Optimal Parametric Feedback Excitation of Nonlinear Oscillators

An optimal parametric feedback excitation principle is sought, found, and investigated. The principle is shown to provide an adaptive resonance condition that enables unprecedentedly robust movement generation in a large class of oscillatory dynamical systems. Experimental demonstration of the theory is provided by a nonlinear electronic circuit that realizes self-adaptive parametric excitation without model information, signal processing, and control computation. The observed behavior dramatically differs from the one achievable using classical parametric modulation, which is fundamentally limited by uncertainties in model information and nonlinear effects inevitably present in real world applications.

Figure 1

(a) $(\gamma ,\mathrm{\Delta }k)$ stability chart for the damped linear oscillator (7) under optimal parametric feedback control. The resonance condition is given by $\mathrm{\Delta }k>\mathrm{\Delta }{k}_{\min }^{\mathrm{opt}}(\gamma )=\frac{1}{2}\pi \gamma -[(3/16)\pi -(1/192){\pi }^3]{\gamma }^3+O({\gamma }^5)$. (b) $(\delta k,\mathrm{\Delta }k;\gamma )$ stability chart showing the behavior of the oscillator under perfectly tuned ($\delta k=0$) and off-tuned ($\delta k\ne 0$) feed-forward modulation (8). (c) Behavior of the oscillator ($\gamma =0.3$, $\mathrm{\Delta }k=0.5$) under optimal feedback excitation, feed-forward modulation, square wave modulation, and sine wave modulation under perfectly tuned conditions ($\delta k=0$, white area) and when off-tuning is present ($\delta k=0.2$, shaded area). The effectiveness and robustness of the feedback controller is evident from these simulations.

Figure 2

(a) Monostable Duffing oscillator (9) ($k_0=1$, $k_3=0.01$, $\mathrm{\Delta }k=0.2$) under optimal parametric feedback control. No model parameters are used to implement this controller. The same plot also shows the behavior of the oscillator under feed-forward parametric excitation implemented using the model-based controller $u_{\mathrm{FF}}(t;\gamma =0,\mathrm{\Delta }k=0.2,\delta k=0)$ (8). This controller is optimal in the present context under the assumption that the nonlinearity is negligible, i.e., $k_3\approx 0$. (b) Bistable Duffing oscillator ($k_0=-1$, $k_3=0.01$, $\mathrm{\Delta }k=0.2$) under optimal parametric feedback control. The dotted lines denote the location of the stable nontrivial and the unstable trivial equilibrium positions of the system (i.e., when $u=u_{\max }$ and $u=u_{\min }$). Self-tuning of the excitation is evident for the feedback control implementation.

Figure 3

(a) Tunable parametrically excited nonlinear oscillator $\ddot{q}+\gamma \dot{q}=F(q,u)$. The force function is given by $F(q,u)=F_{\mathrm{D}}(q)+F_{\mathrm{PE}}(q,u)$, where the first term (implemented with diodes $D_{1,2}$) resembles the bistable Silva-Young oscillator [21] while the second term $F_{\mathrm{PE}}(q,u)=-(k_0+\mathrm{\Delta }ku)q=-\alpha V_k(u)q=-\alpha (V_{k0}+V_{\mathrm{\Delta }k}u)q$ (where $\alpha$ is a circuit parameter) provides parametric excitation. The controller (6), $u=V_u/V_{u\max }=u_{\mathrm{FB}}^{\mathrm{opt}}(q,\dot{q})$, is implemented using a Schmitt trigger comparator. The amplitude of the parametric excitation $V_{\mathrm{\Delta }k}\propto R_1/R_6$ is adjusted by $R_6$. Further, we use an external input $V_{k0}$ to change the static stiffness of this circuit. The values of the analog components are $L=12\mathrm{mH}$, $C=470\mathrm{nF}$, $R=10\mathrm{\Omega }$, $R_1=10\mathrm{k}\mathrm{\Omega }$, $R_2=1\mathrm{M}\mathrm{\Omega }$, $R_3=1.5\mathrm{k}\mathrm{\Omega }$, $R_4=1\mathrm{k}\mathrm{\Omega }$, $R_5=10\mathrm{M}\mathrm{\Omega }$, and $R_6\in [0.001,1]\mathrm{M}\mathrm{\Omega }$. (b) Mean oscillation amplitude vs static stiffness $V_{k0}$ plots under low amplitude $V_{\mathrm{\Delta }k}=1\mathrm{V}$ and high amplitude $V_{\mathrm{\Delta }k}=2\mathrm{V}$ parametric modulation. The same plot includes the static bifurcation diagram, i.e., equilibrium positions of the circuit under no parameter modulation, i.e., $V_{\mathrm{\Delta }k}=0\mathrm{V}$. The solid black lines (dashed gray lines) correspond to the behavior of the system displayed under quasistatic forward (backward) sweep experiments. The error bars on these plots indicate 3 standard deviations. (c) Phase plots and (d) optimal parameter modulation shown for selected experiments.