Restricted feedback control in discrete-time dynamical systems with memory

When an equilibrium state of a physical or biological system suffers a loss of stability (e.g., via a bifurcation), it may be both possible and desirable to stabilize the equilibrium via closed-loop feedback control. Significant effort has been devoted towards using such control to prevent oscillatory or chaotic behavior in dynamical systems, both continuous-time and discrete-time. Regarding control in discrete-time systems, most prior attempts to stabilize unstable equilibria require that the system be perturbed once during each time step. However, there are examples of systems for which this is neither feasible nor possible. In this paper, we analyze a restricted feedback control method for discrete-time systems (restricted in the sense that the controller's perturbations may be applied only in every other time step). We apply our theoretical analysis to a specific example from cardiac electrophysiology in which this sort of restricted feedback control is especially relevant. The example is a useful test case for the theory, and one for which an experimental setup is rather straightforward.

Figure 1

(a) Unrestricted TDAS control domain defined by inequalities (7), assuming $D_1<-1$ and $D_2>0$. (b) Restricted TDAS control domain defined by inequalities (8) under the same assumptions.

Figure 2

Bifurcation diagrams for (11) with parameters chosen from (12). (a) Bifurcation diagram for $E=170\text{ms}$. (b) Bifurcation diagram for $E=250\text{ms}$. (Note the wider horizontal scale.)

Figure 3

Schematic action potentials in a periodically stimulated cardiac cell. (a) Alternans in the absence of control. (b) Introduction of TDAS control in an ON-OFF pattern after the $n$th action potential. Native stimuli are indicated by bold dots, and the artificial stimulus is indicated by a hollow circle.

Figure 4

(a) Control domain (shaded region) for unrestricted TDAS applied to the memoryless restitution mapping (9) and (10). Although the axes have been converted to $\gamma$ versus $B$, this is merely a reproduction of the control domain in Fig. 1 for a specific mapping. (b) Control domain (shaded region) for restricted ON-OFF TDAS applied to the memoryless restitution mapping (9) and (10). The boundary curves are labeled according to (18). As explained in Sec. 3 B, this is not a reproduction of Fig. 1.

Figure 5

Time series for the memoryless model (9) and (10) for $B=440$ ms. During the first 20 beats, no feedback control is applied and alternans occurs. During the next 20 beats, (a) unrestricted TDAS and (b) restricted TDAS are applied with $\gamma =0.2$.

Figure 6

(a) Control domain for (22) using the parameters in (12) and $E=170\text{ms}$. (b) Same as Panel (a) except that $E=250\text{ms}$.

Figure 7

Bifurcation diagrams for (22) with the usual parameters (12) and $\gamma =0.25$. Compare to Fig. 2, the corresponding bifurcation diagrams without control (i.e., $\gamma =0$). (a) For $E=170\text{ms}$, the fixed point $A^{*}$ is stabilized regardless of $B$, and alternans is eliminated. (b) For $E=250\text{ms}$, there is a window of $B$ for which alternans cannot be eliminated even though its amplitude is substantially reduced.

Figure 8

Basins of attraction (shaded) of fixed points stabilized by ON-OFF TDAS. The vertical axis shows $\delta$, the separation between the initial APD and the fixed point $A^{*}$. (a) Memoryless mapping (10), $\gamma =0.1$. (b) Memory model (11), $\gamma =0.1$ and $E=170\text{ms}$. (c) Memory model (11), $\gamma =0.1$ and $E=250\text{ms}$. Panels (d)–(f) are similar, except that $\gamma =0.25$.