NASA/TM–2013–216528

An Indirect System Identification

Technique for Stable Estimation of

Continuous-Time Parameters of the

Vestibulo-Ocular Reflex (VOR)

Sunil L. Kukreja

NASA Dryden Flight Research Center

Ragnar Wallin

OHB Sweden

Richard D. Boyle

NASA Ames Research Center

May 2013
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NASA/TM–2013–216528

An Indirect System Identification

Technique for Stable Estimation of

Continuous-Time Parameters of the

Vestibulo-Ocular Reflex (VOR)

Sunil L. Kukreja

NASA Dryden Flight Research Center

Ragnar Wallin

OHB Sweden

Richard D. Boyle

NASA Ames Research Center

National Aeronautics and

Space Administration

NASA Dryden Flight Research Center, Edwards, California 93524-0001

May 2013
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Abstract

The vestibulo-ocular reflex (VOR) is a well-known dual mode bifurcating system that consists of slow and fast modes associated with nystagmus and saccade, respectively. Estimation of continuous-time parameters of nystagmus and

saccade models are known to be sensitive to estimation methodology, noise and

sampling rate. The stable and accurate estimation of these parameters are critical for accurate disease modelling, clinical diagnosis, robotic control strategies,

mission planning for space exploration and pilot safety, etc.

This paper presents a novel indirect system identification method for the estimation of continuous-time parameters of VOR employing standardised leastsquares with dual sampling rates in a sparse structure. This approach permits

the stable and simultaneous estimation of both nystagmus and saccade data.

The efficacy of this approach is demonstrated via simulation of a continuoustime model of VOR with typical parameters found in clinical studies and in the

presence of output additive noise.

1 Introduction

The eyes play a critical role in maintaining balance. They are directly connected to organs of equilibrium, most importantly the inner ear. Paired structures called semicircular canals behind the ears sense motion and relay information to balance control centres in the brain. Health of the semicircular canals

can be monitored by studying the vestibulo-ocular reflex (VOR). The VOR is a

reflexive eye movement that stabilises images on the retina during head movement. Ocular responses to head perturbations consist of intermingled segments

classified as “slow” (nystagmus) or “fast” (saccade), according to their average

speed characteristics. Changes in VOR activity can be an indication of serious brain or head trauma, which can negatively impact an astronaut or pilot’s

ability to successfully achieve critical mission objectives.

NASA’s Advanced Exploration Systems and Human Research Programs

are critically interested in pioneering new capabilities allowing future human

missions beyond Earth orbit [1,2]. The objective quantification of mechanisms

that control VOR will enhance the capabilities of NASA’s human exploration

missions by developing techniques that permit prediction of spatial disorientation and development of individualised countermeasures for astronaut crews

that must perform critical space operations under varied gravito-inertial conditions such as launch, landing and orbital maneuvering [3].

It has been shown that the VOR can be accurately modelled by the NARMAX (Non-linear AutoRegressive, Moving Average eXogenous) structure [4].

Moreover, a modified extended least-squares (MELS) algorithm was developed to estimate unbiased parameter values of non-linear Hammerstein structure multimode systems. This two-step process adds overhead to data analysis,

which can be significant for sufficiently long data records and batch processing.

However, it can be reduced to one step by exploiting the nature and efficiency

1
of sparse matrices.

Recently, it was demonstrated that the utilisation of a sparse matrix approach is ideally suited for system identification of switched systems with application to the VOR [5]. This study demonstrated that a sparse matrix approach is an efficient technique for the simultaneous analysis of nystagmus and

saccade dynamics. However, several questions remain unsolved in this modelling and identification approach.

The problem of continuous-time system identification from sampled inputoutput data can be divided into two broad approaches: (i) indirect methods,

where a discrete-time model is estimated from sampled data; then an equivalent continuous-time model is calculated and (ii) direct methods, where a

continuous-time model is obtained directly without going through the intermediate step of first determining a discrete-time model; based on concepts

of approximate numerical integration to recreate time-derivatives needed in

continuous-time formulations [6].

Here, we propose to investigate the problem of continuous-time system

identification by developing methodology using an indirect approach. One such

approach relies on matrix preconditioning techniques, such as standardised

least-squares, to improve the spectral properties of the regressor matrix [7, 8].

Often when the clustered spectrum is away from zero it results in rapid and

robust convergence, especially when the preconditioned matrix is close to normal. We deem this will provide more robust solutions and greater consistency

for non-linear biological systems by circumventing issues of implementing numerical derivatives and filter selection required by direct techniques, which

are more challenging in a non-linear framework. Here, we focus on one critical issue, namely, that of mapping the underlying continuous-time system to

discrete-time for estimation, then inverse mapping back to continuous-time to

provide physiological relevance and insight.

The identification of continuous-time VOR parameters is further challenged

due to the slow and fast speed characteristics of the system. This corresponds

to one pole close to the jω-axis in continuous-time or the unit circle in discretetime whilst the other pole is located significantly inside the left hand plane/unit

circle. To avoid aliasing of the fast mode, the system is typically oversampled 5

to 10 times the highest known dynamics. This results in the slow mode moving

close to the unit circle as the sampling interval approaches zero, leading to numerical instability for parameter estimation. To avoid the numerical instability

problem we propose using a dual sampling rate technique and, thus, permitting

stable and robust continuous-time parameter estimation.

The organisation of this paper is as follows. In §2 we formulate the identification problem addressed here. Section 3 introduces a dual sampling rate,

standardised least-squares method in a sparse structure, which permits stable

indirect estimation of continuous-time VOR parameters. This sparse matrix

formulation is capable of simultaneously quantifying both nystagmus and saccade data. Section 4 provides results of the proposed algorithm on a simulated

VOR model whilst §5 provides a discussion of our findings. Section 6 summarises the conclusions of our study.

2
2 Problem Statement

Consider the continuous-time VOR system illustrated in Fig. 1 and characterised by Eqn. 1 as

f

l

[u(n)]

S1

S2

S1

S2

X1(s) Y1(s)

X2(s) Y2(s)

t

✲ t

❄

✻

........ ❄s✲

...

.

Non-linearity

Shared States

Fast-Phase

Saccade

Slow-Phase

Nystagmus

U(s) X(s)

Z(s)

E(s)

L2(s)

L1(s)

t

t

✲

✲

Figure 1. System describing vestibular dynamics. Switch position S1: nystagmus. Switch position S2: saccade

.

f

l

(·) = a+bu(n) +cu2(n) +du3(n), (1)

Y1(s) = K1

τ1s+1

X1(s) + τ1y1(ti)

τ1s+1

=

G1

s+ p1

X1(s) + y1(ti)

s+ p1

; i = 1,··· ,q,

Y2(s) = K2

τ2s+1

X2(s) + τ2y2(t`)

τ2s+1

=

G2

s+ p2

X2(s) + y2(t`)

s+ p2

; ` = 1,··· ,r

where Y1(s) and Y2(s) are first-order dynamics for slow and fast-phase modes,

y1i and y2` represent the initial values at each switch time and q,r are the number of switches [4, 5]. It has been demonstrated that a NARMAX description

of VOR slow and fast-phases is [4]

y(n) = 

y1(n) Dynamic Mode S1

y2(n) Dynamic Mode S2

(2)

y1(n) = β1 +β2y1(n−1) +β3[u(n) +u(n−1)]

+ β4[u

2

(n) +u

2

(n−1)] +β5[u

3

(n) +u

3

(n−1)]

+ κ1δ11(n− j) +···+κiδ1i(n− ji); i = 1,··· ,q

y2(n) = θ1 +θ2y2(n−1) +θ3[u(n) +u(n−1)]

+ θ4[u

2

(n) +u

2

(n−1)] +θ5[u

3

(n) +u

3

(n−1)]

+ λ1δ11(n−k) +···+λ`δ1`(n−k`); ` = 1,··· ,r

3
where δ is the Kronecker impulse function, j, k are the lags on the δ1i

th and

δ2`th impulse and q,r are the number of data segments of sub-system one and

two, respectively. The unknown model parameters are compactly represented

as θ1,2, = [β1,θ1 β2,θ2 β3,θ3 β4,θ4 β5,θ5 κ1,λ1 ···κi,λ`]

T

.

Table 1 illustrates the relationship of the discrete-time (DT) parameters in

Eqn. 2 to the underlying continuous-time (CT) parameters in Eqn. 1 and their

inverse. Notice that the estimated overall gain is a product of the linear system

DT Coefficient Relationship to CT CT Coefficient DT Relationship

β1,θ1

(2G1,2aT)

2+p1,2T

τ1,2

−(β2,θ2)T−T

−2+2(β2,θ2)

β2,θ2

−(−2+p1,2T)

2+p1,2T G1,2a

2(β1,θ1)

(β2,θ2)T+T

β3,θ3

(G1,2bT)

2+p1,2T G1,2ub

4(β3,θ3)

(β2,θ2)T+T

β4,θ4

(G1,2cT)

2+p1,2T G1,2c

4(β4,θ4)

(β2,θ2)T+T

β5,θ5

(G1,2dT)

2+p1,2T G1,2d

4(β5,θ5)

(β2,θ2)T+T

κ1,λ1

(Y1,2(0)p1,2)

2+p1,2T

Table 1. (Left) Forward and (Right) inverse relationship of NARMAX model

parameters to underlying continuous-time parameters.

and static non-linearity gain. Clearly, the system and its parameter mapping

in the forward and inverse direction are non-linear and governed by z =

(2+T s)

(2−T s)

and s =

2

T

(z−1)

(z+1) where T is the sampling interval.

Given this framework, the goal is to estimate the NARMAX model parameters as

min

θ1,2

1

2

k(Z1,2 −φ1,2θ1,2)k

2

2

(3)

where Z1,2 ∈ R

N×1

is a vector of outputs, φ1,2 ∈ R

N×p

is a matrix of regressors

and θ1,2, ∈ R

p×1

is a vector of unknown coefficients. Using the estimate θ1,2

and the relationship on the RHS of Table 1 we calculate the continuous-time

VOR parameters. This approach is known as indirect system identification of

continuous-time parameters.

In addition, notice the expression in Eqn. 3 is a non-sparse matrix approach,

requiring both fast and slow-phase to be analysed separately. However, using a

sparse matrix approach by constructing a diagonal block-oriented data matrix

and exploiting its natural sparseness as



Z1

Z2



=



φ10

0 φ2

 θ1

θ2



(4)

it is possible to simultaneously estimate both nystagmus and saccade parameters.

4
3 Standardised Least-Squares With Dual Sampling

In a clinical setting the VOR is appropriately sampled taking into consideration

fast-phase dynamics. However, a suitable sampling rate for saccade results in

slow-phase dynamics being highly oversampled. Although it is desirable to

oversample system dynamics to avoid aliasing, too high a sampling rate can

lead to numerical instability in system identification resulting in biased estimates [9, 10]. To avoid numerical instability we propose using a dual sampling

rate approach to analyse both modes of VOR. This dual sampling rate method

is easily achieved by modifying the sparse structure in Eqn. 4 as



Z

∗

1

Z2



=



φ

∗

1

0

0 φ2

 θ1

θ2



(5)

where the superscript “*” denotes that the nystagmus signal has been down

sampled to achieve better numerical results. Here, we assume that the fast

mode has been appropriately sampled and does not need to be down sampled

since the user has control over the sampling rate. However, this result is easily generalisable allowing both signals to be analysed at appropriately down

sampled rates.

In many non-linear systems there are large numerical differences of the

regressors due to the basis function(s) used to estimate the static map. The

large differences lead to the regressor matrix being ill-conditioned and results

in unstable matrix inversion and poor parameter estimates [11]. To alleviate

ill-conditioning we propose using standardised least-squares (SLS) in combination with the dual sampling approach proposed in Eqn. 5.

Given a matrix of independent variables φ and of dependent variables Z

compute the mean and standard deviation of each variable, and replace φ and

Z with the centred and standardised variate as

φe =



φ − μφ



Σ

−1

φ

and Ze = (Z− μZ)Σ

−1

Z

(6)

where Σφ is a diagonal matrix of standard deviations with Σφk denoting the

standard deviation of the kth column of φ and μφis a matrix who’s kth column

has all entries equal to the mean of column k of φ. Substituting Eqn. 6 into 5

yields a dual sampling rate SLS formulation in a sparse structure.

"

Ze∗

1

Ze2

#

=

"

φe

∗

1

0

0 φe

2

#"θe1

θe2

#

(7)

In the sequel we use the formulation in Eqn. 7 to estimate continuous-time

parameters of nystagmus and saccade dynamics.

4 Simulations & Results

The accuracy of the dual sampling rate SLS technique with sparsity was validated by simulating a continuous-time VOR model (e.g. Fig. 1) using Simulink.

5
CT System Coeff. Value CT NL Coeff. Value

τ1 15.0 s

τ2 50.0 ms b 1.20

K1 -9.43 c -3.00×10−3

K2 0.222 d -1.50×10−5

T 1.00 ×10−2s

Table 2. Left: Continuous-time coefficient values. τ1: slow-phase timeconstant, τ2: fast-phase time-constant, K1: slow-phase gain, K2: fast-phase

gain and T: sampling interval. Right: Coefficient values of static non-linearity.

b: linear term, c: squared term and d cubic term.

The parameters used in the simulation are shown in Table 2 and representative

of typical values found in experiments [4].

One hundred Monte-Carlo simulations were generated in which the inputoutput realisation was the same but had a unique Gaussian white, zero-mean,

noise sequence added to the output. Excluding the noise free (NF) case the

signal-to-noise ratio (SNR) of the noise sequence was decreased from 20 − 0

dB in increments of 5 dB. The system was perturbed using a sinusoid input (1/6

Hz frequency and 188 deg/s amplitude). A sinusoid input was used because it

is the type of perturbation used in clinical settings. The system input-output

was sampled at 100 Hz.

The system parameters were estimated as outlined in Eqns. 4-7. Specifically, for each input-output realisation, we analysed the VOR model as follows:

1. Single Sampling Rate, Non-Standardised Least-Squares: Both VOR modes

were analysed using a single sampling rate and identified simultaneously

using a sparse structure using traditional least-squares (Eqn. 4),

2. Dual Sampling Rate, Non-Standardised Least-Squares: fast-phase was

sampled at the original rate (100 Hz) whilst slow-phase was down sampled by 10 (10 Hz) and identified simultaneously using a sparse structure

using traditional least-squares (Eqn. 5),

3. Dual Sampling Rate, Standardised Least-Squares: fast-phase was sampled at the original rate (100 Hz) whilst slow-phase was down sampled

by 10 (10 Hz) and identified simultaneously using a sparse structure using SLS (Eqn. 7).

The continuous-time parameters of slow and fast-phase dynamics were estimated using the theoretical relationships in Table 1. Notice that we consider

the linear system to have unity gain and translate the overall gain onto the polynomial basis function giving an estimate as a product of the linear system and

static non-linearity gain. This is necessary because it is impossible to measure

the signal at the output of the static non-linearity. Therefore, we deem that the

best estimate of the linear system gain is a product of the linear system gain

and linear coefficient of the static non-linearity, i.e. G1,2b.

6
The results of this study are shown in Fig. 2. The panels of the left colFigure 2. Left column: Single Sampling Rate, Non-Standardised LeastSquares. Centre column: Dual Sampling Rate, Non-Standardised LeastSquares (100 & 10 Hz). Right Column: Dual Sampling Rate, Standardised

Least-Squares (100 & 10 Hz). Ordinate: STD about mean. Abscissa: Output

SNR = NF, 20, 15, 10, 5, 0 dB, where NF denotes noise free case. (Note that

the abscissa is shown in decreasing SNR which corresponds to increasing noise

amplitude.)

umn illustrate our findings for a non-standardised (traditional) least-squares

approach using a single sampling rate to analyse both modes. The results show

the nystagmus time constant is significantly biased with large variance whilst

the nystagmus gain and saccade parameters are as theoretically expected with

increasing bias and variance for decreasing SNR. The centre column displays

the results of using a dual sampling rate with a non-standardised least-squares

technique. The panels show this method improves bias and variance estimates

for decreasing SNR since it yields theoretically expected trends. However, the

non-monotonic increase in variance for the slow-phase time constant (e.g. 10-0

7
dB) suggests that the regressor matrix tends to becomes ill-conditioned for decreasing SNR. The right column displays the results of using a dual sampling

rate with a standardised least-squares approach. These results demonstrate that

a dual sampling rate approach in combination with SLS significantly improves

bias and variance estimates to expected trends. Nevertheless, the bias and variance of the slow phase gain increase with dual sampling rate and standardisation. This result is expected since in these two cases the gain parameter is

estimated with 10 times less data. Hence, using SLS the slow-phase variance

monotonically increases for decreasing SNR and agrees with theoretically expected results.

5 Discussion

Simulation results presented in §4 demonstrate that a dual sampling rate approach provides improved results to single sampling rate techniques. In addition, when combined with preconditioning offered by SLS, estimates are further improved. Hence, our dual sampling rate SLS technique is a robust and

easily applicable methodology for the analysis of VOR dynamics.

In many continuous and discrete-time parameter estimation problems the

estimates may have incorrect sign due to high variance, possibly due to numerical ill-conditioning. Our proposed technique alleviates this problem by using

matrix preconditioning (compare Fig. 2 (a)-(b) with (c)). This result may have

general applicability for many estimation problems in biology such as for the

analysis of ankle dynamics [12].

The central problem with VOR analysis is due to its time-constants (or

poles) being separated by many orders of magnitude. In our example with a

clinically relevant parameter set, the system poles are separated by 30 orders of

magnitude, which yields in slow-phase being highly oversampled and leading

to numerical instability.

The approach presented here is generalisable to most bifurcating systems

since they have poles that are typically several orders of magnitude apart, resulting in highly oversampled dynamics and unstable parameter estimates.

6 Conclusions

The results demonstrate that our dual sampling rate SLS approach with sparsity

is a robust and easily implementable technique for the analysis or VOR data.

These results suggest that clinical analysis of nystagmus (and saccade) may be

improved, providing clinicians more accurate information for diagnosis, disease modelling, etc. In addition, the robust estimation of VOR may lead to

improved mission planning for astronauts and test-pilots of advanced aircraft.

8
References

1. Anonymous: The NASA Fundamental Space Biology Science Plan 2010-

2020, NASA Headquaters. FSB-10-05-17 ed., 2010, p. 52.

2. Anonymous: Human Research Program, Integrated Research Plan,

NASA Johnson Space Center. HRP-47065 Rev C ed., 2011, p. 18.

3. Lackner, J.; and DiZio, P.: Space motion sickness. Exp. Brain Res.,

vol. 175, 2006, pp. 377–399.

4. Kukreja, S.; Kearney, R.; and Galiana, H.: A Least-Squares Parameter Estimation Algorithm for Switched Hammerstein Systems with Applications

to the VOR. IEEE-TBME, vol. 52, no. 3, 2005, pp. 431–444.

5. Kukreja, S.; Stone, L.; and Boyle, R. D.: A Sparse Matrix Approach for Simultaneous Quantification of Nystagmus and Saccade. Proc. IEEE EMBS,

San Diego, California, vol. 34, September 2012, pp. 5478–5481.

6. Sontag, E. D.: Identification of continuous-time systems. Methodology and

computer implementation, N. Sinha and G. Rao, eds., Kluwer Academic

Publishers, 1991.

7. Trefethen, L. N.; and Bau, D.: Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997.

8. Ljung, L.: System Identification: Theory for the User. Prentice Hall, Inc.,

Upper Saddle River, New Jersey, second ed., 1999.

9. Billings, S.; and Aguirre, L. A.: Effects of the sampling time on the dynamics and identification of nonlinear models. International Journal of

Bifurcation and Chaos, vol. 5, no. 6, 1995, pp. 1541–1556.

10. Worden, K.: On the over-sampling of data for system identification. Mechanical Systems and Signal Processing, vol. 9, no. 3, 1995, pp. 287–297.

11. Hansen, P.; Pereyra, V.; and Scherer, G.: Least Squares Data Fitting with

Applications. John Hopkins University Press, Baltimore, MD, first ed.,

2012.

12. Kukreja, S.; Galiana, H.; and Kearney, R.: NARMAX Representation

and Identification of Ankle Dynamics. IEEE-TBME, vol. 50, no. 1, 2003,

pp. 70–81.

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4. TITLE AND SUBTITLE

An Indirect System Identification Technique for Stable Estimation of

Continuous-Time Parameters of the Vestibulo-Ocular Reflex (VOR)

5a. CONTRACT NUMBER

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6. AUTHOR(S)

Sunil L. Kukreja, Ragnar Wallin and Richard D. Boyle

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14. ABSTRACT

The vestibulo-ocular reflex (VOR) is a well-known dual mode bifurcating system that consists of slow and fast modes associated with

nystagmus and saccade, respectively. Estimation of continuous-time parameters of nystagmus and saccade models are known to be sensitive

to estimation methodology, noise and sampling rate. The stable and accurate estimation of these parameters are critical for accurate disease

modelling, clinical diagnosis, robotic control strategies, mission planning for space exploration and pilot safety, etc.

This paper presents a novel indirect system identification method for the estimation of continuous-time parameters of VOR employing

standardised least-squares with dual sampling rates in a sparse structure. This approach permits the stable and simultaneous estimation of

both nystagmus and saccade data. The efficacy of this approach is demonstrated via simulation of a continuous-time model of VOR with

typical parameters found in clinical studies and in the presence of output additive noise.

15. SUBJECT TERMS

System Identification, Indirect System, Least-Squares, Standardised Least-Squares, Switched Systems, Parametric Models, NARMAX,

Space Motion Sickness, Mal de Debarquement  ́

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