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On the initial function space of time-delayed systems: A time-delayed feedback control perspective
Available online at www.sciencedirect.com
Journal of the Franklin Institute 352 (2015) 3243–3249 www.elsevier.com/locate/jfranklin
On the initial function space of time-delayed systems: A time-delayed feedback control perspective Huailei Wanga,n, Guanrong Chenb a
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China b Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, PR China Received 28 March 2014; received in revised form 7 October 2014; accepted 27 October 2014 Available online 4 November 2014
Abstract We address the problem as if the initial function space of a time-delayed system can be finitedimensional. We argue that the dimension of the initial function space of a time-delayed feedback control system is just identical to that of the corresponding plant which is to be controlled, and so is finitedimensional in many cases. We prove this property by constructing an injection map between the two spaces. We anticipate that this may also be applied to some other systems, such as ecological processes involving time delays without control. We believe that the significance of this discovery not only makes available the physical interpretation and intuitional explanation of the basins of attraction associated with time-delayed feedback control, but also suggests the principle as how to evaluate the robustness of the timedelayed feedback control schemes to external disturbances. We finally visualize and illustrate such basins of attraction via numerical simulation for the classical Duffing system by equipping it with the velocity timedelayed feedback control. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Thanks to the advantages of stabilizing time-periodic states of a dynamical system without the need of precise information about the controlled object, the time-delayed feedback control (TDFC) n
Corresponding author. E-mail address: [email protected] (H.L. Wang).
http://dx.doi.org/10.1016/j.jfranklin.2014.10.021 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
3244
H.L. Wang, G.R. Chen / Journal of the Franklin Institute 352 (2015) 3243–3249
scheme, initiated by Pyragas in 1992 [1], has been developed into an active research direction in applied nonlinear science [2–6]. Although the experimental implementation of TDFC is straightforward, its theoretical analysis proves difficult since the dynamics of a time-delayed system takes place in an infinite-dimensional phase space [7]. To date, the most efficient approach to discussing the performance of TDFC is based on linear stability analysis [8–10], yielding mainly local results whilst general global results are rare [11,12]. One typical challenge is to determine the basins of attraction, even for stabilizing, to a particular equilibrium or periodic orbit of a nonlinear system under TDFC. Studies on the basins of attraction are essential for experimental implementation and engineering applications of the TDFC technique because control performance may largely depend on the basin of attraction of the stabilized state in a nonlinear system [13]. It is known that the essential difficulty for the global analysis of time-delayed systems roots in their infinite dimension of the phase space, therefore a common approach to dealing with the challenge is to find possibilities of reducing the phase space to a finite-dimensional one. Unfortunately, earlier mathematical analysis has revealed that relevant delayed differential equations belong to the category of functional differential equations, also referred to as differential equations in the Banach space, which are of infinite-dimension in nature. In this study we take a new perspective of viewing the initial function space of a TDFC system through a careful analysis of the control process. Our analysis leads to the conclusion that the seemingly infinite-dimensional initial function space of a TDFC system is actually determined by a finite-dimensional initial state space, however this conclusion cannot be extended to all general timedelayed differential equations. Based on the new finding, we can conveniently characterize and visualize the basin of attraction of a TDFC system in a finite-dimensional Euclidean space. We must emphasize that this feature is not just a novel dimensional reduction, but revealing of the very nature of the original initial function space. This is a kind of equivalence in which, unlike other dimensionreduction methods, no dynamical characteristics of the TDFC system will be lost. 2. Finite-dimensional initial state space A diagrammatic view of a typical TDFC system is given in Fig. 1. Let the internal variables of the system be denoted by a state vector xðtÞ. The dynamical equation of the TDFC system can be described by x_ ðtÞ ¼ hðx; uÞ; x A Rn ; uA Rm
ð1Þ
where h is a continuous nonlinear vector function, u is the time-delayed state-feedback control vector, which is defined by a continuous nonlinear vector function g: u ¼ gðxðtÞ; xðt τÞÞ
ð2Þ
where τ40 is the time delay constant. Combining Eqs. (1) and (2) leads to x_ ðtÞ ¼ hðxðtÞ; gðxðtÞ; xðt τÞÞÞ
ð3Þ Plant
g( x (t ), x (t − τ ))
x (t ) Delay x(t − τ )
Fig. 1. Schematic diagram of time-delayed feedback control system.
H.L. Wang, G.R. Chen / Journal of the Franklin Institute 352 (2015) 3243–3249
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which can be simply represented by a continuous map: x_ ðtÞ ¼ fðxðtÞ; xðt τÞÞ
ð4Þ
where fðxðtÞ; xðt τÞÞ hðxðtÞ; gðxðtÞ; xðt τÞÞÞ Eq. (4) is in the general form of a differential-difference equation, which belongs to the category of retarded functional differential equations (RFDEs). The essential difference between such equations and the ordinary differential equations (ODEs) lies in that the state space of an RFDE is a Banach space defined as C ¼ Cð½0; τ; Rn Þ, namely the continuous function space mapping the interval ½0; τ onto the Euclidean space Rn . To determine a specific solution of Eq. (4), one has to define an initial function on the initial time interval t A ½0; τ, instead of an initial state at the instant t ¼ 0 for the ODEs. In other words, an abstract ‘initial point’ of an RFDE is a function defined on the initial time interval ½0; τ. As a stabilized equilibrium of a nonlinear system might not be globally stable due to the nonlinearity, there is a common concern about the control domain of the designed control law, namely to what extent is the disturbance to a system allowed for successfully stabilizing the system to the target state? To answer this question, one has to check the initial functions in interest. However, this is impossible due to the infinite number of initial functions (abstract ‘initial points’) defined on the initial time interval ½0; τ. Yet, we should point out that in some specific scenarios where the situation may not be as general, we may be able to take advantage of some special structures and properties of the system to draw a different conclusion, as follows. Theorem 1. The initial function space of a control system with time-delayed feedback as formulated in Eqs. (1) and (2) can be finite-dimensional, and its dimension is determined by the nature of the plant itself, which is independent of the delayed feedback control. Proof. Differing from the idea developed in [14], which is rather straightforward but incomplete, we consider this problem from another perspective. Inspecting the control diagram in Fig. 1, we first notice that the plant has its own dynamics and may be subject to disturbances from outside before the control is put into action. As soon as the control loop is turned on, we record this very moment as the initial instant t ¼ 0. From then on, in order to implement the delayed control, the measured signals have to be continuously stored in a stack buffer for a period of time τ before the buffer is completely full to shape into the first control force formulated by Eq. (2). That is, not until the moment t ¼ τ does the control force begin to take action, as the buffer has signal data at the moment t but has no data at t τ during the time period t A ½0; τ. However, it can be seen that afterwards, i.e. when t4τ, the control force will take effect continuously with measured signal data continuing to flow in and flow out, causing the stack buffer to be always full.,. From the above analysis of the mechanism of time-delayed control, we can now assert that an initial function of the time-delayed control system is composed of the stored signal data in the buffer, which is just ‘produced’ by the plant itself during the initial time interval t A ½0; τ when the feedback control has not been activated to take effect. Thus, an available initial function for the TDFC system cannot be given arbitrarily as a continuous function, but is determined by the dynamical nature of the plant itself on the initial time interval, which in many cases can be finite-dimensional. □ For the case of a plant described by an ODE with the states in the Euclidean space Rn , the initial state of the plant at t ¼ 0 in Rn determines the initial function on the time interval t A ½0; τ
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in C ¼ Cð½0; τ; Rn Þ, and the initial function on t A ½0; τ determines the entire solution of the TDFC system for all the time afterwards. This implies an injection map from Rn into C ¼ Cð½0; τ; Rn Þ and a one-to-one map between C ¼ Cð½0; τ; Rn Þ and the solution space of the TDFC system, as illustrated by Fig. 2. It is then straightforward to deduce that it is the initial state of the plant in Rn that determines the dynamics of the TDFC system. Therefore, the true initial function space of the TDFC system in this case is merely a subset of the general initial function space, in the sense that one may actually identify the initial time interval ½0; τ with a single initial time instant, t ¼ 0, to produce all the initial functions in interest. From this perspective, the coupled state equation of the TDFC system can be recast to ( x_ ðtÞ ¼ hðxðtÞ; 0Þ; xð0Þ ¼ x0 ; 0r t r τ ð5Þ x_ ðtÞ ¼ hðxðtÞ; uÞ; t Z τ where u ¼ gðxðtÞ; xðt τÞÞ is the time-delayed feedback control. Now, system (5) clearly is a finite-dimensional one, a typical non-delayed switching system with the switching time at t ¼ τ, regardless of the controller u to be used hereafter. 3. Demonstration of finite-dimensional basin of attraction of a TDFC system Consider a simple plant with dynamics governed by the autonomous Duffing equation x€ þ ω20 x þ μx3 ¼ 0; xA R
ð6Þ
where ω0 40, μ40. Clearly, the only steady state of the plant is the equilibrium x ¼ 0, which is stable but not asymptotically stable. Thus, the velocity time-delayed feedback control is introduced to stabilize. The controlled system is as follows [15]: ( x€ þ ω20 x þ μx3 ¼ u; x A R; u A R ð7Þ u ¼ v_xðt τÞ or, in the coupled form, x€ þ ω20 x þ μx3 ¼ v_xðt τÞ
ð8Þ
where the time parameter t is omitted wherever no time delay is involved. According to the above discussions, Eq. (8) can be recast into two switching systems: ( x€ þ ω20 x þ μx3 ¼ 0; 0 r t r τ x€ þ ω20 x þ μx3 ¼ v_xðt τÞ; t Z τ
TDFC system
Plant system
Initial state space in
ð9Þ
Solution
Solution space
R
Initial function space n
C = C ([0, τ ], R )
Fig. 2. Illustration of the sets of initial points in R , the initial function space, the solution space, and their relationships.
H.L. Wang, G.R. Chen / Journal of the Franklin Institute 352 (2015) 3243–3249
It can be equivalently reformulated as the following state space equation: 8 x_ ¼ y > > > 0r t r τ ðaÞ > < y_ ¼ ω20 x þ μx3 ( x_ ¼ y > > > > : y_ ¼ ω2 x þ μx3 þ vyðt τÞ t Z τ ðbÞ 0
3247
ð10Þ
To solve Eq. (10), we first need to set an initial state ðxð0Þ; yð0ÞÞ for Eq. (10)(a), which represents the initial position and initial velocity of the plant respectively, so as to produce the initial function for Eq. (10)(b), and then use the computed initial function to solve Eq. (10)(b). Based on the above analysis, the basin of attraction for an attractor of the TDFC system (10) is in fact a set of initial states in R2 , starting from which the dynamics of the TDFC system will eventually settle down on that attractor. This property not only enables us to completely avoid the difficulty as how to classify initial functions of the TDFC system, but also makes it available for anyone to view the basin of attraction in the plane R2 through the classification of initial state coordinates. To obtain and visualize the results numerically, we scan a constrained area of the plane by a small incremental step and then check the endresults of all these initial states. The system parameters are set as ω0 ¼ 1, μ ¼ 0:1, v ¼ 0:5, τ ¼ 1. The results are illustrated in Fig. 3, wherein x0 and y0 represent the initial position xð0Þ and the initial velocity x_ ð0Þ of the plant, respectively. From Fig. 3, one can see that there are three different basins of attraction for two separate steady states in the concerned area, where the innermost round-shaped steady state is the attracting basin of the equilibrium, while the outer rectangular-shaped steady state is the attracting basin of a periodic motion with amplitude of 18.16, which is newly introduced by the time-delayed feedback. Notably, there is an infinite number of newly introduced periodic solutions for this system [15], each of which has its own basin of attraction. It can be imagined that if the concerned area is enlarged, the basins of attraction for other periodic solutions will emerge and be clearly seen. Notice that the time-delayed controller used in this example is not a global controller, resulting in the effectiveness of the control to depend heavily on the initial states of the plant system when
Fig. 3. Basins of attraction of one equilibrium point and one periodic motion of the TDFC system (10) with the parameters: ω0 ¼ 1, μ ¼ 0:1, v ¼ 0:5, τ ¼ 1.
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all the control parameters are fixed. From Fig. 3, one can see that only if the initial position and initial velocity of the plant both locate in the range of approximately 4 and þ4, can the delayed control stabilize the system to the equilibrium. Beyond that scope, the disturbance may be too large prior to when the controller is turned on thus the delayed controller will fail to do the job and will instead steer the system to a newly introduced periodic motion. It is therefore very important to check the range of the basin of attraction for a successful implementation of the control for a particular steady state of the plant system. The example also confirms that our discovery about the nature of the initial state space of a time-delayed feedback system reveals the underlying mechanism of the TDFC principle. 4. Conclusion A careful investigation on the mechanism of TDFC system shows that the initial function of the TDFC system cannot be given arbitrarily. It in fact has to be generated by the dynamics of the plant itself during the period ½0; τ. If the initial state of the plant is within the Euclidean space Rn , the basin of attraction is finite-dimensional. In this case, the shape and size of the basin of attraction of the stabilized state can be discussed in a finite-dimensional space, which could be an essential aspect for the control performance evaluation of TDFC systems. The conclusion does not change the mathematical nature of a time-delayed system being infinite-dimensional however. Based on this conclusion, we may optimistically anticipate that in some other circumstances, the dynamics of a time-delayed system could also essentially be of finite-dimension, if the historical information characterized by the initial function is rooted in an essentially finitedimensional subsystem. For instance, a delay-involved natural ecological system could be essentially finite-dimensional. Acknowledgment This research was supported by the National Natural Science Foundation of China under Grant 11172126, the State Key Program of National Natural Science Foundation of China under Grant 11032009, and the Hong Kong Research Grants Council under the GRF Grant CityU 1120/14E.
References [1] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170 (6) (1992) 421–428. [2] P. Celka, Experimental of Pyragas's chaos control method applied in Chua's circuit, Int. J. Bifurc. Chaos 4 (6) (1994) 1703–1707. [3] O. Luthje, S. Woflff, G. Pfister, Control of chaotic Taylor-Couette flow with time-delayed feedback, Phys. Rev. Lett. 86 (9) (2001) 1745–1748. [4] P. Parmananda, R. Madrigal, M. Rivera, L. Nyikos, I.Z. Kiss, V. Gaspar, Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control, Phys. Rev. E 59 (5) (1999) 5266–5271. [5] H.G. Schuster (Ed.), Handbook of Chaos Control, Wiley-VCH, Berlin, 1999. [6] G.R. Chen, X.H. Yu, Time-delayed feedback control of chaotic systems, IEEE Trans. Circ. Syst.-I 46 (6) (1999) 767–772. [7] J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [8] M.E. Bleich, J.E.S. Socolar, Controlling spatiotemporal dynamics with time-delayed feedback, Phys. Rev. E 54 (1) (1996) R17–R20.
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[9] H. Nakajima, On analytical properties of delayed feedback control of chaos, Phys. Lett. A 232 (1997) 207–210. [10] W. Just, T. Bernard, M. Ostheimr, E. Reibold, H. Benner, Mechanism of time-delayed feedback control, Phys. Rev. Lett. 78 (2) (1997) 203–206. [11] K. Yamasue, T. Hikihara, Domain of attraction for stabilized orbits in time delayed feedback controlled Duffing systems, Phys. Rev. E 69 (5) (2004) 056209(1-8). [12] W. Just, H. Benner, C. von Loewenich, On global properties of time-delayed feedback control: weakly nonlinear analysis, Phys. D 199 (2004) 33–44. [13] K. Hohne, H. Shirahama, C.-U. Choe, H. Benner, K. Pyragas, W. Just, Global properties in an experimental realization of time-delayed feedback control with an unstable control loop, Phys. Rev. Lett. 98 (21) (2007) 214102(1-4). [14] H.L. Wang, H.Y. Hu, Z.H. Wang, Global dynamics of a Duffing oscillator with delayed displacement feedback, Int. J. Bifurc. Chaos 14 (8) (2004) 2753–2775. [15] H.Y. Hu, Z.H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer-Verlag, New York, 2002.
Journal of the Franklin Institute 352 (2015) 3243–3249 www.elsevier.com/locate/jfranklin
On the initial function space of time-delayed systems: A time-delayed feedback control perspective Huailei Wanga,n, Guanrong Chenb a
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China b Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, PR China Received 28 March 2014; received in revised form 7 October 2014; accepted 27 October 2014 Available online 4 November 2014
Abstract We address the problem as if the initial function space of a time-delayed system can be finitedimensional. We argue that the dimension of the initial function space of a time-delayed feedback control system is just identical to that of the corresponding plant which is to be controlled, and so is finitedimensional in many cases. We prove this property by constructing an injection map between the two spaces. We anticipate that this may also be applied to some other systems, such as ecological processes involving time delays without control. We believe that the significance of this discovery not only makes available the physical interpretation and intuitional explanation of the basins of attraction associated with time-delayed feedback control, but also suggests the principle as how to evaluate the robustness of the timedelayed feedback control schemes to external disturbances. We finally visualize and illustrate such basins of attraction via numerical simulation for the classical Duffing system by equipping it with the velocity timedelayed feedback control. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Thanks to the advantages of stabilizing time-periodic states of a dynamical system without the need of precise information about the controlled object, the time-delayed feedback control (TDFC) n
Corresponding author. E-mail address: [email protected] (H.L. Wang).
http://dx.doi.org/10.1016/j.jfranklin.2014.10.021 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
3244
H.L. Wang, G.R. Chen / Journal of the Franklin Institute 352 (2015) 3243–3249
scheme, initiated by Pyragas in 1992 [1], has been developed into an active research direction in applied nonlinear science [2–6]. Although the experimental implementation of TDFC is straightforward, its theoretical analysis proves difficult since the dynamics of a time-delayed system takes place in an infinite-dimensional phase space [7]. To date, the most efficient approach to discussing the performance of TDFC is based on linear stability analysis [8–10], yielding mainly local results whilst general global results are rare [11,12]. One typical challenge is to determine the basins of attraction, even for stabilizing, to a particular equilibrium or periodic orbit of a nonlinear system under TDFC. Studies on the basins of attraction are essential for experimental implementation and engineering applications of the TDFC technique because control performance may largely depend on the basin of attraction of the stabilized state in a nonlinear system [13]. It is known that the essential difficulty for the global analysis of time-delayed systems roots in their infinite dimension of the phase space, therefore a common approach to dealing with the challenge is to find possibilities of reducing the phase space to a finite-dimensional one. Unfortunately, earlier mathematical analysis has revealed that relevant delayed differential equations belong to the category of functional differential equations, also referred to as differential equations in the Banach space, which are of infinite-dimension in nature. In this study we take a new perspective of viewing the initial function space of a TDFC system through a careful analysis of the control process. Our analysis leads to the conclusion that the seemingly infinite-dimensional initial function space of a TDFC system is actually determined by a finite-dimensional initial state space, however this conclusion cannot be extended to all general timedelayed differential equations. Based on the new finding, we can conveniently characterize and visualize the basin of attraction of a TDFC system in a finite-dimensional Euclidean space. We must emphasize that this feature is not just a novel dimensional reduction, but revealing of the very nature of the original initial function space. This is a kind of equivalence in which, unlike other dimensionreduction methods, no dynamical characteristics of the TDFC system will be lost. 2. Finite-dimensional initial state space A diagrammatic view of a typical TDFC system is given in Fig. 1. Let the internal variables of the system be denoted by a state vector xðtÞ. The dynamical equation of the TDFC system can be described by x_ ðtÞ ¼ hðx; uÞ; x A Rn ; uA Rm
ð1Þ
where h is a continuous nonlinear vector function, u is the time-delayed state-feedback control vector, which is defined by a continuous nonlinear vector function g: u ¼ gðxðtÞ; xðt τÞÞ
ð2Þ
where τ40 is the time delay constant. Combining Eqs. (1) and (2) leads to x_ ðtÞ ¼ hðxðtÞ; gðxðtÞ; xðt τÞÞÞ
ð3Þ Plant
g( x (t ), x (t − τ ))
x (t ) Delay x(t − τ )
Fig. 1. Schematic diagram of time-delayed feedback control system.
H.L. Wang, G.R. Chen / Journal of the Franklin Institute 352 (2015) 3243–3249
3245
which can be simply represented by a continuous map: x_ ðtÞ ¼ fðxðtÞ; xðt τÞÞ
ð4Þ
where fðxðtÞ; xðt τÞÞ hðxðtÞ; gðxðtÞ; xðt τÞÞÞ Eq. (4) is in the general form of a differential-difference equation, which belongs to the category of retarded functional differential equations (RFDEs). The essential difference between such equations and the ordinary differential equations (ODEs) lies in that the state space of an RFDE is a Banach space defined as C ¼ Cð½0; τ; Rn Þ, namely the continuous function space mapping the interval ½0; τ onto the Euclidean space Rn . To determine a specific solution of Eq. (4), one has to define an initial function on the initial time interval t A ½0; τ, instead of an initial state at the instant t ¼ 0 for the ODEs. In other words, an abstract ‘initial point’ of an RFDE is a function defined on the initial time interval ½0; τ. As a stabilized equilibrium of a nonlinear system might not be globally stable due to the nonlinearity, there is a common concern about the control domain of the designed control law, namely to what extent is the disturbance to a system allowed for successfully stabilizing the system to the target state? To answer this question, one has to check the initial functions in interest. However, this is impossible due to the infinite number of initial functions (abstract ‘initial points’) defined on the initial time interval ½0; τ. Yet, we should point out that in some specific scenarios where the situation may not be as general, we may be able to take advantage of some special structures and properties of the system to draw a different conclusion, as follows. Theorem 1. The initial function space of a control system with time-delayed feedback as formulated in Eqs. (1) and (2) can be finite-dimensional, and its dimension is determined by the nature of the plant itself, which is independent of the delayed feedback control. Proof. Differing from the idea developed in [14], which is rather straightforward but incomplete, we consider this problem from another perspective. Inspecting the control diagram in Fig. 1, we first notice that the plant has its own dynamics and may be subject to disturbances from outside before the control is put into action. As soon as the control loop is turned on, we record this very moment as the initial instant t ¼ 0. From then on, in order to implement the delayed control, the measured signals have to be continuously stored in a stack buffer for a period of time τ before the buffer is completely full to shape into the first control force formulated by Eq. (2). That is, not until the moment t ¼ τ does the control force begin to take action, as the buffer has signal data at the moment t but has no data at t τ during the time period t A ½0; τ. However, it can be seen that afterwards, i.e. when t4τ, the control force will take effect continuously with measured signal data continuing to flow in and flow out, causing the stack buffer to be always full.,. From the above analysis of the mechanism of time-delayed control, we can now assert that an initial function of the time-delayed control system is composed of the stored signal data in the buffer, which is just ‘produced’ by the plant itself during the initial time interval t A ½0; τ when the feedback control has not been activated to take effect. Thus, an available initial function for the TDFC system cannot be given arbitrarily as a continuous function, but is determined by the dynamical nature of the plant itself on the initial time interval, which in many cases can be finite-dimensional. □ For the case of a plant described by an ODE with the states in the Euclidean space Rn , the initial state of the plant at t ¼ 0 in Rn determines the initial function on the time interval t A ½0; τ
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in C ¼ Cð½0; τ; Rn Þ, and the initial function on t A ½0; τ determines the entire solution of the TDFC system for all the time afterwards. This implies an injection map from Rn into C ¼ Cð½0; τ; Rn Þ and a one-to-one map between C ¼ Cð½0; τ; Rn Þ and the solution space of the TDFC system, as illustrated by Fig. 2. It is then straightforward to deduce that it is the initial state of the plant in Rn that determines the dynamics of the TDFC system. Therefore, the true initial function space of the TDFC system in this case is merely a subset of the general initial function space, in the sense that one may actually identify the initial time interval ½0; τ with a single initial time instant, t ¼ 0, to produce all the initial functions in interest. From this perspective, the coupled state equation of the TDFC system can be recast to ( x_ ðtÞ ¼ hðxðtÞ; 0Þ; xð0Þ ¼ x0 ; 0r t r τ ð5Þ x_ ðtÞ ¼ hðxðtÞ; uÞ; t Z τ where u ¼ gðxðtÞ; xðt τÞÞ is the time-delayed feedback control. Now, system (5) clearly is a finite-dimensional one, a typical non-delayed switching system with the switching time at t ¼ τ, regardless of the controller u to be used hereafter. 3. Demonstration of finite-dimensional basin of attraction of a TDFC system Consider a simple plant with dynamics governed by the autonomous Duffing equation x€ þ ω20 x þ μx3 ¼ 0; xA R
ð6Þ
where ω0 40, μ40. Clearly, the only steady state of the plant is the equilibrium x ¼ 0, which is stable but not asymptotically stable. Thus, the velocity time-delayed feedback control is introduced to stabilize. The controlled system is as follows [15]: ( x€ þ ω20 x þ μx3 ¼ u; x A R; u A R ð7Þ u ¼ v_xðt τÞ or, in the coupled form, x€ þ ω20 x þ μx3 ¼ v_xðt τÞ
ð8Þ
where the time parameter t is omitted wherever no time delay is involved. According to the above discussions, Eq. (8) can be recast into two switching systems: ( x€ þ ω20 x þ μx3 ¼ 0; 0 r t r τ x€ þ ω20 x þ μx3 ¼ v_xðt τÞ; t Z τ
TDFC system
Plant system
Initial state space in
ð9Þ
Solution
Solution space
R
Initial function space n
C = C ([0, τ ], R )
Fig. 2. Illustration of the sets of initial points in R , the initial function space, the solution space, and their relationships.
H.L. Wang, G.R. Chen / Journal of the Franklin Institute 352 (2015) 3243–3249
It can be equivalently reformulated as the following state space equation: 8 x_ ¼ y > > > 0r t r τ ðaÞ > < y_ ¼ ω20 x þ μx3 ( x_ ¼ y > > > > : y_ ¼ ω2 x þ μx3 þ vyðt τÞ t Z τ ðbÞ 0
3247
ð10Þ
To solve Eq. (10), we first need to set an initial state ðxð0Þ; yð0ÞÞ for Eq. (10)(a), which represents the initial position and initial velocity of the plant respectively, so as to produce the initial function for Eq. (10)(b), and then use the computed initial function to solve Eq. (10)(b). Based on the above analysis, the basin of attraction for an attractor of the TDFC system (10) is in fact a set of initial states in R2 , starting from which the dynamics of the TDFC system will eventually settle down on that attractor. This property not only enables us to completely avoid the difficulty as how to classify initial functions of the TDFC system, but also makes it available for anyone to view the basin of attraction in the plane R2 through the classification of initial state coordinates. To obtain and visualize the results numerically, we scan a constrained area of the plane by a small incremental step and then check the endresults of all these initial states. The system parameters are set as ω0 ¼ 1, μ ¼ 0:1, v ¼ 0:5, τ ¼ 1. The results are illustrated in Fig. 3, wherein x0 and y0 represent the initial position xð0Þ and the initial velocity x_ ð0Þ of the plant, respectively. From Fig. 3, one can see that there are three different basins of attraction for two separate steady states in the concerned area, where the innermost round-shaped steady state is the attracting basin of the equilibrium, while the outer rectangular-shaped steady state is the attracting basin of a periodic motion with amplitude of 18.16, which is newly introduced by the time-delayed feedback. Notably, there is an infinite number of newly introduced periodic solutions for this system [15], each of which has its own basin of attraction. It can be imagined that if the concerned area is enlarged, the basins of attraction for other periodic solutions will emerge and be clearly seen. Notice that the time-delayed controller used in this example is not a global controller, resulting in the effectiveness of the control to depend heavily on the initial states of the plant system when
Fig. 3. Basins of attraction of one equilibrium point and one periodic motion of the TDFC system (10) with the parameters: ω0 ¼ 1, μ ¼ 0:1, v ¼ 0:5, τ ¼ 1.
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all the control parameters are fixed. From Fig. 3, one can see that only if the initial position and initial velocity of the plant both locate in the range of approximately 4 and þ4, can the delayed control stabilize the system to the equilibrium. Beyond that scope, the disturbance may be too large prior to when the controller is turned on thus the delayed controller will fail to do the job and will instead steer the system to a newly introduced periodic motion. It is therefore very important to check the range of the basin of attraction for a successful implementation of the control for a particular steady state of the plant system. The example also confirms that our discovery about the nature of the initial state space of a time-delayed feedback system reveals the underlying mechanism of the TDFC principle. 4. Conclusion A careful investigation on the mechanism of TDFC system shows that the initial function of the TDFC system cannot be given arbitrarily. It in fact has to be generated by the dynamics of the plant itself during the period ½0; τ. If the initial state of the plant is within the Euclidean space Rn , the basin of attraction is finite-dimensional. In this case, the shape and size of the basin of attraction of the stabilized state can be discussed in a finite-dimensional space, which could be an essential aspect for the control performance evaluation of TDFC systems. The conclusion does not change the mathematical nature of a time-delayed system being infinite-dimensional however. Based on this conclusion, we may optimistically anticipate that in some other circumstances, the dynamics of a time-delayed system could also essentially be of finite-dimension, if the historical information characterized by the initial function is rooted in an essentially finitedimensional subsystem. For instance, a delay-involved natural ecological system could be essentially finite-dimensional. Acknowledgment This research was supported by the National Natural Science Foundation of China under Grant 11172126, the State Key Program of National Natural Science Foundation of China under Grant 11032009, and the Hong Kong Research Grants Council under the GRF Grant CityU 1120/14E.
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