This paper discusses a novel approach to the control of chaos based on the use of the adaptive minimal control synthesis algorithm. The strategies presented are based on the explicit exploitation of different properties of chaotic systems including the boundedness of the chaotic attractors and their topological transitivity (or ergodicity). It is shown that chaos can be exploited to synthesize more efficient control techniques for nonlinear systems. For instance, by using the ergodicity of the chaotic trajectory, we show that a local adaptive control strategy can be used to synthesize a global controller. An application is to the swing-up control of a double inverted pendulum.
Over the past 15 years, much research effort has been spent within the area of controlling bifurcations and chaos. Numerous techniques have been presented in the literature to control nonlinear chaotic dynamical systems. Many of the approaches presented were aimed at showing that chaos can indeed be controlled in a nonlinear dynamical system by using state feedback controllers, sometimes based on classical control theoretic design methodologies (for an extensive literature review, see the recent books edited by Chen & Yu (2003) and Chen et al. (2003)).
The paper by Ott et al. (1990) suggested for the first time in the literature that chaos can actually be exploited in control applications. A large number of papers followed, where researchers presented different modifications of what started to be known as the Ott–Grebogi–Yorke method for chaos control. All the schemes presented performed well, when validated numerically or via carefully arranged laboratory experiments. Nevertheless, almost none of the schemes made it through to realistic engineering applications. The main reason was the lack of robustness of these schemes and lack of a consistent control of theoretic framework to account for their asymptotic stability, time of convergence and other issues, which are typically taken into account for control design purposes.
The aim of this paper is to present an approach to chaos control, which was developed over the past few years at the University of Bristol (Stoten & di Bernardo 1996). The idea is to show that (i) novel adaptive control strategies can be effectively used to control chaotic systems, and (ii) some properties of chaos, namely boundedness and topological transitivity, can be used to obtain a rigorous proof of asymptotic stability and to solve global control problems by using local control strategies. We make use of an innovative model reference adaptive control scheme, which relies on minimal knowledge of the plant and requires no off-line computation of the controller gains: the minimal control synthesis (MCS) algorithm, first proposed by Stoten in 1989. Similar control schemes can also be obtained by considering other adaptive controllers in the main loop, such as, for example, those described in Andrievsky et al. (1988), Narendra & Annaswamy (1988) and Sastry & Bodson (1989).
The methodology is first illustrated using a Duffing oscillator as a simple representative example and then showing a novel application to solve the swing-up problem of a double pendulum, by adding no extra actuators on its joints. The swing-up control of the double pendulum systems has been the subject of recent research. For example, a strategy based on the application of an extra torque between the pendulum links was presented in Bradshaw & Shao (1996) and references therein. Most of these strategies are based on the use of an extra control torque at the joint between the two links.
Only recently, strategies to stabilize the pendulum-inverted position have been presented that do not rely on any control torque between the links. The most successful strategy so far seems to be the one presented by Astrom & Furuta (1996), which is based on controlling the system energy. Here, we show that chaos-based adaptive control can be used to successfully swing-up the pendulum by means of a local adaptive control strategy complemented with a chaotic swing-up phase, based on the exploitation of the topological transitivity of chaotic dynamics.
The rest of the paper is outlined as follows. An overview of the MCS adaptive algorithm is presented in §2. In §3, an extension of the strategy is presented, whereby an additional adaptive switching action renders the origin asymptotically stable even in the presence of fast nonlinear terms acting on the plant. The proof of asymptotic stability is obtained by exploiting the boundedness of chaotic evolutions in order to prove boundedness of the nonlinear perturbations on the error system. Further exploitation of chaos for control system design is presented in §4, where the topological transitivity (or ergodicity) of the chaotic solutions is used for control system design. The application of this methodology to the swing-up problem of a double pendulum is then discussed in §5 showing its feasibility to achieve global control using local control techniques.
2. Overview of the minimal control synthesis algorithm
The MCS algorithm was first proposed to control an n-state system of the form:(2.1)the control aim being to track the evolution of a given reference model described by(2.2)i.e. under MCS control, we must ensure that(2.3)with .
The original MCS strategy allows the pair A, B to have unknown parameters, but assumes a known phase canonical structure given bywhere b is assumed to be strictly positive. A similar structure is also required for the matrices Am and Bm in (2.2).
The control signal is then chosen as the time-dependent state feedback strategy:(2.4)where is the feedback gain, the forward gain and the reference signal. In equation (2.4), n+1 time-varying gains are computed according to the following adaptation laws:(2.5)(2.6)where(2.7)with Ce determined from the positive-definite solution, P, of the Lyapunov equation, (with Q being positive definite), is given asand α and β are positive scalar adaptation weights (usually chosen empirically, see Stoten & Benchoubane 1990a).
It should be noted that in equations (2.5) and (2.6), the initial conditions can be set to zero. This is one advantage of the MCS algorithm with respect to the original MRAC approach of Landau (1979), where an initial guess for the control gain was required to prove the stability (see Landau 1979 for further details). (It should be noted that the non-zero initial condition on the error might impart non-zero gains at t=0+ via the proportional part of the control algorithm.)
In fact, the removal of any assumption on the initial conditions of the gains greatly reduces the amount of information needed by the controller (2.4). Hence the name ‘MCS algorithm’ given to this strategy; in order to achieve the control objective, (2.3), the MCS algorithm relies neither on the knowledge of the system parameters appearing in the matrices A and B, nor any a priori information on the control gains K and KR.
It follows that the MCS algorithm is a flexible adaptive control technique. The original proofs of stability and robustness of the algorithm presented in Stoten & Benchoubane (1990b) and Stoten & Benchoubane (1992) were extended by Stoten & di Bernardo (1996) to include the case of systems that are not in canonical form and have nonlinear terms acting on them. This allowed the successful exploitation of MCS controllers for chaos control and synchronization. Other adaptive algorithms for control of chaos based on speed-gradient algorithms were also proposed (e.g. Fradkov & Progromsky 1996, 1998). The advantage of the MCS is its easy implementation and flexibility.
(a) The extended minimal control synthesis algorithm
Although the MCS algorithm was initially developed for the control of structurally linear systems, several theoretical and practical investigations have shown that its control action is effective even in the presence of plant nonlinearities, external disturbances and parameter variations. In particular, we reported in Stoten & di Bernardo (1996) that MCS can be successfully applied to control or synchronize chaotic evolutions of nonlinear dynamical systems.
However, the original MCS control law, (2.4), does not contain any action designed to cope directly with the nonlinearities acting on the systems involved. Therefore, in order to deal with highly nonlinear dynamical systems, the original MCS control strategy needs to be modified.
A first modification to MCS was suggested in Stoten & Benchoubane (1992), where the extended minimal control synthesis (EMCS) strategy was introduced. According to the EMCS algorithm, a discontinuous switching action is added to the standard MCS in order to achieve global asymptotic stability of the error dynamics in the presence of rapidly varying disturbances. This yields a clear improvement on the standard MCS performance, but the amplitude of the switching action has to be determined off-line. This is against the philosophy of MCS synthesis procedure. To overcome this disadvantage of EMCS, an empirical method for the choice of such an action was proposed in Stoten & Benchoubane (1992). Such an empirical choice, though, often selects a larger amplitude than required, yielding an unwanted waste of control energy.
In this paper, the MCS-based controller family is further expanded. In the spirit of the MCS synthesis procedure, it is proposed that the amplitude of discontinuous switching action, characterizing the EMCS law, is adaptively determined. Thus, a new purely adaptive EMCS strategy is presented, whose gains are estimated online through some appropriately stated adaptive laws. We will show that it is possible to prove global asymptotic stability of the new EMCS error dynamics, and also the boundedness of all adaptively estimated gains.
In Stoten & Benchoubane (1992) and Stoten & di Bernardo (1996), it was shown that the MCS algorithm guarantees asymptotic stability of the error dynamics even in the presence of a nonlinear disturbance, given that such a perturbation is of a vanishing type (Khalil 1992). If we now assume that a generic rapidly varying disturbance,is acting on the error equation, the standard MCS will no longer guarantee asymptotic stability. Instead, the EMCS algorithm should be used to retain asymptotic stability. As proposed in Stoten & Benchoubane (1992), the EMCS control equation is(2.8)where the gains K(t) and KR(t) are chosen as defined in equations (2.5) and (2.6).
On comparing equation (2.8) with the original MCS law, equation (2.4), we can see that EMCS contains the additional discontinuous switching action N sgn(ye), whose amplitude is determined by the real positive constant N. This action is introduced ad hoc to achieve global asymptotic stability of the error dynamics in the presence of rapidly varying disturbances. In particular, if we derive the error dynamics, xe=xm−x, we get (assuming the presence of the unknown nonlinear term d):(2.9)and substituting equation (2.8) into (2.9), we obtain(2.10)which can be rewritten, by means of the same algebraic manipulation outlined in Stoten & di Bernardo (1996), as(2.11)where(2.12)(2.13)(2.14)(2.15)and [⋯]n represents the nth row of a vector or matrix.
According to hyperstability theory, the error, xe, as given in equation (2.11) is globally asymptotically stable, if and only if the triple Am, Be, Ce is strictly positive real and the following integral inequality is satisfied for all t2>t1(2.16)with being independent of t2.
Strict positive realness of the triple Am, Be, Ce can be derived via the Kalman–Yakubovich lemma (Landau 1979).
Moreover, condition (2.16) can be rewritten as(2.17)where(2.18)and(2.19)From the developments reported in Stoten & di Bernardo (1996) and di Bernardo & Stoten (1997a,b), we can conclude similarly thatwith , K and KR chosen as in equations (2.5) and (2.6). Moreover, writing , (2.19) can be rewritten asIf N is chosen so that(2.20)for all t2>t1, thenand condition (2.16) is satisfied with . It should be noted that if N is chosen to be positive, then condition (2.20) can be reformulated as:(2.21)Hence, if equation (2.21) holds, the global asymptotic stability of the error dynamics of the EMCS algorithm is proven even in the presence of rapidly varying disturbances.
The most sensitive feature of the EMCS procedure is the choice of the constant N, which has to be selected according to the condition (2.20). Finding a suitable value for N would require knowledge of a few system parameters, such as an estimate of b and an upper bound of the external disturbance d1. This is counter to the philosophy of MCS and EMCS algorithms, which were designed to rely on a minimal knowledge of the plant parameters. Hence, in Stoten & Benchoubane (1992), an empirical method for the choice of N was proposed, based on the saturation limits of the control signal. This can yield an unwanted waste of control energy, since N can be an overestimate of the required discontinuous signal amplitude.
3. The new extended minimal control synthesis algorithm
To overcome the limitations of the original EMCS, outlined in §2, we propose a new purely adaptive EMCS algorithm, described by the control law(3.1)where K(t) and KR(t) are still chosen according to equations (2.5), (2.6) and(3.2)with being a positive adaptation weight.
By comparing equation (3.1) with (2.8), we can see that the constant N, characterizing the EMCS control law, has been substituted by a gain KN(t) generated according to equation (3.2). The original EMCS control strategy is therefore modified into a purely adaptive control law, where the amplitude of discontinuous switching action is also adaptively estimated.
global asymptotic stability of the error dynamics is guaranteed; and the amplitude of the discontinuous switching action converges to a finite value K*, namely
(3.3)To prove global asymptotic stability, the same procedure outlined in §2 can be followed by using KN(t) instead of the constant value N. In doing this, condition (2.20) becomes:(3.4)and since by definition KN(t)>0, equation (2.21) can be rewritten as(3.5)From equation (3.2), we can deduce that KN(t) is monotonically increasing (i.e. ); hence, for any constant value , there exists a positive value of γ and a time instant t*, such that KN(t)>D, for all t>t*. If we now setwe can then conclude that for all t>t1, condition (3.5) remains satisfied together with the condition (2.17). This proves the global asymptotic stability of the error dynamics for the new, purely adaptive EMCS as given by equation (3.1).
It should be noted that the bounded input–bounded output property of hyperstable systems (Landau 1979) guarantees boundedness of the error dynamics in the standard MCS algorithm, i.e. the error remains bounded until t=t1 even in the presence of the nonlinear disturbance d(x, t) and the new switching action. The role of the additional switching action in the new EMCS is to provide an auxiliary contribution to the control law in order to cope with any rapidly varying external disturbances.
Finally, because of the global asymptotic stability of the error dynamics, we obtainand from equation (3.2), it follows that .
In view of the above, we can conclude that the new EMCS control equation (3.1) can guarantee both global asymptotic stability of the error dynamics and the boundedness of the adaptively estimated switching contribution. It should be noted that this strategy does not require any off-line computation of the amplitude of the switching action and is now in the spirit of the original MCS algorithm.
(a) New extended minimal control synthesis control of a chaotic system
In Stoten & di Bernardo (1996), it was shown that the standard MCS can be applied to the control of a Duffing oscillator or the synchronization of two identical Chua circuits. In both the cases presented, though, the error equation contained a vanishing nonlinear perturbation, since the plant and reference model were characterized by the same kind of nonlinearity. In the case of the Duffing oscillator, for instance, both the plant and reference model had a cubic nonlinear term acting on them.
If a linear reference model is considered instead, simulations showed that convergence of the error system is still obtained with the standard MCS, although a residual tracking error remains, as shown in figure 1. Increasing the adaptive weights, α and β, can further reduce the residual error at the expense of noise propagation in practical implementations.
In fact, the controlled chaotic Duffing oscillator can be represented by the nonlinear system(3.6)wherewith q=1.8 and ω=1.8.
Hence, by choosing a linear reference model, such as:wherewith , the error dynamics, (2.11), will contain a non-vanishing cubic nonlinear perturbation and hence global asymptotic stability cannot be proven by means of the standard MCS approach. Figures 1 and 2 show the corresponding MCS error trajectories and gain evolutions, respectively. There is clearly a consistent residual tracking error.
We will now show that by using the new EMCS control strategy, outlined in this paper, it is possible to achieve global asymptotic stability of the error system even in the presence of the cubic nonlinearity. In particular, we form the control signal according to equations (3.1) and (3.2) with γ=10. It should be noted that the control parameters α, β and γ were chosen here by trial and error.
As shown in figure 3, when the new EMCS control is activated, the error on both the system states decays smoothly towards zero. This clearly shows the effectiveness of the EMCS control strategy in dealing with nonlinear disturbances. The controller adaptive gains are shown in figures 4 and 5.
By comparing figures 2 and 4, we can see how the use of an adaptively estimated switching contribution affects the evolution of the feedback and feed-forward gains, K(t) and KR(t). While the standard MCS gains represented in figure 2 continue to oscillate, the new EMCS gains, shown in figure 4, quickly converge to lower values because of the presence of the extra switching action term. In addition, although all gains have zero initial conditions, the non-zero initial condition on the error xe immediately imparts non-zero gains at t=0+ via the proportional part of the algorithm.
Finally, in figure 6, the standard MCS control input is compared with the EMCS strategy. Although the two inputs are bounded within the same limits, as expected, the EMCS control input is affected by the chattering introduced by the switching action. It is relevant to point out that, as shown in Ryan (1991), this can be reduced by appropriately smoothing the discontinuous action in equation (3.1) at the expense of a small bounded error.
4. Using topological transitivity: the local minimal control synthesis strategy
Up to this point, it has been shown that the MCS and MRAC in general can be successfully used to control chaotic systems or synchronize their evolution. In doing this, though, no property of chaotic evolution, except the boundedness of their attractors, has been exploited. As mentioned in Vincent (1997), chaos can be explicitly used for the design of innovative control schemes. In this section, we will propose an alternative modification to the MCS in order to exploit the ergodicity1 of chaotic trajectories.
First, we assume that the error trajectory, , obtained as the difference between the reference model evolution and plant, is also evolving along a chaotic attractor, which includes the origin. Hence, we can deduce that the error evolution will belong to a compact set of the error phase plane. This is often the case in many systems of practical significance. When the problem of controlling chaos is considered, the plant is usually in a chaotic regime, while the reference model evolves along a periodic solution embedded in it. As a consequence, the error evolution will also evolve along a chaotic attractor in the error phase plane that includes the origin.
The aim of our control scheme is to render the origin of the error phase space an asymptotically stable equilibrium point. Using a standard MCS control, such an objective would be achieved globally by starting the adaptive control strategy at t=0. The adaptive gain will then evolve and compensate for the nonlinear disturbances acting on the plant. Boundedness of chaotic trajectories would then be exploited to achieve control.
It should be noted that during the system evolution, the uncontrolled error trajectories, xe(t), will enter a ball at the origin of arbitrarily small radius, δ, because of the ergodicity of chaotic evolution. We will then exploit this additional property of chaotic systems in order to improve our original control strategy.
Once the desired unstable periodic orbit to be stabilized is chosen, the MCS control will be initialized when the chaotic evolution of the error system enters a sufficiently close neighbourhood of the origin. As pointed out earlier, this will be satisfied at some finite time, because of the ergodicity of chaotic trajectories.
The MCS algorithm will then start an adaptive action to stabilize the plant onto the desired unstable periodic orbit. As previously discussed, this procedure requires minimal knowledge of the systems involved and there is no need for off-line identification.
The controller, though, relies on the plant under open-loop control to free-fall, until certain conditions are met, which is not usually desirable from a control-engineering viewpoint. In this case, the chaotic nature of the resulting system under open-loop control guarantees that the system evolution will remain bounded to a compact set of phase space. Thus, by allowing the system to be chaotic during the transient phase, the adaptive control effort is further reduced from that described earlier, since the control starts only when the plant and the reference model states are sufficiently close. It should also be noted that this implies the possibility of exploiting chaos to use controllers that can guarantee only local stability to achieve global stability of the error system.
Finally, we note that the original stability proof for the MCS algorithm as presented in Stoten & Benchoubane (1990a) is valid for this modification of the strategy, i.e. without loss of generality, we can assume that the chaotic evolution of the plant will take it into a neighbourhood of the desired orbit, where the linearization is valid. Hence, the problem will be in stabilizing a linearized system to an unstable equilibrium, which the original MCS algorithm is capable of solving without any major change (given that the linearization satisfies the constraints required by the MCS adaptive technique).
(a) Example: local minimal control synthesis control of a Duffing oscillator
As a representative problem, we consider the problem of controlling a chaotic Duffing oscillator to the periodic solution that it exhibits when q=0.62. We then consider a reference Duffing oscillator evolving along the solution we wish to track. According to the new local MCS strategy, after the transients have decayed, we monitor the error evolution and start the controller when the error trajectories first enter the ball at the origin of arbitrary value δ=10−3.
Figure 7 shows the plant states before and after the control. After 30 s, the error trajectories were monitored and the MCS started when they entered the desired ball of the origin. As shown, the control objective was achieved after a short transient and the plant states started evolving along the desired periodic trajectory. The gain evolutions are depicted in figure 8 over a longer time range. It should be noted that as the control objective is achieved, the feedback gains, , approach zero. Hence, once again, the control is dissipative and only a feed-forward action is necessary to compensate the forcing input responsible for the chaotic behaviour.
5. Swing-up control of a double inverted pendulum
We will now present an application of the novel MCS-based strategy presented in §4 to control a nonlinear double pendulum system as shown in figure 9. The system consists of a carriage and two rigid links, which form the upper and lower pendulum arms. The lower pendulum has mass m1, length l1 and a rotational friction coefficient c1 at the pivot O. The upper pendulum has mass m2, length l2 and rotational friction coefficient c2 at the joint with the other link. The carriage has mass m3 and translational friction coefficient c3. We assume that a horizontal force u(t) on the carriage is the only control input acting on the system and the carriage position x and the lower and upper pendulum angles θ and β, respectively, are generalized coordinates.
The system can then be represented by the following equations:(5.1)whereFrom now on, we will assume that , m3=0.5, l1=0.25, l2=0.25, , c3=3.8 following the design presented in Perry (2000).
The aim of the control action will be to choose the lateral force on the cart, u, in order to stabilize the unstable inverted position corresponding to θ=β=0. It should be noted that the linearized system about this equilibrium position can be shown to be completely observable and controllable (Perry 2000). Hence, a local control strategy can be used to stabilize the inverted double pendulum, if the initial conditions are close enough to its inverted position.
Thus, we propose to split the control problem into two phases. First, we will consider the problem of swinging-up the pendulum to a neighbourhood of the desired target equilibrium. Then, we will synthesize a local control strategy to stabilize the inverted equilibrium point. We will show that this can be effectively done using the strategy presented in §4.
(a) Phase 1: swing-up
Let us assume that without loss of generality, we start the system in its stable equilibrium position . The aim of the swing-up phase is to move the system away from its stable equilibrium up to a sufficiently close neighbourhood of the unstable inverted equilibrium. The state position at which this phase ends and control switches to a stabilizing linear controller will be the state values within the neighbourhood of θ=β=0, where the states of the system are suitable for control. This will be referred to as the switching point. In general, we would set this point to be where the pendula enter the linear control basin and term the time taken to reach this point as the swing-up time, ts.
We look for a robust swing-up method characterized by a small swing-up time and bounded control inputs, which does not require any additional torque acting between the pendulum links. To achieve this, we propose to use the fact that the forced double pendulum is a well-known chaotic system (e.g. Strogatz 1998). Specifically, we require the swing-up control to render the closed-loop system chaotic and, because of the topological transitivity of the attractor, to bring the system trajectory within a sufficiently small neighbourhood of the target equilibrium point. In fact, we know that the trajectory of a chaotic system densely fills a compact set of the phase space, which means that after some time ts, the system trajectory will enter any arbitrary phase space neighbourhood covered by the chaotic attractor. Thus, if this includes the origin, there exists a time ts, where the system enters a neighbourhood of the inverted equilibrium state.
To make the system chaotic during swing-up, we select u as a sinusoidal force of appropriate amplitude and frequency with no extra torque acting between the links. Figure 10 shows the resulting chaotic motion of the closed-loop system. As shown in figure 10, the open-loop chaotic motion of the system successfully swings up the links from the natural equilibrium position to a sufficiently small neighbourhood of the inverted position. It should be noted that θ and β enter the desired neighbourhood after ts≈2.4 s (the discontinuities of the angular evolutions are due to the fact that the angular values are all projected onto the interval (−π,π)).
In order to simplify the next phase of the control action, we choose the switching time so that both the states of interest are within a 15° range from the origin with their velocities being close to the origin, i.e. . Moreover, to facilitate the control action, we require each pendulum link to be moving towards the desired equilibrium position, i.e. . In general, we require the accelerations of the links to be small to facilitate their control onto the inverted equilibrium. As shown in figure 10, the chaotic trajectory of the system during the swing-up phase reaches the phase space neighbourhood, where these constraints are satisfied after about 2.38 s. At this time instant, the control strategy switches to the next phase, where a local MCS adaptive controller is used to ‘catch’ the system trajectory and render the origin asymptotically stable.
(b) Phase II: minimal control synthesis catching control
Once the system trajectory has entered a sufficiently close neighbourhood of the origin, system (5.1) can be linearized and takes the form:(5.2)where the matrices A and B can be explicitly computed in terms of the system parameters and are not reported here for the sake of brevity. The problem then is controlling the linearized system onto the origin. This can be done using the MCS adaptive algorithm as explained in §4. To further minimize the overall control effort, we chose to use an optimal adaptive MCS controller, which was recently presented in di Bernardo & Santini (2005) in order to enhance the overall robustness of the resulting closed-loop scheme.
The evolution of the global control scheme obtained by the combined use of the swing-up chaotic control phase and the local control phase is shown in figure 11 and the corresponding control input is shown in figure 12. Figure 11 clearly shows that by applying the control strategy presented here, the double pendulum is indeed stabilized to the desired inverted position by applying a force on the cart. The pendula settle down to the desired position in about 4.5 s with the cart returning to its rest position within 14 s.
The control laws presented in this paper further extend the range of application of MCS-based controllers. The case of rapidly varying nonlinearities acting on the plant is taken into account and a new ad hoc modification of the EMCS algorithm is presented, i.e. the new adaptive EMCS algorithm. This is described by a control equation characterized by a discontinuous switching action, whose amplitude is adaptively generated. In the spirit of the MCS class of controllers, the new EMCS strategy does not rely on any knowledge of the plant parameters. Moreover, the limitations of the original EMCS algorithm are overcome, since no prior estimate of the nonlinearities acting on the plant is required. It has also been suggested that the MCS algorithm can be modified using the topological transitivity of chaotic trajectories. Thus, a new method of engineering chaos has been presented. The control strategy proposed makes effective use of the ergodicity of chaotic evolution together with its boundedness. In fact, it was possible to use a control scheme, mainly developed for structurally linear dynamical systems, to achieve the control of a structurally nonlinear system with minimal knowledge of the system itself. Neither knowledge of the plant nonlinear terms nor of its dynamic parameters is required at any stage of the synthesis (other than a a priori assumption that it is evolving along an ergodic, bounded, chaotic attractor). The method was used to propose a novel strategy to solve the problem of swinging-up a double pendulum stabilizing its inverted position in the absence of any extra torque between the pendulum links.
One contribution of 15 to a Theme Issue ‘Exploiting chaotic properties of dynamical systems for their control’.
↵In this context, we refer to ‘ergodicity’ as the property of densely filling a bounded set of the phase space exhibited by the chaotic evolutions of the systems under investigation.
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