Control of unstable steady states by time-delayed feedback methods

We show that time-delayed feedback methods, which have successfully been used to control unstable periodic orbits, provide a tool to stabilize unstable steady states. We present an analytical investigation of the feedback scheme using the Lambert function and discuss effects of both a low-pass filter included in the control loop and nonzero latency times associated with the generation and injection of the feedback signal.

Figure 1

Control of an unstable focus with $\lambda =0.5$ and $\omega =\pi$ in the configuration space for different values of the feedback gain $K$. Panels (a)–(d) correspond to $K=0$,0.2,0.25, and 0.3, respectively. The time delay $\tau$ of the TDAS control scheme is chosen as 1, corresponding to $\tau =T_0∕2=\pi ∕\omega$.

Figure 2

(Color online) (a) Largest real part of the complex eigenvalues $\Lambda$ vs $\tau$ for $\lambda =0.5$ and $\omega =\pi$ for different $K$. Some lower eigenvalues are also displayed for $K=0.3$ (green online). (b) Time series of the $x$ component of the unstable focus: The solid line (red online) corresponds to $x\left(t\right)$, the dashed line (green online) to the delayed $x$ component $x(t-\tau )$ with $\tau =1$. The parameters of the unstable focus and the control scheme are as in panel (d) of Fig. 1.

Figure 3

(Color online) Domain of control in the $K\text{−}\tau$ plane and largest real part of the complex eigenvalues $\Lambda$ as a function of $K$ and $\tau$ according to Eq. (7). The two-dimensional projection at the bottom shows combinations of $\tau$ and $K$, for which $\mathrm{Re}\left(\Lambda \right)$ is negative and thus the control successful [panel (a): $\lambda =0.5$ and $\omega =\pi$; panel (b): $\lambda =0.1$ and $\omega =\pi$].

Figure 4

Largest real part of the eigenvalues $\Lambda$ vs $\tau$ for $\lambda =0.5$, $\omega =\pi$, and $K=0.3$ as given by Eq. (23). The solid, dashed, dotted, and dash-dotted curves correspond to a latency time of $\delta =0$, 0.1, 0.2, and 0.3, respectively.

Figure 5

Minimal feedback gain $K$ vs relative latency $\delta ∕\tau$ for $\lambda =0.5$ and $\omega =\pi$ according to Eq. (24). The shaded areas show the domain of control for suitably chosen $\tau$.

Figure 6

(Color online) Domain of control in the $K\text{−}\tau$ plane for different latency times [panels (a) and (c): $\delta =0.1$; panels (b) and (d): $\delta =0.3$]. The shaded areas indicate combinations of $\tau$ and $K$, for which the largest real part of the complex eigenvalues $\Lambda$ is negative and thus control is successful. The value of $\mathrm{Re}\left(\Lambda \right)$ is indicated by the greyscale (color online). The parameters of the unstable focus are chosen as $\omega =\pi$ in all panels and $\lambda =0.5$ in (a) and (b) and $\lambda =0.1$ in (c) and (d).

Figure 7

Minimal feedback gain $K$ vs cutoff frequency $\alpha$ for $\lambda =0.5$ and $\omega =\pi$ according to Eq. (31). The shaded area shows the domain of control.

Figure 8

(Color online) Largest real part of the complex eigenvalues $\Lambda$ vs $\tau$ for $\lambda =0.5$, $\omega =\pi$, and $K=0.3$ as given by Eq. (28). The solid, dashed, dotted, and dash-dotted curves correspond to a cutoff frequency of $\alpha =1000$, 10, 7, and 4, respectively.

Figure 9

(Color online) Domain of control in the $K\text{−}\tau$ plane for different cutoff frequencies [panels (a) and (c): $\alpha =1000$; panels (b) and (d): $\alpha =7$]. The shaded areas indicate combinations of $\tau$ and $K$, for which the largest real part of the complex eigenvalues $\Lambda$ is negative and thus control is successful. The value of $\mathrm{Re}\left(\Lambda \right)$ is indicated by the greyscale (color online). The parameters of the unstable focus are chosen as $\omega =\pi$ in all panels and $\lambda =0.5$ in (a) and (b) and $\lambda =0.1$ in (c) and (d).