Chaos Control for Modified Shinriki Oscillator
UPOs embedded in strange Attractor
The strange attractor of the Shinriki oscillator contains (infinitely many) unstable periodic orbits (UPOs). One of the goals of chaos control is to stabilize them. In the figure the attractor is shown in its full 3 dimensional phase space together with two (stabilized) UPOs: Period 2 (blue) and period 1 (black). Data are from actual measurements, not from simulation!
Simple TDFC scheme
Time-Delay-Feedback Control
One of the simplest ways to control a chaotic system is Time Delay Feedback Control (TDFC). From point of view of signal processing this corresponds to a periodic notch-filter, as shown to the left. The advantage of this type of control is - besides its simplicity of implementation- that the system dynamics of the chaotic oscillator needs not to be known in detail. Only the period duration n needs to be known exactly, in order to obtain a truly vanishing control signal. To this end, the sampling frequency must be choosen rather high, allowing for fine tuning of the delay time. A more sophisticated controller, including also automatic period locking, is discussed below.
TDFC Measurement Results
Measurement GUI for TDFC
GUI for TDFC measurement. In the first picture, a spectrogram of the Shinriki-State-Signal x1 is displayed. Clearly, the orbit becomes perfectly periodic. The phase image contains both, the trajectories from the chaotic attractor (0...1sec) and the embedded period one UPO (from data 1..2sec). Lower right are Fourier spectra, clearly showing discrete lines as proof for periodicity.
Spectrogram of control signal
The corresponding spectrogram of the control signal shows some "strong action" shortly after activating control (t = 1sec), followed by a quick decay of energy. Note that the total power after having stabilized the periodic orbit is well below -80dB (!), wheras the state signal x1 has power of around -10dB.
Control Signal in Time Domain
Similar scenario as above, showing a "zoom" of the control signal in time domain. At sample time 500 the controller was activated, most of the "work" is done after a few periods (of the UPO to be stabilized).
Signal Processing Implementation of TDFC
Signal Processing View of TDFC: Overview
Programmable Comb Filter
Automatic Frequency Control
The simple time delay implementation discussed above has the disadvantage that the sampling rate must be choosen rather high, in order to have a low enough granularity for the delay time. From point of view of signal processing, a sampling frequency which catches, say, the 8th harmonic of the periodic signal, is completely sufficient, since the spectra decay relatively fast with frequency.
The controller can thus be realized as a programmable comb filter with notches at Ω, 2Ω, 3Ω,... . Each notch is implemented as a biquad section comprising a zero-pole pair. The fundamental frequency Ω is then controlled by an AFC (automatic frequency control) algorithm, which operates in the baseband (double-line arrows are complex signals). With this circuit we can easily stabilize all kinds of UPOs, also with higher periods and also in a wider range of parameters R1 and R2. The AFC locks in a relatively wide frequency-range around the nominal point Ω0.