Variable-delay feedback control of unstable steady states in retarded time-delayed systems

We study the stability of unstable steady states in scalar retarded time-delayed systems subjected to a variable-delay feedback control. The important aspect of such a control problem is that time-delayed systems are already infinite-dimensional before the delayed feedback control is turned on. When the frequency of the modulation is large compared to the system’s dynamics, the analytic approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed-delay system. Otherwise, the stability domains are obtained by a numerical integration of the linearized variable-delay system. The analysis shows that the control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form and the frequency of the delay modulation.

Figure 1

Diagram in the complex plane $z$, associated with the derivation of the parametric representation of the stability boundary in $\left(T_1,T_2\right)$ plane.

Figure 2

Representative samples of phase plots $x\left(t\right)$ vs. $x\left(t-T_1\right)$ for the uncontrolled Mackey-Glass system at different values of $T_1$: (a) $T_1=4$—the trajectory is attracted to the stable equilibrium point $x_3^{\ast }=-1$; (b) $T_1=8$—the trajectory approaches a limit cycle; (c) $T_1=15$—the attractor has evolved into a period-2 cycle; (d) $T_1=23$—chaos. The simulations were performed using the MATLAB routine dde23 for integrating delay-differential equations with constant delays.

Figure 3

(Color online) Domains of successful VDFC control in the $\left(K,T_2\right)$ plane for the unstable equilibria $x_{2,3}^{\ast }=\pm 1$ in the chaotic Mackey-Glass system $\left(T_1=23\right)$. The control delay $\tau \left(t\right)$ is modulated with a saw tooth wave, and the values of the modulation amplitudes are: (a) $\epsilon =0$ (TDFC); (b) $\epsilon =0.5$, (c) $\epsilon =1$; (d) $\epsilon =2$. Combinations of $K$ and $T_2$ where VDFC successfully stabilizes the fixed points $x_{2,3}^{\ast }=\pm 1$ are plotted in gray tones (color tones online). Note the shifts of the origin along the $T_2$ axes by an amount equal to $\epsilon$ due to the limitation $T_2\ge \epsilon$.

Figure 4

(Color online) [(a), (b)] Stability domains of $x_{2,3}^{\ast }=\pm 1$ in the $\left(K,T_2\right)$ plane for the VDFC-controlled Mackey-Glass system with $T_1=23$. The delay modulation is in a form of a sine-wave with $\epsilon =1$ (panel a) and $\epsilon =2$ (panel b). [(c), (d)] Corresponding stability domains for a square-wave modulation. Note the shifts of the origin along the $T_2$ axes by an amount equal to $\epsilon$ due to the limitation $T_2\ge \epsilon$.

Figure 5

(Color online) Domains of successful control in the $\left(T_1,T_2\right)$ plane for the unstable fixed points $x_{2,3}^{\ast }=\pm 1$ in the Mackey-Glass system. The feedback gain is fixed at $K=0.5$. (a) Stability diagram for $\epsilon =0$ (TDFC). (b)–(d) Respective stability diagrams for saw tooth, sine and square-wave modulations with $\epsilon =2$ (VDFC). Note that the minimum value of the $T_2$ axis in panels (b)–(d) is $T_2=2$ due to the limitation $T_2\ge \epsilon$.

Figure 6

VDFC applied to the chaotic Mackey-Glass system using different modulations of the delay-time $\tau \left(t\right)$. The parameters of the uncontrolled system are: $a=0.2$, $b=0.1$, $c=10$, $T_1=23$. [(a), (b)] Time plots of the variable $x\left(t\right)$ and the feedback signal $u\left(t\right)$ for a saw-tooth-wave modulation, indicating a successful control of the unstable fixed point at $x_2^{\ast }=+1$. [(c), (d)] Stabilization of the unstable equilibrium at $x_3^{\ast }=-1$ with a sine-wave modulation. [(e), (f)] Time series for a square-wave modulation stabilizing the unstable point at $x_3^{\ast }=-1$. In each case, the control parameters were: $K=2$, $\epsilon =2$, $T_2=18$, and $\nu =5$. The control was activated at $t=500$. The total time span shown in each panel is 1500 time units. The simulations were performed using the MATLAB routine ddesd for integrating delay-differential equations with general delays.

Figure 7

The stability domains in $\left(K,T_2\right)$ plane related to the unstable steady states $x_{2,3}^{\ast }=\pm 1$ of the chaotic Mackey-Glass system $\left(T_1=23\right)$ for a low frequency modulation of the time delay $\tau \left(t\right)$. The delay modulation is in a form of a saw tooth wave with $\epsilon =2$, and the value of the modulation frequency is: (a) $\nu =1.4$, (b) $\nu =1.8$, (c) $\nu =1.9$, (d) $\nu =2.0$, (e) $\nu =2.2$, (f) $\nu =2.4$, (g) $\nu =2.6$, (h) $\nu =3.0$. The combinations $\left(K,T_2\right)$ leading to a successful stabilization of the unstable equilibria $x_{2,3}^{\ast }=\pm 1$ are marked in black. The characteristic eigenfrequencies of the uncontrolled system that lie in this interval of $\nu$ are: 1.43, 1.71, 1.98, 2.25, 2.52, and 2.80. Note the appearance of the resonance islands at the right of the main structure as $\nu$ becomes smaller than $\nu =3.0$. Also note the shifts of the origin along the $T_2$ axes by an amount equal to $\epsilon$ due to the limitation $T_2\ge \epsilon$.

Figure 8

A sample of the low-frequency control domains in $\left(K,T_2\right)$ plane corresponding to the unstable steady states $x_{2,3}^{\ast }=\pm 1$ of the chaotic Mackey-Glass system $\left(T_1=23\right)$ for a square-wave modulation with $\epsilon =2$. The value of the modulation frequency is: (a) $\nu =1.7$, (b) $\nu =2.8$, (c) $\nu =3.0$, (d) $\nu =4.0$, (e) $\nu =8.0$, (f) $\nu =10.0$, (g) $\nu =11.0$, (h) $\nu =12.0$. The range of $\nu$ in which the resonance islands exist consists of several distinguished intervals encompassing some of the eigenfrequencies of the uncontrolled system: 1.71, 3.07, 7.99, 10.99. Note that although the modulation frequency $\nu$ in panel (b) coincides with one of the eigenfrequencies of the uncontrolled system $\left(\approx 2.8\right)$, the resonance islands are hardly noticeable in this case.