Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

    Abstract

    This paper proposes a new control method applicable for a class of non-autonomous dynamical systems that naturally exhibit coexisting attractors. The central idea is based on knowledge of a system's basins of attraction, with control actions being applied intermittently in the time domain when the actual trajectory satisfies a proximity constraint with regards to the desired trajectory. This intermittent control uses an impulsive force to perturb one of the system attractors in order to switch the system response onto another attractor. This is carried out by bringing the perturbed state into the desired basin of attraction. The method has been applied to control both smooth and non-smooth systems, with the Duffing and impact oscillators used as examples. The strength of the intermittent control force is also considered, and a constrained intermittent control law is introduced to investigate the effect of limited control force on the efficiency of the controller. It is shown that increasing the duration of the control action and/or the number of control actuations allows one to successfully switch between the stable attractors using a lower control force. Numerical and experimental results are presented to demonstrate the effectiveness of the proposed method.

    1. Introduction

    In recent years, control of nonlinear dynamical systems has been among the most active fields of research due to its diverse applications in science and engineering. Most of the existing works on control of such systems have been focused on controlling chaos. Examples include stabilizing unstable periodic orbits embedded in chaotic attractors [1,2], exploiting the properties of chaos for control [3,4] and synchronizing two identical [5,6] or different chaotic systems [7]. However, little attention has been paid to the control of systems that exhibit multi-stability or coexistence of several attractors, despite the fact that they are observed abundantly in different fields [8,9]. Control of coexisting attractors in such systems is vital, because these attractors can be extremely sensitive to perturbations due to their fractally interwoven basins of attraction [8], and a mechanism for protecting against noise-induced basin hopping is always desirable.

    This paper investigates controlling coexisting attractors for a class of non-autonomous dynamical systems, because there is no universally approved method to control such dynamical systems. It is known that the Ott–Grebogi–Yorke (OGY) method [10] was originally designed for stabilizing an unstable periodic orbit embedded in a chaotic attractor by introducing small perturbations to system parameters. In [11,12], the OGY method was modified to control coexisting attractors by adding a slow periodic modulation that led to the crisis of undesired attractors and consequently settling down of the chaotic transients to an adjacent surviving attractor. Martinez-Zerega et al. [13] have shown that the time-delayed feedback control method proposed by Pyragas [2] may induce multiple coexisting attractors where a harmonic modulation of the feedback can make the system monostable. There are also some other older approaches in controlling systems with coexisting attractors. For instance, Lai [14] proposed building a bush-like structure of paths to the desired attractor and to stabilize a trajectory on some paths allowing asymptotical approaches to the target. Pecora & Carroll [15] used chaotic signals with a periodic courier to control a system with coexisting attractors, whereas Yang et al. [16] combined noise and a bias signal. Jiang [17] has shown that the coexisting attractors can be selected by using periodic driving, and the switch between different attractors can be achieved through a feedback control that contains dynamical features of the desired attractor. It is notable that in a number of papers on controlling coexisting attractors, including [11,12,18], the control objective was achieved by crisis. However, destroying coexisting attractors can be difficult for some dynamical systems [19,20], because any small perturbations of the system parameters can result in the emergence of new complex basins of attraction. It is therefore important not to change the existing structure of the solutions. In this paper, we propose a method for intermittent control of stable coexisting attractors without changing the original dynamical system and its basins of attraction.

    In the control system literature, the term ‘intermittent control’ introduced by Ronco et al. [21] has been adopted by a number of control methods [22,23] for different control purposes. It is worth noting that the concept of ‘act-and-wait’ control was proposed in Insperger [24], which was a form of intermittent control applied periodically [25]. Later on, this method was broadly used to synchronize chaotic systems with time delays [26,27]. The basic idea of ‘act-and-wait’ control is to periodically switch on and off the feedback control in the acting and the waiting periods, respectively. The concept of our method is similar, in which we aim to switch between stable coexisting attractors by applying an intermittent control, and we also focus on the appropriate selection of the control force which is activated when a certain criterion is satisfied.

    The method we proposed is applied to the well-known Duffing and impact oscillators in order to show its applicability to both smooth and non-smooth systems. The impact oscillator is a non-smooth system exhibiting rich behaviour close to grazing [28], where a large number of coexisting attractors has been observed both numerically and experimentally [8]. Experimental studies of such systems are further complicated by the presence of chaos, and fractal boundaries of the basins of attraction. Recent work by our group [20] has shown different bifurcation scenarios when the excitation amplitude was set to different values. In some cases, the occurrence of coexisting attractors was manifested on the bifurcation diagrams through a discontinuous transition from one orbit to another via a boundary crisis, which appeared to be closely related to the grazing events.

    The paper is organized as follows. In §2, the control method is introduced and its stability is studied. In §3, numerical and experimental results for control of a Duffing oscillator and an impact oscillator are presented. In §4, a constrained control method is considered for the intermittent control law. In §5, the neighbourhood boundary of the intermittent control law is briefly investigated. Finally, conclusions are drawn in §6.

    2. Mathematical formulation of the intermittent control method

    (a) Governing equations

    Consider a non-autonomous system expressed as a second-order ordinary differential equation

    Display Formula
    2.1
    where x is the state variable, Inline Formula is a nonlinear function, p (t) is a harmonic excitation and u(t) is an external control input. Assume that multiple stable solutions exist at the selected parameter values, and that equation (2.1) has a solution in the form of a desired stable periodic orbit when no external control action is applied (u(t)=0), so that the system can be written as
    Display Formula
    2.2
    The control objective is then formulated so as to design a closed-loop intermittent controller u(t) that can drive the system from one stable periodic orbit to another.

    Let us define the error variables e1 and e2 to monitor the state difference between the actual and the desired trajectories

    Display Formula
    2.3
    and
    Display Formula
    2.4
    By subtracting equation (2.2) from equation (2.1), the error equation is obtained
    Display Formula
    2.5
    where Inline Formula.

    In order to achieve the control objective, we need to define an intermittent control law, which may be written as

    Display Formula
    2.6
    where kp and kd are positive linear gains of the controller, and δ is a small number defining the neighbourhood boundary of the intersection at which the displacement error between the actual and the desired trajectories is sufficiently small.

    The intersection here is defined as the equality of the displacements of the actual and desired orbits (i.e. e1=0). When the trajectory lies in the neighbourhood of the intersection, these displacements are sufficiently close to each other (i.e. |e1|≤δ). Therefore, it is clear from equation (2.6) that the control signal is applied in an ‘act-and-wait’ manner, i.e. starting from the moment when this condition is satisfied and until it no longer is.

    The parameter δ defines the size of this neighbourhood, which can be chosen based on the system to be controlled. The influence of the selection of δ on controlling coexisting attractors is discussed in §5 of this paper.

    It is important to emphasize that the main difference between feedback linearization control laws [1,7] and the proposed control law is that the former are continuous feedback control methods, whereas the latter is an intermittent feedback control, which aims not to change the underlying dynamics of the system or its coexisting attractors. Also, previously proposed intermittent control laws [29,22] used the periodically time-driven control, whereas our method is event-driven control.

    (b) Stability analysis

    Let us now consider the stability of the proposed method.

    Theorem 2.1

    The intermittent controller (2.6) guarantees that the non-autonomous system (2.1) asymptotically converges to the desired periodic orbit (2.2) during the intervals where |e1|≤δ with an appropriate selection of the proportional and derivative gains kpand kd, respectively.

    Proof.

    Consider a positive-definite Lyapunov function

    Display Formula
    2.7
    where λ is a positive scalar. A sufficient condition for equation (2.7) to be positive definite is
    Display Formula
    2.8
    Applying the intermittent control law (2.6) to the error equation (2.5), we can obtain
    Display Formula
    2.9
    Then, substitution of equation (2.9) into the time derivative of V (e1,e2) yields the following results: for |e1|>δ, Inline Formula is indeterminate; for |e1|≤δ, the time derivative of the Lyapunov function V (e1,e2) is given as
    Display Formula
    2.10
    Because 0<λ<kd, we have that the time derivative of V (e1,e2) is negative definite, which guarantees an asymptotical convergence for the closed-loop system (2.1) to the desired periodic orbit (2.2), i.e.
    Display Formula
    2.11
     ■

    It is worth noting that the proposed control law (2.6) is only acting during short intervals when |e1|≤δ, and the error variables e1 and e2 converge to zero asymptotically. The rate of convergency of the Lyapunov function can be controlled by an appropriate choice of kp and kd, or by applying the subsequent intermittent control during the transient behaviour until the system settles down on the desired periodic orbit. In the first case, the system state (Inline Formula) is driven to the basin of attraction of the desired orbit within one short interval. In the latter case, the system state (Inline Formula) may not be driven to the desired state by one control action, but the system will eventually settle down onto the desired orbit after subsequent control actions. The principle of the proposed intermittent control law will be studied in detail in §2c.

    It is important to stress that the choice of the neighbourhood defined by δ is vital, because it differentiates the proposed intermittent control law from the continuous one [1] such that, if δ is sufficiently small, the proposed control law will produce an impulse-like control signal in short time intervals; if δ is infinity, then the proposed method will perform as a continuous controller [1].

    In general, there are three possible outcomes of applying the intermittent control law (2.6) depending on the stability of the actual attractor to be controlled: (i) a successful control from the actual orbit to the desired orbit during one control action; (ii) a successful control from the actual orbit to the desired orbit via intermediate transient states; (iii) a failure to reach the desired orbit due to the strong stability of the actual attractor. This paper will consider cases (i) and (ii) in §3 and §4, respectively. A possible solution to avoid failure of the control method (case (iii)) will be discussed in §5.

    (c) Principle of the intermittent control

    The central idea of the intermittent control is to use an impulsive control action to perturb one of the system's stable solutions in order to switch to another stable solution. This is carried out by bringing the perturbed state into the desired basin of attraction. Figure 1 shows this in detail. Figure 1c demonstrates schematically the actual and desired trajectories showing the cross section in time where the neighbourhood condition is satisfied. In figure 1d, an example of basins of attraction is presented where two stable solutions, shown in the additional windows of figure 1c, coexist. When the displacements of the actual and the desired orbits in the time domain are sufficiently close to each other, i.e. |e1|≤δ, as shown in figure 1a,b, the control action is applied for a time τ, which depends on how long the displacements of the two orbits remain in the specified neighbourhood. Because we do not want to change the basins of attraction, an effective impulsive control signal is applied only between tb and tb+τ, as shown in figure 1b, where tb is the time when the neighbourhood condition is first satisfied. Again, our aim is to drive the actual orbit towards the desired orbit during the time interval [tb,tb+τ].

    Figure 1.

    Figure 1. (a) Trajectories of the displacements of actual (black line) and desired (red line) trajectories in the time domain. (b) A neighbourhood of the crossings of actual and desired trajectories that satisfy |e1|≤δ (tb is the time when the neighbourhood condition is initially satisfied, and τ is the duration of the control action). (c) Schematic of actual and desired trajectories showing the cross section in time where the neighbourhood condition is satisfied, and additional windows showing both trajectories on the phase plane. (d) Basins of attraction of the two coexisting orbits. (e) Basins of attraction recalculated at the phase shift where the neighbourhood condition is satisfied. (Online version in colour.)

    Let us now consider the error equation (2.9) within the time interval between tb and tb+τ rewritten as below,

    Display Formula
    2.12
    It can be seen from equation (2.12) that the error dynamics is described by a second-order differential equation in which the rate of decay of the error is controlled by the damping coefficient, which in this case is the derivative gain, kd. Therefore, the following lemma is formulated.

    Lemma 2.2

    For any positive scalar ϵ>0, the errors of the displacement e1 and the velocity e2 can be driven towards zero, so that |e1|<δ and Inline Formula, within the time interval between tb and tb+τ, if the gain kd is chosen as

    Display Formula
    2.13

    Proof.

    Since we can easily choose the value of δ to make the displacement error e1 smaller than the velocity error e2, the following approximation is made from equation (2.12):

    Display Formula
    2.14
    Solving equation (2.14), we obtain
    Display Formula
    2.15
    As the controller is required to drive Inline Formula to decay to ϵ at tb+τ, then for any ϵ>0 we have a linear gain kd such that
    Display Formula
    2.16
    By solving (2.16), we have satisfied the criterion (2.13) for kd. ■

    To estimate the value of ϵ, the basins of attraction recalculated at the appropriate phase shift (see figure 1e) have to be used instead of those shown in figure 1d. Generally, the boundaries between the basins of attraction for the actual and the desired orbits are known. If we know the actual state (Inline Formula) in the basin of attraction and its closest state (Inline Formula) on the boundary of this basin when the condition |xaxp|≤δ is satisfied, then we can estimate the required change in the velocity to switch to the other attractor. Since driving the system to any state within the basin of attraction of the desired orbit will achieve our goal, ϵ can be estimated using Inline Formula. Where the basin boundary is fractal, ϵ must be sufficiently small such that the ϵ ball around the desired trajectory contains no initial conditions belonging to the actual trajectory.

    3. Examples of the control of smooth and non-smooth systems

    In this section, two examples are given to demonstrate the effectiveness of the proposed method. To show its versatility, control of a Duffing oscillator and an impact oscillator representing smooth and non-smooth dynamical systems is presented.

    (a) Duffing oscillator

    The first example is the Duffing oscillator, which is known to have many coexisting attractors. The Duffing oscillator is described as

    Display Formula
    3.1
    and we have considered the following parameters: k=0.9, Γ=1.9 and ω=1.2. At these parameter values, the system has two coexisting attractors, depicted in figure 2a. They are period-1 small (green dot) and large (red dot) amplitude responses. The control objective is to drive the period-1 response with small amplitude to the period-1 response with large amplitude, and vice versa. Choosing the control parameters as δ=0.05, ϵ=10−5, kp=1 and kd=20, we introduce the following intermittent controller:
    Display Formula
    3.2
    where e1(t)=y(t)−yd(t).
    Figure 2.

    Figure 2. Numerical simulation of the Duffing oscillator response with the intermittent control calculated at k=0.9, Γ=1.9 and ω=1.2. The control parameters were chosen as δ=0.05, ϵ=10−5, kp=1 and kd=20. (a) Basins of attraction recalculated at the phase shift where the neighbourhood condition is satisfied. (b) Trajectory of the applied excitation and the intermittent control (switched on between t=45.41 and 45.49) as a function of time, and trajectory of the response as a function of time and on the phase plane. (Online version in colour.)

    The results of the numerical simulations are shown in figure 2b. As can be seen from figure 2b, the system first settled down on the small-amplitude periodic response. Then, the controller (3.2) was switched on at t=40.00. The first time when displacements of the two trajectories get close to each other is at t=45.41 (i.e. the neighbourhood condition is satisfied), and the control action was applied between t=45.41 and 45.49 s. The amplitude of the control signal was 30.11, which was sufficient to switch the trajectories and, as a result, the Duffing oscillator was successfully driven from the period-1 response with small amplitude to the period-1 response with large amplitude after a short transient. Here, we can also observe the numerical verification of the assumption (2.14). As can be seen from figure 2b, the transient behaviour from y(t) to yd(t) is very short during the control period, which is due to the fact that e1 was sufficiently small at the instant when the control was applied. Also, the intermittent controller produced an impulsive force that was able to generate a large change of velocity Inline Formula over a short time interval.

    (b) Impact oscillator

    The second example is an impact oscillator with the following equation of motion:

    Display Formula
    3.3
    where y is the mass displacement and H(⋅) is the Heaviside step function. To obtain two coexisting periodic attractors [20], the system parameters are chosen as ξ=0.01, β=29, g=1.26, Γ=1.0385 and ω=0.686. These two attractors along with their basins of attraction are shown in figure 3a. The first one is a period-1 response with one impact per period of excitation (green dot), and the second is a superharmonic period-2 response with two impacts per period of excitation (yellow dot). Again, the control objective is to drive the small-amplitude response with one impact to the large-amplitude response with two impacts. The control parameters are chosen as δ=1.2, ϵ=10−5, kp=1, kd=5, and the intermittent controller is written as
    Display Formula
    3.4
    where γ=(yg)H(yg)−(ydg)H(ydg).
    Figure 3.

    Figure 3. Numerical simulation of the impact oscillator response with the intermittent control calculated at ξ=0.01, β=29, g=1.26, Γ=1.0385 and ω=0.686. The control parameters were chosen as δ=1.2, ϵ=10−5, kp=1 and kd=5. (a) Basins of attraction recalculated at the phase shift where the neighbourhood condition is satisfied. (b) Trajectory of the applied excitation and the intermittent control (switched on between t=2.20 and 2.21) as a function of time, and trajectory of the response as a function of time and on the phase plane. (Online version in colour.)

    The results of the numerical simulations are shown in figure 3b, where the nature frequency of the system wn=9.35 Hz was used for scaling to compare experimental and numerical results. In this case, the intermittent controller (3.4) was switched on at t=2.00. Here, the neighbourhood condition was satisfied at 2.20 s, and the control action was applied in the time interval between 2.20 and 2.21 s. As can be seen, after the control action, the period-1 response with one impact per period was driven to the period-2 response with two impacts as required. It is noted that the control period of the impact oscillator is shorter than the control period of the Duffing oscillator. This is partly because of the higher forcing frequency, which results in a faster dynamic response for the impact oscillator. Additionally, the velocity error e2 is larger when |e1|≤δ in comparison with the previous case, so the time duration available for the control action is shorter since the attractors transition this region quicker. This will generally be true for systems with a higher degree of nonlinearity. To compensate this effect, δ can be simply adjusted so that the duration of the control action is tuned as required. Therefore, for the impact oscillator, δ was increased from 0.05 to 1.2. Again, it can be observed from figure 3b that the displacement of the impact oscillator was virtually unchanged, whereas the velocity experienced a large change due to the impulsive force.

    To verify the efficiency of the proposed control method in the laboratory environment, experiments on the impact oscillator were carried out at the Centre for Applied Dynamics Research at the University of Aberdeen, UK. A schematic of the experimental set-up can be found in Ing et al. [8]. The oscillator rig was mounted on an electrodynamic shaker, providing harmonic excitation and the intermittent control. Because the mass is excited through the base, the parameter Γ in equation (3.3) is equal to 2 in the experiment, where A is the excitation amplitude of the base. Displacement of the oscillator y was detected by an eddy current probe displacement transducer. The data were collected by a Labview data acquisition system and one analogue output channel of the system generated the control signal p*(t)+u*(t). This signal was amplified and supplied to the electrodynamic shaker in order to provide harmonic excitation and the intermittent control p (t)+u(t).

    An example of the experimental results is shown in figure 4. The excitation frequency and the amplitude were chosen to be the same as for the numerical simulation, and the neighbourhood boundary was set to δ=1. The external control action was activated for a short interval between 2.48 and 2.49 s, when the actual and the desired displacements were equal. Within this short interval, an impulsive signal of −0.5 V was generated by the Labview data acquisition system. As can be seen, after a short transient behaviour, the system settled down to the desired period-2 orbit with two impacts per period.

    Figure 4.

    Figure 4. Experimental time histories and phase portraits of the intermittent control of the impact oscillator at ξ=0.01, β=29, g=1.26, Γ=1.0385 and ω=0.686 with the following control parameters: δ=1, ϵ=10−5, kp=1 and kd=5. (a) Time history of the excitation and the intermittent control p*(t)+u*(t) (volts), and trajectories of the responses on the phase plane (see insets). (b) Time history of the response displacement (mm).

    4. Constrained control

    In this section, the strength of the intermittent control force is considered. Additional simulation and experimental results are presented and discussed here to analyse the requirements for the proposed control method.

    In reality, it is impossible to create an actuator with unlimited power and without energy consumption limitations, and, therefore, the strength of the control input needs to be limited. Here, we propose a constrained intermittent control law described as

    Display Formula
    4.1
    where Inline Formula is the maximal control input.

    The main principle of the constrained intermittent control (4.1) can be explained with the help of the basins of attraction, as shown in figure 5. A successful control effort can be defined as a transition of the system state from s1 to s2, as depicted in figure 5a. If we consider a case when the required control signal does not exceed Inline Formula, then there is sufficient energy brought to the system by the controller for a large change of velocity from s1 to s2 in a short time interval. After an impulsive perturbation at s1, the system state is driven to s2, which has the same phase as s1. Let us now consider another case when the required control signal exceeds Inline Formula. Here, the maximal control input Inline Formula is applied, which leads to a transition from s1 to an intermediate state s1 as shown in figure 5b. The system state is then driven to s2 by subsequent control actions applied during the transient behaviour of the system when the condition |e1|≤δ is satisfied.

    Figure 5.

    Figure 5. The transitions of the system state schematically presented on its basins of attraction for the unconstrained and constrained intermittent control. (a) A direct transition from the actual attractor (state s1) to the desired attractor (state s2) is shown, indicating that the system is driven from an actual orbit to the desired orbit after one control action. (b) An indirect transition from the state s1 to the state s2 via an intermediate state s1 is presented, showing that the system is driven from an actual orbit to a transient state, and then to the desired orbit by using the constrained intermittent control. (Online version in colour.)

    The application of this method to continuous and discontinuous systems will be shown next using the Duffing oscillator and the impact oscillator as examples.

    (a) Duffing oscillator

    Numerical simulations of the Duffing oscillator responses with the constrained intermittent control have been carried out and typical results are shown in figure 6. Here, the maximal control force is limited to Inline Formula, whereas the system parameters and the control parameters remain the same as for the unconstrained case.

    Figure 6.

    Figure 6. Numerical simulation of the Duffing oscillator response with the constrained intermittent control input computed for k=0.9, Γ=1.9 and ω=1.2. The control parameters were set to δ=0.05, ϵ=10−5, kp=1, kd=20 and Inline Formula. (a) Time history of the excitation and the constrained intermittent control, and trajectories of the coexisting orbits on the phase plane (see insets). (b) Time history of the response displacement.

    As can be seen from figure 6, the constrained intermittent control law was switched on at t=40.00 s. The first constrained impulsive force was activated in the time interval between 40.18 and 40.26 s. There were, in total, five constrained impulsive inputs applied during the transient behaviour of the system. The velocity Inline Formula was not forced to the desired velocity Inline Formula immediately as shown in figure 2, but gradually during five consecutive control intervals.

    A comparison of the phase trajectories computed for the unconstrained and the constrained intermittent controllers is given in figure 7. It is clear that the unconstrained control led to a large control input as can be deduced from figure 7a, which shows very fast change of the velocity. On the other hand, as can be seen from figure 7b, it is observed that the constrained control force had a limited effect on the velocity change compared with the unconstrained one, but the velocity was gradually driven to the desired value. It is obvious that the constrained intermittent control features transient dynamic behaviour, which is the key in achieving the control objective.

    Figure 7.

    Figure 7. Numerical simulation of the Duffing oscillator response with the unconstrained and the constrained control inputs computed for k=0.9, Γ=1.9 and ω=1.2. The control parameters were set to δ=0.05, ϵ=10−5, kp=1 and kd=20. (a) Trajectory on the phase plane when the unconstrained intermittent control was applied. (b) Trajectory on the phase plane when the constrained intermittent control was applied with Inline Formula. Control actions are indicated by red arrows. (Online version in colour.)

    Furthermore, from the viewpoint of energy, we can conclude that the desired trajectory of the system can be achieved either by injecting once a sufficient amount of energy or by injecting many times small portions of energy into the system intermittently. An optimization issue may arise here on how to minimize the energy injected into the system for optimal control of coexisting attractors. However, this matter is beyond the scope of this paper.

    (b) Impact oscillator

    Similar to the Duffing system, the constrained intermittent control was applied to the impact oscillator. Examples of numerical and experimental results are shown in figures 8 and 9, respectively.

    Figure 8.

    Figure 8. Numerical simulation of the impact oscillator response with the constrained intermittent control computed for ξ=0.01, β=29, g=1.26, Γ=1.0385 and ω=0.686. The control parameters were chosen as δ=1.2, ϵ=10−5, kp=1, kd=5 and Inline Formula. (a) Time history of the excitation and the constrained intermittent control, and trajectories of the coexisting orbits on the phase plane (see insets). (b) Time history of the response displacement.

    Figure 9.

    Figure 9. Experimental results for the impact oscillator with the constrained intermittent control at ξ=0.01, β=29, g=1.26, Γ=1.0385 and ω=0.686. The control parameters were set to δ=1, ϵ=10−5, kp=1, kd=5 and Inline Formula. (a) Time history of the excitation and the intermittent control, p*(t)+u*(t) (volts), and trajectories of the responses on the phase plane (see insets). (b) Time histories of the response and the desired orbit displacements (mm) shown by black solid and red dashed lines, respectively. After several control actions, the oscillator was driven to the period-2 response as required. Looking at the timing of the control action in part (a), it is clear that the constrained control was applied in the neighbourhood of the crossings of the actual and the desired trajectories (see part (b)). (Online version in colour.)

    As presented in figure 8, the maximal intermittent control input was limited to Inline Formula, and the constrained intermittent control was switched on at t=1.00 s. The first control input was applied between 1.01 and 1.02 s, which was then followed by three subsequent control inputs during the transient behaviour of the oscillator. Compared with the numerical result depicted in figure 3, the resulting velocity of the oscillator from the constrained control is much smaller than the resulting velocity in the case of the unconstrained control, but the constrained control was intermittently applied to the system until the system was driven to the desired orbit successfully. So, there can be a trade-off for the proposed intermittent control law between the amplitude of forcing and the duration of the total control action. If the control input is sufficiently large, then the velocity of the oscillator can be brought to the desired value in a short time interval. Otherwise, the insufficient control input will drive the system state through a few intermediate states as depicted in figure 5b, and finally bring it to the desired state. In this case, however, the total duration of the control action will be larger.

    The experimental results using Inline Formula V shown in figure 9 are used to verify the effectiveness of the constrained intermittent control law in the laboratory environment. As can be seen in figure 9, the desired period-2 response marked by the red dashed line is in the same phase as the actual period-1 response. Due to the limitations of the experimental set-up, the constrained control force was not turned on exactly at the same time as for the numerical simulation (figure 8), but it was applied when the actual and the desired trajectories would have crossed. It can be observed that the oscillator was successfully driven from the actual period-1 response to the desired period-2 response by the constrained intermittent control inputs activated during the transient behaviour of the system.

    5. Influence of the control action duration

    It has already been emphasized that the choice of the neighbourhood boundary δ is critical as it influences the required magnitude and the duration of the control input. So, in this section, we will further investigate the relationship between the magnitude and the time interval of a single control action for the proposed control method by considering various values of δ.

    A comparative numerical simulation of the Duffing system is presented in figure 10 to show the efficacy of the constrained control input with broader switching duration at the neighbourhood of the intersection. The neighbourhood boundaries of the constrained and the unconstrained controls were set to δ=0.2 and δ=0.05, respectively. The constrained control with Inline Formula was applied to the time interval between 45.32 and 45.6 s, and the unconstrained control was applied between 45.41 and 45.49 s. Comparing these two results, both systems were driven to an identical desired periodic orbit almost at the same time. The control action also can be observed from the phase trajectories in the subplot of figure 10. It is seen that both control actions led to similar phase trajectories. Therefore, we can conclude that both the constrained and unconstrained control inputs can achieve our goal of controlling coexisting attractors by a proper selection of the neighbourhood boundary δ. However, the constrained control significantly reduces the maximal control input that is important for practical applications.

    Figure 10.

    Figure 10. Comparison between the unconstrained and the constrained intermittent controls of the Duffing oscillator computed for k=0.9, Γ=1.9 and ω=1.2. The control parameters were chosen as ϵ=10−5, kp=1 and kd=20. The black line marks the unconstrained control for δ=0.05 and the red line marks the constrained control for Inline Formula and δ=0.2. (a) Time histories of the unconstrained and the constrained intermittent control. (b) Time histories of the response displacements. After the unconstrained and the constrained control actions, the oscillator was driven to the large-amplitude response in both cases as required. The subplots indicate that the constrained control action has broader control duration than the unconstrained control action, but both actions have the same effect on the response as shown in the phase portrait. (Online version in colour.)

    It should be noted that once the trajectory is perturbed by the control action, so that its new state still belongs to the basin of the actual orbit, there is an ‘attracting force’ that brings the trajectory back to this orbit. The stronger the stability of the actual orbit, the shorter the time to bring the trajectory back. For controlling orbits that have strong stability, two strategies can be applied: (i) to increase the linear gain kd of the intermittent control law (2.6), which can potentially result in a higher magnitude of the intermittent control force; (ii) to increase the neighbourhood boundary δ, which can enlarge the switching duration of the intermittent control force.

    6. Concluding remarks

    In this paper, a new method for controlling coexisting attractors occurring in a class of non-autonomous systems was proposed. The method was applied to both smooth and non-smooth dynamical systems, where the Duffing and the piecewise linear impact oscillators were studied as examples. The central idea of this new method is based on knowledge of a system's basins of attraction, with control actions being applied intermittently in the time domain when the actual trajectory satisfies a proximity constraint with regards to the desired trajectory. This intermittent control uses an impulsive force to perturb one of the system attractors to switch the system response onto another attractor.

    The method has been implemented successfully to control the Duffing oscillator and the piecewise linear impact oscillator, and the effectiveness of the method is confirmed by both the numerical and experimental results. Although these relatively simple systems were chosen as examples, the method is also applicable to a broad range of smooth and non-smooth dynamical systems where coexistence of attractors occurs. For practical purposes, the constrained intermittent control was considered, and it was applied to the two example systems. Further considerations have been given to the strength of the intermittent control force and the duration of control action. It was shown that, for the constrained control, increasing the duration of control action led to a decrease in the number of actions required.

    Footnotes

    One contribution of 17 to a Theme Issue ‘A celebration of mechanics: from nano to macro’.

    References