System identification of propagating wave segments in excitable media and its application to advanced control

The present paper proposes a scheme for controlling wave segments in excitable media. This scheme consists of two phases: in the first phase, a simple mathematical model for wave segments is derived using only the time series data of input and output signals for the media; in the second phase, the model derived in the first phase is used in an advanced control technique. We demonstrate with numerical simulations of the Oregonator model that this scheme performs better than a conventional control scheme.

Figure 1

Feedback system with PI control [23].

Figure 2

Snapshots and time series of wave segments [$K_{\mathrm{P}}=-3\times {10}^{-3}$, $K_{\mathrm{I}}=-2\times {10}^{-3}$, $z\left(0\right)=0.093$].

Figure 3

Block diagram of a linear system around the steady state of the PI control system.

Figure 4

Time series of the reference signal $\widehat{r}\left(t\right)$ and width $\widehat{w}\left(t\right)$ around the steady state ($w_0=4$ and $U_0=0.0913$) for system identification with starting time $\underline{T}=15$, finishing time $\overline{T}=30$, and amplitude $\mathrm{\Delta }{w}_0=0.1$.

Figure 5

Time series of the predicted output signal $\mathrm{\Delta }w({\tilde{\mathbf{\theta }}}_m,t)$ (black line) and the observed output signal $\mathrm{\Delta }\widehat{w}\left(t\right)$ (gray line) for (a) $m=1$, (b) $m=2$, and (c) $m=3$.

Figure 6

Snapshots and time series of wave segments in the medium where there are two narrow channels. The parameters, initial conditions, and gains are the same as in Fig. 2.

Figure 7

Control system with feed-forward compensation.

Figure 8

Control performances of the PI control system without the feed-forward compensator $C_{\mathrm{F}}\left(s\right)$ as shown in Fig. 1 and with the feed-forward compensator as shown in Fig. 7.

Figure 9

Snapshots and time series of wave segments with compensator (12) in the medium with two narrow channels. The parameters, initial conditions, and gains are the same as in Fig. 6.