the method of harmonic oscillators, Phys. Rev.E 92 (2) (2015) 022925. [9] A. A. Olyaei, C. Wu, Controlling chaos using a system of harmonic os-cillators, Phys. Rev. E 91 (1) (2015) 012920.[10] V. Noviˇcenko, K. Pyragas, Phase-reduction-theory-based treatment of ex-tended delayed feedback control algorithm in the presence of a small timedelay mismatch, Phys. Rev. E 86 (2) (2012) 026204.[11] A. S. Purewal, C. M. Postlethwaite, B. Krauskopf, Effect of delay mis-match in Pyragas feedback control, Phys. Rev. E 90 (5) (2014) 052905.[12] H. Ma, V. Deshmukh, E. Butcher, V. Averina, Controller design for lineartime-periodic delay systems via a symbolic approach, in: Proc. Amer.Contr. Conf., 2003, pp. 2126–2131.[13] E. A. Butcher, H. Ma, E. Bueler, V. Averina, Z. Szabo, Stability of lin-ear time-periodic delay-differential equations via Chebyshev polynomi-als, Int. J. Numer. Meth. Engng. 59 (2004) 895–922.[14] T. Insperger, Act-and-wait concept for continuous-time control systemswith feedback delay, IEEE Trans. Contr. Sys. Tech. 14 (5) (2006) 974–977.[15] T. Insperger, G. Stepan, Act-and-wait control concept for discrete-timesystems with feedback delay, IET Cont. Theory Applications 1 (3) (2007)553–557.[16] T. Insperger, L. Kovacs, P. Galambos, G. Stepan, Increasing the ac-curacy of digital force control process using the act-and-wait concept,IEEE/ASME Trans. Mechatronics 15 (2) (2010) 291–298.[17] K. Konishi, H. Kokame, N. Hara, Delayed feedback control based on theact-and-wait concept, Nonlinear Dynamics 63 (3) (2011) 513–519.[18] V. Pyragas, K. Pyragas, Act-and-wait time-delayed feedback control ofnonautonomous systems, Phys. Rev. E 94 (1) (2016) 012201.[19] P. Montagnier, R. J. Spiteri, A Gramian-based controller for linear peri-odic systems, IEEE Trans. Autom. Contr. 49 (8) (2004) 1380–1385.[20] B. Zhou, G.-R. Duan, Periodic Lyapunov equation based approaches tothe stabilization of continuous-time periodic linear systems, IEEE Trans.Autom. Contr. 57 (8) (2012) 2139–2146.[21] C. E. De Souza, A. Trofino, An LMI approach to stabilization of lineardiscrete-time periodic systems, Int. J. Contr. 73 (8) (2000) 696–703.[22] M. A. F. Mohd Taib, T. Hayakawa, A. Cetinkaya, Delayed feedback con-trol for linear time-varying periodic systems in act-and-wait fashion, in:Proc. 5th IFAC Int. Workshop on Periodic Control Systems, 2013, pp.11–16.[23] D. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas, Prince-ton University Press: Princeton, 2011.[24] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields, Springer: New York, 2002.[25] C. Chicone, Ordinary Differential Equations with Applications, Springer:New York, 2006.[26] R. Leine, H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Me-chanical Systems, Springer-Verlag: Berlin, 2013.[27] L. Meirovitch, Methods of Analytical Dynamics, Dover: New York,2003.[28] H. K. Khalil, Nonlinear Systems, Prentice Hall: New Jersey, 2002.[29] T. Insperger, G. Stepan, On the dimension reduction of systems with feed-back delay by act-and-wait control, IMA J. Math Contr. Inform. 27 (2011)457–473.