Adaptive time-delayed stabilization of steady states and periodic orbits

We derive adaptive time-delayed feedback controllers that stabilize fixed points and periodic orbits. First, we develop an adaptive controller for stabilization of a steady state by applying the speed-gradient method to an appropriate goal function and prove global asymptotic stability of the resulting system. For an example we show that the advantage of the adaptive controller over the nonadaptive one is in a smaller controller gain. Second, we propose adaptive time-delayed algorithms for stabilization of periodic orbits. Their efficiency is confirmed by local stability analysis. Numerical examples demonstrate the applicability of the proposed controllers.

Figure 1

Solutions $x_1\left(t\right)$ (blue solid line) and $x_2\left(t\right)$ (green dashed line) of the closed-loop system of Eqs. (5) and (11), with $A$ given by Eq. (15) and $P$ and $Q$ given by Eq. (16). The parameters are $\lambda =0.5$, $\omega =\pi$, $\tau =1$, and $\gamma =1$. The initial conditions are $x_1\left(0\right)=10$, $x_2\left(0\right)=-7$, and $k\left(\tau \right)=0$.

Figure 2

Evolution of $k\left(t\right)$ for the system from Fig. 1. The limit value is ${lim}_{t\to \infty }k\approx 2.4$.

Figure 3

Norms of five different solutions $y\left(t\right)$ of the closed-loop system given by Eqs. (17), (18), and (19) with randomly chosen initial conditions ${[y_1\left(0\right),y_2\left(0\right)]}^T\in [-2,2]\times [-2,2]$ and $k\left(\tau \right)=0$. The parameters are $\varepsilon =0.2$, $\tau =\frac{\pi }{\omega }\approx 3.16$, and $\gamma =1$.

Figure 4

Evolution of adaptation gain $k\left(t\right)$ for five different initial conditions as given in Fig. 3.

Figure 5

($k-\lambda$) plane: blue shaded region, region of stability of (non-adaptive) time-delayed feedback control [10]; $k_{*}$, critical coupling strength above which the system is stable for all $\lambda \in [0,1]$; and $k=0.153$, value reached by adaptive control according to Eq. (11).

Figure 6

Evolution of $k\left(t\right)$ (blue solid line) for the system given by Eqs. (5) and (11) with $A$ given by Eq. (15). Value $k_{*}=0.5$ (red dashed line). $\lambda =0.1$. Other parameters are as given in Fig. 1. The initial conditions are $x\left(0\right)=2$, $y\left(0\right)=0$, and $k\left(\tau \right)=0$.

Figure 7

Maximum real part (red dashed line) and corresponding imaginary part (blue solid line) of Floquet exponent $\Lambda$ obtained by numerically solving Eq. (23). Parameters are as given in Fig. 1.

Figure 8

Phase portrait of the system given by Eqs. (20) and (21) for $t\ge \tau$. Blue solid lines are the trajectories for ten different initial conditions chosen randomly from $[-1,1]\times [-1,1]$. The red dashed line corresponds to the periodic solution $[x_{*}\left(t\right),y_{*}\left(t\right)]$ given by Eq. (22). Parameters are as given in Fig. 1.

Figure 9

(a) Evolution of $k\left(t\right)$ (blue solid line) for ten different initial conditions chosen randomly from $[-1,1]\times [-1,1]$; the value $k_{*}=\frac{\lambda }{2}$ (red dashed line). Parameters are as given in Fig. 1. (b) Zoom of panel (a).

Figure 10

Phase portrait of the system given by Eqs. (20) and (24) for $t\ge \tau$. Blue solid lines are the trajectories forten different initial conditions chosen randomly from $[-5,5]\times [-5,5]$. The red dashed line corresponds to a cycle of radius $a=3$. $\alpha =0.01$. Other parameters are as given in Fig. 1.

Figure 11

Evolution of $\gamma \left(t\right)$ (blue solid line) for the system given by Eqs. (20) and (24). The red dashed line is the value of ${\gamma }_{*}=\frac{\lambda }{2a^2}$. Parameters and initial conditions are as given in Fig. 10.

Figure 12

Maximum real part (red dashed line) and corresponding imaginary part (blue dashed line) of Floquet exponent $\Lambda$ obtained by numerically solving Eq. (26). $\alpha =0.01$ and $a=3$. Other parameters are as given in Fig. 1.