Design of active network filters as hysteretic sensors

Abstract

This work aims to propose and design a class of networks of coupled linear and nonlinear oscillators, in which short bursts of exogenous excitation result in sustained endogenous network activity that returns to a quiescent state only after a characteristic time and along a different path than when originally excited. The desired hysteretic behavior is obtained through the coupling of self-excited oscillations with purposely designed rate laws for slowly varying nodal parameters, governed only by local interactions in the network. The proposed architecture and the sought dynamics take inspiration from complex biological systems that combine endogenous energy sources with a paradigm for distributed sensing and information processing. In this paper, the network design problem considers arbitrary topologies and investigates the dependence of the desired response on model parameters, as well as on the placement of a single nonlinear node in an otherwise linear network. Perturbation analysis in various asymptotic parameter limits is used to define the proposed internal dynamics. Parameter continuation techniques validate the asymptotic results numerically and demonstrate their robustness over finite ranges of parameter values. Both approaches suggest a nontrivial dependence of the optimal distribution of nonlinearity on the network topology.

Appendices

Appendix A: Nomenclature

$\delta$ :

A small offset for damping dynamics

$\epsilon$ :

Scaling parameter of linear and nonlinear damping in the network model

$\eta$ :

Nonlinear damping parameter of the nonlinear oscillator in the network model

$\mu$ :

Damping parameter for linear oscillator at each node when $\zeta _k=\mu$ for all k

$\nu$ :

Linear damping parameter of the nonlinear oscillator in the network model

$\omega _k$ :

Natural frequency of k-th mode in the network model

$\phi (\epsilon t)$ :

Phase of $\mathcal {O}(1)$ solution in multiple-scale analysis

$\tau$ :

A large time scale for damping dynamics

$\tilde{\zeta }$ :

A damping parameter in bifurcation analysis, $\tilde{\zeta }=\sum _{k\ne Q}\frac{K_{Q,k}K_{k,Q}}{\epsilon ^2\zeta _k K_{Q,Q}}$

$\tilde{C}(x)$ :

Modal damping matrix

$\tilde{K}$ :

Modal stiffness matrix

$\zeta$ :

A damping parameter in bifurcation analysis, $\zeta =\sum _{k\ne Q}P^2_{k,I}\zeta _k$

$\zeta _k$ :

Damping parameter for linear oscillator at node k in the network model

$\zeta _{\text {HB}}$, $\tilde{\zeta }_{\text {HB}}$, $\mu _{\text {HB}}$:

Damping parameter at Hopf bifurcation

$\zeta _{\text {SN}}$, $\tilde{\zeta }_{\text {SN}}$, $\mu _{\text {SN}}$:

Damping parameter at saddle-node bifurcation

$A(\epsilon t)$ :

Amplitude of $\mathcal {O}(1)$ solution in multiple-scale analysis

$A_k(t)$ :

Amplitude of node k in the network

C(u):

Damping matrix of the network model

f :

Amplitude vector for exogenous excitation

F(t):

Time-dependent exogenous excitation vector

$I_N$ :

$N\times N$ identity matrix

K :

Stiffness matrix of the network model, determined by network topology

L :

The Laplacian matrix of network

P :

An orthogonal matrix whose columns span the eigenspace of the network Laplacian

Q :

The index of the nonlinear oscillator node in the network model

u(t):

Nodal displacement vector in the network model

x :

Modal coordinate vector

Appendix B: Multiple-scales perturbation analysis for the 4-node network with small linear damping

In this appendix, we present a detailed derivation of the slow-flow dynamics in Eqs. (7) and (8) in the limit of $\epsilon \ll 1$ using the method of multiple scales [19]. For the case of constant $\zeta _2,\zeta _3,\zeta _4\sim \mathcal {O}(1)$, we seek to arrive at a zeroth-order (in $\epsilon$) description of the displacement vector u that mirrors the form of the corresponding linear normal-mode, albeit with slowly varying amplitude and phase. To this end, we assume two time scales $t_0=t$ and $t_1=\epsilon t$ and, per Eq. (6), a response of the form

$$\begin{aligned} u&=\frac{A(t_1)}{\sqrt{2}}\begin{pmatrix}1\\ 0\\ 0\\ -1\end{pmatrix}\cos \left( 2t_0+\phi (t_1)\right) +\epsilon v(t_0,t_1) \end{aligned}$$

(61)

and compute derivatives using the relationship

$$\begin{aligned} \frac{d}{dt}=\frac{\partial }{\partial t_0}+\epsilon \frac{\partial }{\partial t_1}. \end{aligned}$$

(62)

Using this notation, we obtain

$$\begin{aligned} \dot{u}_1&=-\sqrt{2}A(t_1)\sin \left( 2t_0+\phi (t_1)\right) \nonumber \\&\qquad +\epsilon \frac{A'(t_1)}{\sqrt{2}}\cos \left( 2t_0+\phi (t_1)\right) \nonumber \\&\qquad -\epsilon \phi '(t_1)\frac{A(t_1)}{\sqrt{2}}\sin \left( 2t_0+\phi (t_1)\right) \nonumber \\&\qquad +\left( \frac{\partial }{\partial t_0}+\epsilon \frac{\partial }{\partial t_1}\right) \epsilon v_1(t_0,t_1) \end{aligned}$$

(63)

and so on. Substitution into the governing equation (1) with $F=0$ results in the perfect cancellation of all $\mathcal {O}(1)$ terms. A similar cancellation of all terms of $\mathcal {O}(\epsilon )$ requires that

$$\begin{aligned} \frac{\partial ^2 v}{\partial t_0^2} + K v = q(t_0,t_1) \end{aligned}$$

(64)

where

$$\begin{aligned} q_1&= \frac{5 A(t_1)^5 }{8 \sqrt{2}}s_1+\frac{15 A(t_1)^5 }{16 \sqrt{2}}s_3 +\frac{5 A(t_1)^5 }{16 \sqrt{2}}s_5\nonumber \\&\qquad -\frac{5 A(t_1)^3 }{2 \sqrt{2}}s_1 -\frac{5 A(t_1)^3 }{2 \sqrt{2}}s_3+2 \sqrt{2} A(t_1) \phi ' c_1\nonumber \\&\qquad -\sqrt{2} A(t_1) s_1+2 \sqrt{2} A'(t_1) s_1 \end{aligned}$$

(65)

$$\begin{aligned} q_2&= q_3=0 \end{aligned}$$

(66)

$$\begin{aligned} q_4&= -\sqrt{2} A(t_1) \zeta _4 s_1-2 \sqrt{2} A(t_1) \phi ' c_1\nonumber \\&\qquad -2 \sqrt{2} A'(t_1) s_1, \end{aligned}$$

(67)

where we use the shorthand $c_k=\cos \left( 2kt_0+k\phi (t_1)\right)$ and $s_k=\sin \left( 2kt_0+k\phi (t_1)\right)$. It follows that

$$\begin{aligned} \frac{\partial ^2 w}{\partial t_0^2}&+ 4w = \frac{5}{16} A^5(t_1) s_1+\frac{15}{32} A^5(t_1) s_3+\frac{5}{32} A^5(t_1) s_5\nonumber \\&\qquad -\frac{5}{4} A^3(t_1) s_1-\frac{5}{4} A^3(t_1) s_3+A(t_1) \zeta _4 s_1\nonumber \\&\qquad +4 A(t_1) \phi '(t_1) c_1-A(t_1) s_1+4 A'(t_1) s_1, \end{aligned}$$

(68)

where $w=(v_1-v_4)/\sqrt{2}$. For a consistent approximation, the secular terms on the right-hand side proportional to $\cos (2t_0+\phi (t_1))$ and $\sin (2t_0+\phi (t_1))$, respectively, must cancel, resulting in the conditions

$$\begin{aligned} 4 A(t_1)\phi '(t_1)=0 \end{aligned}$$

(69)

and

$$\begin{aligned} 4 A'(t_1)-(1-\zeta _4)A(t_1)-\frac{5}{4}A^3(t_1)+\frac{5}{16}A^5(t_1)=0 . \end{aligned}$$

(70)

Appendix C: 4-node network with small linear damping and $Q=2$, 3, and 4, respectively

The existence of a subcritical Hopf bifurcation and an associated branch of limit cycles found in the case of $Q=1$ carries over also to other configurations of the network with the nonlinear oscillator located at node 2, 3, or 4. Indeed, when the nonlinear oscillator is located at node 4 with $\zeta _1,\zeta _2,\zeta _3\sim \mathcal {O}(1)$, the network symmetry results in an identical set of results with $\zeta _4$ replaced by $\zeta _1$. In contrast, if the nonlinear oscillator is located at node 2 with $\zeta _1,\zeta _3,\zeta _4\sim \mathcal {O}(1)$, the complex exponential rates in Eq. (6) become

$$\begin{aligned} \begin{aligned} \lambda _{1,2}&=\pm {j}+\frac{1-\zeta _1-\zeta _3-\zeta _4}{8}\epsilon +\mathcal {O}(\epsilon ^2),\\ \lambda _{3,4}&=\pm \sqrt{2}{j}-\frac{\zeta _1+4\zeta _3+\zeta _4}{12}\epsilon +\mathcal {O}(\epsilon ^2),\\ \lambda _{5,6}&=\pm 2{j}-\frac{\zeta _1+\zeta _4}{4}\epsilon +\mathcal {O}(\epsilon ^2),\\ \lambda _{7,8}&=\pm \sqrt{5}{j}+\frac{9-\zeta _1-\zeta _3-\zeta _4}{24}\epsilon +\mathcal {O}(\epsilon ^2). \end{aligned} \end{aligned}$$

(71)

In this case, with $\zeta :=\zeta _1+\zeta _3+\zeta _4$, $u=0$ is asymptotically stable for $\zeta >9$ and unstable for $\zeta <9$. Substitution of the ansatz

$$\begin{aligned} \begin{aligned}&u_1(t)=\frac{1}{2\sqrt{3}}A(\epsilon t)\cos \left( \sqrt{5}t+\phi (\epsilon t )\right) +\epsilon v_1(t),\\&u_2(t)=-\frac{\sqrt{3}}{2}A(\epsilon t)\cos \left( \sqrt{5}t+\phi (\epsilon t )\right) +\epsilon v_2(t),\\&u_3(t)=\frac{1}{2\sqrt{3}}A(\epsilon t)\cos \left( \sqrt{5}t+\phi (\epsilon t )\right) +\epsilon v_3(t),\\&u_4(t)=\frac{1}{2\sqrt{3}}A(\epsilon t)\cos \left( \sqrt{5}t+\phi (\epsilon t )\right) +\epsilon v_4(t) \end{aligned} \end{aligned}$$

(72)

into the fully nonlinear governing equations then yields the differential equation

$$\begin{aligned} A'=\frac{1}{24}(9-\zeta )A+\frac{45}{64}A^3-\frac{135}{512}A^5 \end{aligned}$$

(73)

governing the slow dynamics of the amplitude A with nonzero equilibria obtained from

$$\begin{aligned} A^2=\frac{4}{3}\pm \frac{4}{9}\sqrt{\frac{81-4\zeta }{5}}, \end{aligned}$$

(74)

i.e., for $\zeta \in [0,81/4]$ with two co-existing solutions on the interval $\zeta \in [9,81/4)$. We obtain the equivalent of Eq. (10), for example, by letting

$$\begin{aligned} \zeta _{1,3,4}'=\tau ^{-1}\left( -\zeta _{1,3,4}+\frac{1}{3}\delta +3+\frac{45}{16}A^2\right) , \end{aligned}$$

(75)

since these imply that

$$\begin{aligned} \zeta '=\tau ^{-1}\left( -\zeta +\delta +9+\frac{135}{16}A^2\right) . \end{aligned}$$

(76)

Finally, with the nonlinear oscillator located at node 3 with $\zeta _1,\zeta _2,\zeta _4\sim \mathcal {O}(1)$, the complex exponential rates in Eq. (6) become

$$\begin{aligned} \begin{aligned} \lambda _{1,2}&=\pm {j}+\frac{1-\zeta _1-\zeta _2-\zeta _4}{8}\epsilon +\mathcal {O}(\epsilon ^2),\\ \lambda _{3,4}&=\pm \sqrt{2}{j}+\frac{4-\zeta _1-\zeta _4}{12}\epsilon +\mathcal {O}(\epsilon ^2),\\ \lambda _{5,6}&=\pm 2{j}-\frac{\zeta _1+\zeta _4}{4}\epsilon +\mathcal {O}(\epsilon ^2),\\ \lambda _{7,8}&=\pm \sqrt{5}{j}+\frac{1-\zeta _1-9\zeta _2-\zeta _4}{24}\epsilon +\mathcal {O}(\epsilon ^2). \end{aligned} \end{aligned}$$

(77)

In this case, with $\zeta :=\zeta _1+\zeta _4$, $u=0$ is asymptotically stable for $\zeta >4$ and unstable for $\zeta <4$. Substitution of the ansatz

$$\begin{aligned} \begin{aligned}&u_1(t)=\frac{1}{\sqrt{6}}A(\epsilon t)\cos \left( \sqrt{2}t+\phi (\epsilon t )\right) +\epsilon v_1(t),\\&u_2(t)=\epsilon v_2(t),\\&u_3(t)=-\sqrt{\frac{2}{3}}A(\epsilon t)\cos \left( \sqrt{2}t+\phi (\epsilon t )\right) +\epsilon v_3(t),\\&u_4(t)=\frac{1}{\sqrt{6}}A(\epsilon t)\cos \left( \sqrt{2}t+\phi (\epsilon t )\right) +\epsilon v_4(t) \end{aligned} \end{aligned}$$

(78)

into the fully nonlinear governing equations then yields the differential equation

$$\begin{aligned} A'=\frac{1}{12}(4-\zeta )A+\frac{5}{9}A^3-\frac{5}{27}A^5 \end{aligned}$$

(79)

governing the slow dynamics of the amplitude A with nonzero equilibria obtained from

$$\begin{aligned} A^2=\frac{3}{2}\pm \frac{3}{2}\sqrt{\frac{9-\zeta }{5}}, \end{aligned}$$

(80)

i.e., for $\zeta \in [0,9]$ with two co-existing solutions on the interval $\zeta \in [4,9)$. We obtain the equivalent to Eq. (10), for example, by letting

$$\begin{aligned} \zeta _{1,4}'=\tau ^{-1}\left( -\zeta _{1,4}+\frac{1}{2}\delta +2+\frac{5}{3}A^2\right) , \end{aligned}$$

(81)

since these imply that

$$\begin{aligned} \zeta '=\tau ^{-1}\left( -\zeta +\delta +4+\frac{10}{3}A^2\right) . \end{aligned}$$

(82)

Appendix D: 4-node network with large linear damping and $Q=2$, 3, and 4, respectively

When the nonlinear oscillator is located at node 4 with $\zeta _1,\zeta _2,\zeta _3\sim \mathcal {O}(1/\epsilon ^2)$, the analysis from Sect. 2.2 still applies, albeit with $\zeta _4$ and $\tilde{\zeta }_4$ replaced by $\zeta _1$ and $\tilde{\zeta }_1$, respectively. If, instead, the nonlinear oscillator is located at node 2 with $\zeta _1,\zeta _3,\zeta _4\sim \mathcal {O}(1/\epsilon ^2)$, we obtain the complex exponential rates

$$\begin{aligned} \begin{aligned}&\lambda _{1,2,3}\approx -\epsilon \zeta _{1,3,4}+\mathcal {O}(1),\\&\lambda _{4,5,6}\lesssim 0, \\&\lambda _{7,8}=\pm 2{j}+\frac{1}{8}\left( 4-\tilde{\zeta }_1-\tilde{\zeta }_3-\tilde{\zeta }_4\right) \epsilon +\mathcal {O}(\epsilon ^2), \end{aligned} \end{aligned}$$

(83)

where $\tilde{\zeta }_1:=1/\epsilon ^2\zeta _1$, $\tilde{\zeta }_3:=1/\epsilon ^2\zeta _3$, and $\tilde{\zeta }_4:=1/\epsilon ^2\zeta _4$. With $\tilde{\zeta }:=\tilde{\zeta }_1+\tilde{\zeta }_3+\tilde{\zeta }_4$, it follows that the trivial equilibrium is asymptotically stable for $\tilde{\zeta }>4$ and unstable for $\tilde{\zeta }<4$, with $\tilde{\zeta }=4$ corresponding to a Hopf bifurcation out of which emanates a branch of periodic orbits approximated in the small-amplitude limit by the normal-mode oscillations $u_2=A\cos (2t+\phi )$, $u_1(t)=u_3(t)=u_4(t)=0$ for constant amplitude A and phase $\phi$. A consistent multiple-scale ansatz now yields the amplitude equation

$$\begin{aligned} A'=\frac{1}{8}\left( 4-\tilde{\zeta }\right) A+\frac{5}{4}A^3-\frac{5}{8}A^5 \end{aligned}$$

(84)

with nontrivial equilibria at

$$\begin{aligned} A^2=1\pm \sqrt{\frac{9-\tilde{\zeta }}{5}}, \end{aligned}$$

(85)

i.e., for $\tilde{\zeta }\in (0,9]$ with two co-existing solutions on the interval $\tilde{\zeta }\in [4,9)$. We obtain the equivalent of Eq. (10), for example, by letting

$$\begin{aligned} \tilde{\zeta }'_{1,3,4}=\tau ^{-1}\left( -\tilde{\zeta }_{1,3,4}+\frac{1}{3}\delta +\frac{4}{3}+\frac{5}{3}A^2\right) , \end{aligned}$$

(86)

since these imply that

$$\begin{aligned} \tilde{\zeta }'=\tau ^{-1}\left( -\tilde{\zeta } + \delta + 4 + 5 A^2\right) . \end{aligned}$$

(87)

Finally, when the nonlinear oscillator is located at node 3 with $\zeta _1,\zeta _2,\zeta _4\sim \mathcal {O}(1/\epsilon ^2)$, we obtain the complex exponential rates

$$\begin{aligned} \begin{aligned}&\lambda _{1,2,3}\approx -\epsilon \zeta _{1,2,4}+\mathcal {O}(1),\\&\lambda _{4,5,6}\lesssim 0, \\&\lambda _{7,8}=\pm {j}\sqrt{2}+\frac{1}{4}\left( 2-\tilde{\zeta }_2\right) \epsilon +\mathcal {O}(\epsilon ^2), \end{aligned} \end{aligned}$$

(88)

where $\tilde{\zeta }_2:=1/\epsilon ^2\zeta _2$. It follows that the trivial equilibrium is asymptotically stable for $\tilde{\zeta }_2>2$ and unstable for $\tilde{\zeta }_2<2$, with $\tilde{\zeta }_2=2$ corresponding to a Hopf bifurcation out of which emanates a branch of periodic orbits approximated in the small-amplitude limit by the normal-mode oscillations $u_3=A\cos (\sqrt{2}t+\phi )$, $u_1(t)=u_2(t)=u_4(t)=0$ for constant amplitude A and phase $\phi$. A consistent multiple-scale ansatz now yields the amplitude equation

$$\begin{aligned} A'=\frac{1}{4}\left( 2-\tilde{\zeta }_2\right) A+\frac{5}{4}A^3-\frac{5}{8}A^5 \end{aligned}$$

(89)

with nontrivial equilibria at

$$\begin{aligned} A^2=1\pm \sqrt{\frac{9-2\tilde{\zeta }_2}{5}}, \end{aligned}$$

(90)

i.e., for $\tilde{\zeta }_2\in (0,9/2]$ with two co-existing solutions on the interval $\tilde{\zeta }_2\in [2,9/2)$. We obtain the equivalent of Eq. (10), for example, by letting

$$\begin{aligned} \tilde{\zeta }'_{2}=\tau ^{-1}\left( -\tilde{\zeta }_{2}+\delta +2+\frac{5}{2}A^2\right) . \end{aligned}$$

(91)