Multi-dimensional Feedback Particle Filter for Coupled Oscillators

Adam K. Tilton, Prashant G. Mehta and Sean P. Meyn

Abstract— This paper presents a methodology for state es-timation of coupled oscillators from noisy observations. Themethodology is comprised of two parts: modeling and estima-tion. The objective of the modeling is to express dynamics interms of the so-called phase variables. For nonlinear estimation,a coupled-oscillator feedback particle filter is introduced.

The filter is based on the construction of a large population ofoscillators with mean-field coupling. The empirical distributionof the population encodes the posterior distribution of the phasevariables. The methodology is illustrated with two numericalexamples.

I. INTRODUCTIONThis paper is concerned with the problem of estimat-

ing phases of coupled oscillators from noisy observationdata. The motivation comes from biology where rhythmicityunderlies several important behaviors – e.g., periodic gaitsexhibited by animals during locomotion (walking, hopping,running etc); cf., [5]. The importance of the phase estimationproblem is discussed in [13] and in several references therein.An important motivation for us comes from the problemof controlling rhythmic behaviors – e.g., by using feedbackcontrol to stabilize walking gaits [4]. For such applications,estimation is seen as an intermediate step towards control.

The methodology proposed here comprises of two parts:1. Modeling: The objective of the modeling is to obtain areduced order model of the dynamics. The reduced ordermodel comprises of weakly coupled oscillators, expressed intheir phase variables; cf., [7], [8], [1].

As an example, consider a dynamical system with twoweakly coupled oscillators: The reduced order model hastwo phase variables, θ1(t) ∈ [0,2π) and θ2(t) ∈ [0,2π),representing the instantaneous phase of the two oscillatorsat time t. The dynamics evolve according to,

dθ1(t) = ω1 dt + εI12(θ1(t),θ2(t))dt, (1a)dθ2(t) = ω2 dt + εI21(θ1(t),θ2(t))dt, (1b)

where ω1,ω2 are the frequency parameters and I12(·), I21(·)are the interaction functions.2. Estimation: The objective of the estimation problem isto construct a filter for estimating the instantaneous phase((θ1(t),θ2(t)) in the example above) given a time-history ofnoisy observations. A Bayesian procedure is described forsynthesis of the coupled oscillator feedback particle filter.

Financial support from the AFOSR grant FA9550-09-1-0190 and the NSFCPS grant 0931416 is gratefully acknowledged.

A. K. Tilton and P. G. Mehta are both with the Coordinated Sci-ence Laboratory and the Department of Mechanical Science and En-gineering at the University of Illinois at Urbana-Champaign (UIUC)atilton2@illinois.edu; mehtapg@illinois.edu

S. P. Meyn is with the Department of Electrical and Computer Engineer-ing at the University of Florida meyn@ece.ufl.edu.

The filter comprises of a large population of oscillatorscoupled through a mean-field coupling. The empirical distri-bution of the population encodes the posterior distribution ofthe phase variables. The distribution is continuously updatedbased on noisy observations. Our approach is influenced bythe conceptual framework in Lee and Mumford’s work [12],who proposed a Bayesian inference framework for the visualcortex based on particle filtering.

For the model described in (1a, 1b) the coupled oscil-lator particle filter is comprised of N stochastic processes{(θ i1(t),θ i2(t)) : 1≤ i≤ N}, where the value (θ i1(t),θ i2(t)) ∈[0,2π)× [0,2π) is the state of the ith particle at time t. Thedynamics evolve according to,

dθ i1(t) = ω1 dt + εI12(θ i1(t),θ

i2(t))dt + dU

i1(t),

dθ i2(t) = ω2 dt + εI21(θ i1(t),θ

i2(t))dt + dU

i2(t),

where (U i1(t),Ui2(t)) is the control input. It is chosen such

that the empirical distribution of the population (for large N)approximates the posterior distribution of the phase variables(θ1(t),θ2(t)). A constructive design procedure for controlsynthesis is described, based on the feedback particle filtermethodology [18], [17].

The two-state model is used here for illustrative purposes;the methodology is applicable to models with arbitrarilymany phase variables.

The contributions of this paper are summarized as follows:The paper is amongst the first to explicitly use phase modelsto construct nonlinear estimators. This is important becausea single oscillator is a model of the repetitive firing of asingle neuron; cf., [7]. A coupled oscillator particle filtercan thus be realized as a neuronal network. The activityof the network is encoded in the phase variables – e.g.,{θ i1(t)}Ni=1 and {θ i2(t)}Ni=1 encode the posterior distributionof the (hidden) phase variables θ1(t) and θ2(t), respectively.

Apart from this paper, the problem of estimating phasevariables appears in our own prior work [15], which appliesthe feedback particle filter to estimating a single phasevariable associated with a walking gait cycle. In [13], acomputational method, Phaser, is described based on takinga Hilbert transform and applying a Fourier series basedcorrection (see also [6]). Dynamical systems techniques forrecovering phase via delay coordinate embedding appearin [14]. In [10], an approach for invariant reconstruction ofphase equations from measurement data is discussed.

The outline of the remainder of this paper is as follows.The problem statement appears in Sec. II. The methodologyis comprised of two parts, modeling and estimation, whichare discussed in Sec. III-Sec. IV, respectively. Two illustra-tive numerical examples are described in Sec. V.

2013 American Control Conference (ACC)Washington, DC, USA, June 17-19, 2013

978-1-4799-0176-0/$31.00 ©2013 AACC 2421

II. PROBLEM STATEMENT

A. Preliminaries and Notation

The dynamic model describes r weakly coupled oscillatorsystems. The state of the kth-system is denoted as Xk(t)∈Rd .The uncoupled dynamic model of the kth-system is given by,

dXk(t) = ãk(Xk(t))dt, (3)

where ãk is a C1 function. The system is assumed to have anisolated asymptotically stable periodic orbit (limit cycle) withperiod 2π/ωk. Denote the limit cycle solution as XkLC(θk(t))where θk(t) = θk(0)+ωkt mod 2π . Denote the set of pointson the limit cycle as Pk.

By construction, the map XkLC : [0,2π) → Pk ⊂ Rd isinvertible and denote the inverse map as φ̃ k. The inversemap φ̃ k : Pk→ [0,2π) is defined as

φ̃ k(x) = θ iff XkLC(θ) = x,

for x ∈Pk, θ ∈ [0,2π).The inverse map is extended to a small neighborhood N k

of the limit cycle Pk by using the notion of asymptoticphase: Let Φk : N k×R→N k be the flow map so Φk(x0, t)is the solution of (3) from initial condition x0 ∈N k. Theasymptotic phase of the point x0 ∈N k is defined as the phaseof the point y0 ∈Pk such that ‖Φk(x0, t)−Φk(y0, t)‖ → 0as t → ∞. Denote the extended map as φ k where note φ k :N k→ [0,2π) and φ k = φ̃ k on the periodic orbit Pk.

The instantaneous phase is defined in terms of the ex-tended map:

θk(t) = φ k(Xk(t)), for Xk(t) ∈N k. (4)

The set of points with same phase are called isochrons.The phase model reduction is standard; cf., [1] and refer-

ences therein.

Example 1: Consider the Hopf oscillator model. The stateX(t) = (x(t),y(t)) ∈ R2 and the dynamics are given by,

dX(t) = ωJX(t)dt + γ(1− r2)X(t)dt, (5)

where J =[

0 −11 0

], ω = 1, γ = 0.1 and r2 = (x(t))2 +

(y(t))2.The system has a limit cycle, XLC(θ) = (cos(θ),sin(θ)),

where θ = ωt mod 2π . The set P = {(x,y) ∈ R2 : x2 +y2 = 1}. For points (x,y) ∈N , a small neighborhood of P ,the phase φ(x,y) = arctan(y/x), i.e., the isochrons are raysoriginating at origin. Using (4), the instantaneous phase isgiven by

θ(t) = φ(X(t)) = arctan(

y(t)x(t)

), for (x(t),y(t)) ∈N .

B. Problem Statement

For the coupled system, the state is denoted as X(t) :=(X1(t),X2(t), . . . ,Xr(t)). The dynamical system is given by,for k = 1,2, . . . ,r:

dXk(t) = ãk(Xk(t))dt + εr

∑l=1

b̃kl(Xk(t),Xl(t))dt, (6)

where ãk, b̃kl are C1 functions. The coupling parameter ε isassumed to be small enough such that Xk(t) ∈N k, a smallneighborhood of the limit cycle Pk of the uncoupled system,for all k = 1,2, . . . ,r and t ≥ 0.

The observation model is given by,

dZ(t) = h̃(X(t))dt +σW dW (t),

where Z(t) ∈ Rm is the observation process, {W (t)} is aWiener process, σW ∈R is the standard deviation parameter,and h̃( ·) is a C1 function; h̃ is a column vector whose j-th coordinate is denoted as h̃ j (i.e., h̃ = (h̃1, h̃2, . . . , h̃m)T ).By scaling, we assume without loss of generality that thecovariance matrix associated with {W (t)} is an identitymatrix.

The objective of the filtering problem is to estimate theposterior distribution of θ(t) := (θ1(t),θ2(t), . . . ,θr(t)) giventhe filtration (time history of observations) Zt := σ(Zs :s ≤ t). Note that the phase θk(t) is well-defined by (4) forXk(t) ∈N k . The posterior is denoted by p∗, so that for anymeasurable set A⊂ [0,2π)r,∫

ϑ∈Ap∗(ϑ , t) dϑ = P{θ(t) ∈ A |Zt}.

The filter is infinite-dimensional since it defines the evolu-tion, in the space of probability measures, of {p∗( · , t) : t ≥0}.

Example 2: Consider the coupled Hopf oscillator modelof the form (6) with r = 2. The state X(t) = (X1(t),X2(t))is four dimensional, with Xk(t) = (xk(t),yk(t)) for k = 1,2.The dynamics of the coupled model is given by,

dX1(t) =(ωJX1(t)+ γ(1− r21)X1(t)+ ε(X1(t)−X2(t)))

)dt,

(7a)

dX2(t) =(ωJX2(t)+ γ(1− r22)X2(t)+ ε(X2(t)−X1(t))

)dt,(7b)

where r2k = (xk(t))2 +(yk(t))2 for k = 1,2, and other param-

eters are as in Example 1.The observation process is two-dimensional Z(t) =

(Z1(t),Z2(t)), and the observation model is given by,

dZ1(t) = x1(t)dt +σW dW1(t), (8a)dZ2(t) = x2(t)dt +σW dW2(t), (8b)

where W1,W2 are mutually independent standard Wienerprocesses, and σW is the standard deviation parameter.

We revisit this example in Sec. V-B.

2422

III. MODELING: PHASE MODELS

A. Signal model

For the uncoupled limit cycling system (3), the state ofthe kth oscillator is θk(t) ∈ [0,2π), and the dynamics evolveaccording to,

dθk(t) = ωk mod 2π.

The solution of (3) is expressed as Xk(t) = XkLC(θk(t)).The modeling technique that is used in this paper is based

on interpreting the original weakly coupled system (6), asa perturbation of this ideal uncoupled system. It can beshown that the approximation Xk(t) = XkLC(θk(t)) + O(ε)holds for (6). That is, to O(ε), the effect of weak couplingis manifested through changes in the phase variable only.

The state of the reduced order model is θ(t) =(θ1(t),θ2(t), . . . ,θr(t))∈ [0,2π)r, and the dynamics (to O(ε))evolve according to,

dθk(t) = ωk + εr

∑l=1

Ikl(θk(t),θl(t)) mod 2π, (9)

for k = 1,2 . . . ,r and t ≥ 0. The interaction terms Ikl(θk,θl)are obtained from the corresponding terms bkl(Xk,Xl) in (6).An algorithm for the same, based on a standard singularperturbation approach (see [1]), appears in Appendix VII.

Example 3: For the single-Hopf model (5) in Example 1,the state is θ(t) ∈ [0,2π) and the dynamics are given by,

dθ(t) = 1dt mod 2π.

In terms of θ(t), the (limit cycle) solution of (5) is given byX(t) = XLC(θ(t)) = (cos(θ(t)),sin(θ(t))).

For the coupled model (7a, 7b) in Example 2, the stateθ(t) = (θ1(t),θ2(t)) ∈ [0,2π)2 and the dynamics are givenby,

dθ1(t) = 1+ ε sin(θ1(t)−θ2(t)) mod 2π,dθ2(t) = 1+ ε sin(θ2(t)−θ1(t)) mod 2π.

This reduced order phase model is derived as part of theExample 5 in Appendix VII. In terms of θ(t), the (limitcycle) solution of the coupled full-order model (7a, 7b) isgiven by,

X1(t) = X1LC(θ1(t))+O(ε) = (cos(θ1(t)),sin(θ1(t)))+O(ε),X2(t) = X2LC(θ2(t))+O(ε) = (cos(θ2(t)),sin(θ2(t)))+O(ε).

B. Observation model

On the limit cycle, Xk(t) = XkLC(θk(t)) +O(ε). We thusdefine an observation function on the limit cycle as,

h(θ) := h̃(XLC(θ)),

for θ = (θ1,θ2, . . . ,θr) ∈ [0,2π)r where XLC(θ) :=(X1LC(θ1),X

2LC(θ2), . . . ,X

rLC(θr)).

Example 4: Consider the observation model (8a, 8b)introduced in Example 2: the observation function

h̃((x1,y1),(x2,y2)) =[

x1x2

]. So,

h(θ1,θ2) = h̃(X1LC(θ1),X2LC(θ2)) =

[cos(θ1)cos(θ2)

].

IV. ESTIMATION: FEEDBACK PARTICLE FILTERConsider the multivariable filtering problem:

dθ(t) = a(θ(t))dt + dB(t), mod 2π, (10a)dZ(t) = h(θ(t))dt + dW (t), (10b)

where θ(t) = (θ1(t),θ2(t), . . . ,θr(t))∈ [0,2π)r is the state attime t, Z(t) ∈ Rm is the observation process, a( ·), h( ·) are2π-periodic C1 functions, and {B(t)}, {W (t)} are mutuallyindependent Wiener processes of appropriate dimension. Byscaling, we assume without loss of generality that the covari-ance matrices associated with {B(t)}, {W (t)} are identitymatrices.

Recall here that our objective is to obtain the posteriordistribution p∗ of θ(t) given the filtration Zt .

Remark 1: The signal model (10a) is allowed to be moregeneral than (9) primarily to simplify the notation. Note,however, that the formulation allows one the option toinclude process noise in oscillator models.

A. Feedback Particle FilterIn our prior work [18], [17], [16], a feedback control-based

approach to the solution of the nonlinear filtering problemwas introduced. The resulting algorithm is referred to as thefeedback particle filter.

The feedback particle filter is a controlled system. Thestate of the filter is {θ i(t) : 1 ≤ i ≤ N}: The value θ i(t) =(θ i1(t),θ

i2(t), . . . ,θ

ir(t)) ∈ [0,2π)r is the state for the ith

particle at time t. Note that the ith particle comprises ofr-oscillators. The dynamics of the ith particle have thefollowing gain feedback form, expressed in its Stratonovichform,

dθ i(t) = a(θ i(t))dt + dBi(t)+K(θ i(t), t)◦ dIi(t), mod2π,

where {Bi(t)} are mutually independent standard Wienerprocesses, and Ii is similar to the innovation process thatappears in the nonlinear filter,

dIi(t)=:dZ(t)− 12(h(θ i(t))+ ĥ(t))dt,

where ĥ(t) :=E[h(θ i(t))|Zt ]. In a numerical implementation,we approximate ĥ≈ N−1 ∑Ni=1 h(θ i(t)) =: ĥ(N).

The gain function K is obtained as a solution to anEuler-Lagrange boundary value problem (E-L BVP): Forj = 1,2, . . . ,m, the function φ j is a solution to the second-order differential equation,

∇ · (p(θ , t)∇φ j(θ , t)) =−(h j(θ)− ĥ j)p(θ , t),∫[0,2π)r

φ j(θ , t)p(θ , t)dθ = 0,(11)

2423

where p denotes the conditional distribution of θ i(t) givenZt . In terms of these solutions, the gain function is givenby,

[K]l j(θ , t) =∂φ j∂θl

(θ , t) .

Note that the gain function needs to be obtained for eachvalue of time t.

It is shown in [17] that the feedback particle filter isconsistent with the nonlinear filter, given consistent ini-tializations p( · ,0) = p∗( · ,0). Consequently, if the initialconditions {θ i(0)}Ni=1 are drawn from the initial distributionp∗( · ,0) of θ(0), then, as N→ ∞, the empirical distributionof the particle system approximates the posterior distributionp∗( · , t) for each t (see [16], [17] for details).

Notation: L2([0,2π)r; p) is used to denote the Hilbertspace of 2π-periodic functions on [0,2π)r that are square-integrable with respect to density p(·, t); Hk([0,2π)r; p) isused to denote the Hilbert space of functions whose first k-derivatives (defined in the weak sense) are in L2([0,2π)r; p).Denote

H10 ([0,2π)r; p) :=

{φ ∈ H1

∣∣∣ ∫ φ(ϑ)p(ϑ , t)dϑ = 0} .A function φ j ∈ H10 ([0,2π)r; p) is said to be a weak

solution of the BVP (11) if∫∇φ j(ϑ , t) ·∇ψ(ϑ)p(ϑ , t)dϑ

=∫(h j(ϑ)− ĥ j)ψ(ϑ)p(ϑ , t)dϑ ,

for all ψ ∈ H1([0,2π)r; p).Denoting E[·] := ∫ ·p(ϑ , t)dϑ , the weak form of the

BVP (11) can also be expressed as follows:

E[∇φ j ·∇ψ] = E[(h j− ĥ j)ψ], ∀ ψ ∈ H1([0,2π)r; p). (12)

This representation is useful for the numerical algorithmdescribed next.

B. Synthesis of Gain Function

In this section, a Galerkin algorithm is described to obtainan approximate solution of (11). Since there are m uncoupledBVPs, without loss of generality, we assume scalar-valuedobservation in this section, with m = 1, so that K = ∇φ .The time t is fixed. The explicit dependence on time issuppressed for notational ease (That is, p(θ , t) is denotedas p(θ), φ(θ , t) as φ(θ) etc.).

The function φ is approximated as,

φ(θ) =L

∑l=1

κlψl(θ),

where {ψl(θ)}Ll=1 are the Fourier basis functions on [0,2π)r.The gain function K is given by,

K(θ) = ∇φ(θ) =L

∑l=1

κl∇ψl(θ).

The finite-dimensional approximation of (12) is to chooseconstants {κl}Ll=1 such that

L

∑l=1

κlE[∇ψl ·∇ψ] = E[(h− ĥ)ψ], ∀ ψ ∈ S, (13)

where S := span{ψ1,ψ2, . . . ,ψL} ⊂ H1([0,2π)r; p).Denoting [A]kl = E[∇ψl ·∇ψk], bk = E[(h− ĥ)ψk], κ =

(κ1,κ2, . . . ,κL)T , the finite-dimensional approximation (13)is expressed as a linear matrix equation:

Aκ = b. (14)

The matrix A and vector b are approximated by using onlythe particles:

[A]kl = E[∇ψl ·∇ψk]≈1N

N

∑i=1

∇ψl(θ i(t)) ·∇ψk(θ i(t)),

bk = E[(h− ĥ)ψk]≈1N

N

∑i=1

(h(θ i(t))− ĥ)ψk(θ i(t)),

where recall ĥ≈ N−1 ∑Ni=1 h(θ i(t)).The important point to note is that the gain function is

expressed in terms of averages taken over the population.

V. EXAMPLES

A. Single Hopf oscillator

Fig. 1. Observations over one cycle for example 1.

Consider the filtering problem for the single-Hopf model(see Example 1):

dX(t) = ωJX(t)dt + γ(1− r2)X(t),dZ(t) = h̃(X(t))dt +σW dW (t),

where X(t) = (x(t),y(t))∈R2, r2 = x2(t)+y2(t), h̃(x,y) = x,and parameters ω = 1, γ = 0.1 and σW = 0.1. Fig. 1 depictsthe trajectory x(t) and noisy observations Y (t). At eachdiscrete time, these are obtained as Y (t) := ∆Zt/∆t, where∆t denotes the discrete time step, and ∆Zt = Z(t+∆t)−Z(t).In numerical simulations, ∆t = 0.01 is chosen.

Recall that the limit cycle solution is given by X(t) =XLC(θ(t)) = (cos(θ(t)),sin(θ(t))), where θ(t) = θ(0)+ωtmod 2π . The objective is to approximate the posterior of theinstantaneous phase θ(t) given Zt .

2424

t0 t132

t

0

π2

π

3π2

2π

θθi(t) ±2σ True State: θ Estimate: θ̂

(a)

t1 t2 t3 t4t

0

π2

π

3π2

2π

θ

±2σ Estimate: θ̂ True State: θ

(b)

0.0 0.5 1.0 1.5 2.0

p(θ)

0

π2

π

3π2

2π

θ

p(θ, t2) p(θ, t3)

(c)

Fig. 2. Summary of numerical results for example 1: (a) Initial transients as particles converge starting from a uniform initial distribution; (b) Comparisonof estimated mean {θ̂(t)(N)} with the true state {θ(t)}, where the shaded region show ± two standard deviations; and (c) Representative empiricaldistributions at two times, t2 and t3.

The feedback particle filter is given by,

dθ i(t) = ωi dt +σB dBi(t)

+K(θ i(t), t)◦(

dZ(t)− h(θi(t))+ ĥ

2

), mod 2π,

(16)where h(θ i(t)) = cos(θ i(t)) and ĥ = N−1 ∑Nj=1 cos(θ j(t)).For the exact filter, ωi = ω and σB = 0. In numericalsimulations, we include a small noise (σB > 0) and pick ωifrom a uniform distribution on [ω−δ ,ω +δ ]. Such a modelis more realistic from an implementation standpoint, eventhough it is not exact.

The gain function K(θ , t) = ∂φ∂θ (θ , t) is obtained via thesolution of the E-L equation:

∂∂θ

(p(θ , t)∂φ∂θ

(θ , t)) =−(h(θ)− ĥ)p(θ , t).

An approximate solution is obtained by using the Galerkinmethod described in Sec. IV-B with basis functions{cos(θ),sin(θ)}. In terms of basis functions,

K(θ , t) =−κ1(t)sin(θ)+κ2(t)cos(θ),where κ(t) := [κ1(t),κ2(t)] are obtained at each time bysolving the linear matrix equation (see (14)):[

1N ∑ j sin

2(θ j(t)) − 1N ∑ j sin(θ j(t))cos(θ j(t))− 1N ∑ j sin(θ j(t))cos(θ j(t)) 1N ∑ j cos2(θ j(t))

][κ1(t)κ2(t)

]=[

1N ∑ j(h(θ

j(t))− ĥ)cos(θ j(t))1N ∑ j(h(θ

j(t))− ĥ)sin(θ j(t))

]. (17)

We next discuss the result of numerical experiments. Theparticle filter model is given by (16) with gain functionK(θ i(t), t), obtained using solution of (17). The number ofparticles N = 1,000 and their initial condition {θ i(0)}Ni=1 wassampled from a uniform distribution on circle [0,2π). Thenoise parameter σB = 0.1 and the frequencies ωi are sampledfrom a uniform distribution on [1−δ ,1+δ ] with δ = 0.5.

Fig. 2 summarizes some of the results of the numericalsimulation. The initial transients due to the initial conditionconverge rapidly – as shown in part (a). A single cycle froma time-window after transients have converged is depicted inpart (b): the figure compares the sample path of the actualstate θ(t) (as a solid line) with the estimated mean θ̂ (N)(t)

(as a dashed line). The shaded area indicates ± two standarddeviation bounds. The density at two time instants is depictedin part (c). These results show that the filter is able to trackthe phase accurately.

Note that at each time instant t, the estimated mean,the bounds and the density p(θ , t) shown here are allapproximated from the ensemble {θ i(t)}Ni=1.

B. Coupled Hopf oscillator

Consider next the filtering problem for the coupled Hopfmodel. The problem is described in Example 2: The signalmodel is the 4-dimensional coupled Hopf model (7a, 7b) andthe observation model (8a, 8b) is two-dimensional.

Starting from an arbitrary initial condition, one asymp-totically obtains an anti-phase limit cycle solution whereXk(t)≈ XLC(θk(t)) = (cos(θk(t)),sin(θk(t))) for k = 1,2 andθ1(t) and θ2(t) are out of phase. Fig. 3 (a) depicts thissolution for ε = 0.05.

The objective is to approximate the posterior of the phaseθ1(t),θ2(t) given Zt .

The feedback particle filter is given by,

dθ 1i (t) = ω1,i dt + ε sin(θ11 (t)−θ i2(t))dt +σB dBi1(t)

+K1(θ i(t), t)◦(

dZ(t)− h(θi(t))+ ĥ(t)

2

), mod 2π,

dθ i2(t) = ω2,i dt + ε sin(θ12 (t)−θ i1(t))dt +σB dBi2(t)

+K2(θ i(t), t)◦(

dZ(t)− h(θi(t))+ ĥ(t)

2

), mod 2π,

where θ i(t) = (θ i1(t),θi2(t)) ∈ [0,2π)2,

h(θ i(t)) =[

h1(θ i1(t))h2(θ i2(t))

]=

[cos(θ i1(t))cos(θ i2(t))

],

and ĥ(t) = N−1 ∑Nj=1 cos(θ j(t)). As in the preceding sec-tion, we include small noise (σB > 0) and pick frequenciesω1,i,ω2,i from a uniform distribution on [ω−δ ,ω +δ ].

The gain function,[K1(θ1,θ2, t)K2(θ1,θ2, t)

]=

[ ∂φ1∂θ1

(θ1,θ2, t) ∂φ2∂θ1 (θ1,θ2, t)∂φ1∂θ2

(θ1,θ2, t) ∂φ2∂θ2 (θ1,θ2, t)

],

2425

t1 t2 t3 t4t

− 2

− 1

0

1

2x

(a)

0 π2

π 3π2

2π

θ1

0

π2

π

3π2

2π

θ 2

True State: θ Estimate:θ̂

(b)

0 π2

π 3π2

2π

θ1

0

π2

π

3π2

π

θ 2p(θ

1)

p(θ2 )

-

-

(c)

Fig. 3. Summary of numerical results for example 2: (a) Convergence of trajectories to the anti-phase limit cycle solution; (b) Comparison of (θ1(t),θ2(t))with the estimated mean (θ̂ (N)1 (t), θ̂

(N)2 (t)); and (c) Representative empirical distribution at a fixed time instant (where the true state θ1 = π and θ2 = 0).

where each column is obtained by solving the E-L equation:

∇ · (p(θ , t)∇φ j(θ , t)) =−(h j(θ)− ĥ j)p(θ , t), for j = 1,2.

An approximate solution is obtained by using the Galerkinmethod described in Sec. IV-B with basis functions{sin(θ1),cos(θ1),sin(θ2),cos(θ2)}. This leads to a 4× 4linear matrix problem (14), whose solution is obtained nu-merically. As in the preceding section, the calculation of thegain function requires averages taken over the population.

We next discuss the result of numerical experiments. Thenumber of particles N = 1,000 and their initial condition{θ i1(0)}Ni=1 and {θ i2(0)}Ni=1 was sampled from a uniformdistribution on the torus [0,2π)2. The noise parameter σB =0.1 and frequencies are sampled uniformly from [0.5,1.5].

Fig. 3 (b) compares the sample path of the actual state(θ1(t),θ2(t)) (as a solid line) with the estimated mean(θ̂ (N)1 (t), θ̂

(N)2 (t)) (as a dashed line). The standard deviation

bounds are tight: the density at a single time instant isdepicted in part (c). These results show that the filter is ableto track the phase variables on the torus accurately.

VI. CONCLUSIONS

The application of singular perturbation techniques to ob-tain a reduced order model provides an approach to nonlinearfilter design for a coupled oscillator model. This techniqueis general, and is expected to find many applications.

The importance of coupled oscillators as models of bothmovement in animals (periodic gaits), and of central patterngenerators in nervous system provided the impetus to thiswork; cf. [11], [3], [9], [2]. The filtering viewpoint espousedin this paper is useful because it connects naturally tofeedback control of periodic behavior. This is a subject ofongoing work.

In addition, we are currently applying these techniques tophase estimation in the power grid, which is of significantinterest in industry. Reliable power system operations requirethe balance of supply and demand on a second-by-secondbasis: The frequency must be kept at 60Hz, and consistencyof the phase of generation at neighboring nodes in the gridmust be maintained. Control requires state estimation ofprecisely the kind described in this paper.

VII. APPENDIX: MODEL REDUCTION

In this section, an algorithm is described to obtain thereduced order phase model (9) starting from the weaklycoupled dynamical system model (6). We use the notationof Sec. II-A.

For the kth-system (3), the phase response curve (PRC),denoted as Pk(θ), is defined to be the 2π-periodic solutionof the adjoint equation:

ωkdPkdθ

(θ)+(Dak(XkLC(θ)))T Pk(θ) = 0, for θ ∈ [0,2π),

where Dak(XkLC(θ)) is the Jacobian of the vector-field ak

evaluated on the limit cycle XkLC(θ), and Pk(θ) satisfies thenormalizing condition < Pk(0),ak(XkLC(0)) >= 1, where <·, ·> denotes the standard inner product in Rd .

In terms of PRC, the interaction term in (9) is obtained as

Ikl(θk,θl) =< Pk(θk),bkl(XkLC(θk),XlLC(θl))> . (19)

The derivation relies on a singular perturbation argument;cf., [7], [8], [1].

Example 5: Consider the single-Hopf model (5) in Ex-ample 1. The limit cycle solution is given by X1LC(θ) =(cos(θ),sin(θ)), and a1(X1LC(θ)) = (−sin(θ),cos(θ)). Theadjoint equation is

dP1dθ

(θ)+AP1(θ) = 0,

where

A =[

−2cos2(θ) 1−2sin(θ)cos(θ)−1−2sin(θ)cos(θ) −2sin2(θ)

],

and the normalizing constraint

< P1(0),a1(X1LC(0))>=< P1(0),(0,1)>= 1.

The solution to the adjoint equation, the phase responsecurve

P1(θ) = (−sin(θ),cos(θ)).

Consider now the coupled model (7a, 7b) in Example 2with b12 = X1−X2. Using (19), the interaction term for the

2426

phase model is obtained as,

I12(θ1,θ2) =[−sin(θ1) cos(θ1)

][ cos(θ1)− cos(θ2)sin(θ1)− sin(θ2)

]= sin(θ1−θ2).

REFERENCES[1] E. Brown, J. Moehlis, and P. Holmes. On the phase reduction and

response dynamics of neural oscillator populations. Neural Comp.,16:673–715, 2004.

[2] Z. Chen, L. Zhu, and T. Iwasaki. Autonomous locomotion of multi-link mechanical systems via natural oscillation pattern. In In Proc. ofIEEE Conference on Decision and Control, pages 6034–6039, 2010.

[3] R. J. Full and D. E. Koditschek. Templates and anchors: neurome-chanical hypotheses of legged locomotion on land. J. Exp. Biol.,202(23):3325–3332, December 1999.

[4] J. W. Grizzle, J.-H. Choi, H. Hammouri, and B. Morris. On Observer-Based Feedback Stabilization of Periodic Orbits in Bipedal Locomo-tion. Methods and Models in Automation and Robotics (MMAR 2007),August 2007. Plenary Talk.

[5] P. Holmes, R. J. Full, D. Koditschek, and J. Guckenheimer. Thedynamics of legged locomotion: models, analyses, and challenges.SIAM Rev., 48(2):207–304, 2006.

[6] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng,N. C. Yen, C. C. Tung, and H. H. Liu. The empirical mode decompo-sition and the Hilbert spectrum for nonlinear and non-stationary timeseries analysis. Proceedings of the Royal Society of London. SeriesA: Mathematical, Physical and Engineering Sciences, 454(1971):903–995, March 1998.

[7] E. M. Izhikevich. Dynamical Systems in Neuroscience: The Ge-ometry of Excitability and Bursting (Computational Neuroscience),chapter 10. The MIT Press, 1 edition, November 2006.

[8] N. Kopell and G. B. Ermentrout. Symmetry and phaselocking in chainsof weakly coupled oscillators. Commun. Pure Appl. Math., 39:623–660, 1986.

[9] N. Kopell and G.B. Ermentrout. Coupled oscillators and the designof central pattern generators. Math. Biosci., 90:87–109, 1988.

[10] B. Kralemann, A. Pikovsky, and M. Rosenblum. Reconstructing phasedynamics of oscillator networks. Chaos, 21(2):025104, 2011.

[11] P. S. Krishnaprasad. Motion control and coupled oscillators. InProceedings of Symposium on Motion, Control and Geometry, Boardof Mathematical Sciences, National Academy of Sciences, Washington,DC, 1995.

[12] T. S. Lee and D. Mumford. Hierarchical Bayesian inference inthe visual cortex. Journal of the Optical Society of America, A,20(7):1434–1448, 2003.

[13] S. Revzen and J. M. Guckenheimer. Estimating the phase of synchro-nized oscillators. Physical Review E, 78(5):051907, 2008.

[14] T. Schreiber. Interdisciplinary application of nonlinear time seriesmethods. Phys. Rep., 308:1–64, 1999.

[15] A. K. Tilton, E. T. Hsiao-Wecksler, and P. G. Mehta. Filtering withrhythms: Application to estimation of gait cycle. In Proc. of AmericanControl Conference, pages 3433–3438, June 2012.

[16] T. Yang, R. S. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariablefeedback particle filter. In Proc. of IEEE Conference on Decision andControl, pages 4063–4070, Dec 2012.

[17] T. Yang, P. G. Mehta, and S. P. Meyn. Feedback particle filter withmean-field coupling. In Proc. of IEEE Conference on Decision andControl, pages 7909–7916, December 2011.

[18] T. Yang, P. G. Mehta, and S. P. Meyn. A mean-field control-oriented approach to particle filtering. In Proc. of American ControlConference, pages 2037–2043, June 2011.

2427