An observability result related to active sensing Eduardo D. Sontag1 Debojyoti Biswas2 and Noah J. Cowan23 1Department of Electrical and Computer Engineering and Department of

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An observability result related to active sensing

Eduardo D. Sontag1, Debojyoti Biswas2, and Noah J. Cowan2,3

1Department of Electrical and Computer Engineering and Department of

Bioengineering, Northeastern University, Boston, Massachusetts, United States

2Laboratory for Computational Sensing and Robotics, Johns Hopkins University,

Baltimore, Maryland, United States.

3Department of Mechanical Engineering, Johns Hopkins University,

Baltimore, Maryland, United States.

1 Abstract

For a general class of translationally invariant systems with a speciﬁc category of nonlinearity in the

output, this paper presents necessary and suﬃcient conditions for global observability. Critically, this

class of systems cannot be stabilized to an isolated equilibrium point by dynamic output feedback.

These analyses may help explain the active sensing movements made by animals when they perform

certain motor behaviors, despite the fact that these active sensing movements appear to run counter

to the primary motor goals. The ﬁndings presented here establish that active sensing underlies the

maintenance of observability for such biological systems, which are inherently nonlinear due to the

presence of the high-pass sensor dynamics.

2 Introduction

Active sensing is the process of expending energy, typically through movement, for the purpose of

sensing [1–3]. Animals use this strategy to enhance sensory information across sensory modalities e.g.,

echolocation [4,5], whisking [6,7] and other forms of touch [8,9], electrosense [10–12], and vision [13,14].

It is well established that conditions of decreased sensory acuity leads to increased active movements

[5,11,12, 14–21] but its actual role in relation to task-level control remains underexplored. The ubiquity

of active sensing in nature motivates us to explore the mathematical conditions that might necessitate

active sensing. Our theory is that active sensing is at least in part borne out of the needs of nonlinear

state estimation. We hypothesize that animals—through active sensing—generate time-varying motor

commands that continuously stimulate their sensory receptors so that the system states can be estimated

with satisfactory error bounds from the sensor measurements. In essence, these movements aim to

maintain the observability of the system.

A dominant paradigm in control systems engineering involves designing state feedback and state estima-

tion independently, an approach can be applied successfully to a wide range of system designs. Indeed,

for linear plants corrupted by Gaussian noise, there is a separation principle: it is not only satisfactory

to separate state estimation from the task-level control design, but, in fact, it is optimal to perform

this decomposition. In particular, the linear-quadratic-Gaussian (LQG) controller decomposes into a

linear-quadratic regulator (LQR) applied to the optimal state estimate which comes from a Kalman ﬁl-

ter. Critically, the Kalman ﬁlter does not depend on the LQR cost function, and the LQR gains do not

depend on the sensor noise and process noise. Conceptually speaking, “active sensing” is the opposite

arXiv:2210.03848v1 [math.DS] 7 Oct 2022

approach to applying a separation principle: control inputs are speciﬁcally designed to excite sensory

receptors, presumably in service to the state estimator. This may be, at least in part, because biological

sensory systems often stop responding to persistent (i.e. “DC”) stimuli, via sensory “adaptation” [22–26]

or “perceptual fading” [27,28].

In this paper, we formalize a class of nonlinear systems that have a simple high-pass sensory output that

mimics sensory adaptation or perceptual fading. Under some simpliﬁed modeling assumptions, reviewed

below, this implies that linear observability is lost, which means the usual LQG-style framework does not

apply. However, under some interesting modeling conditions, nonlinear observability persists. Critically,

nonlinear observability does not necessarily aﬀord a separation principle: the control signal may need

to contain ancillary energy that is expressly for the purpose of state estimation, and may be in conﬂict

with task goals. Indeed, the energy expended for active sensing movements do not necessarily directly

serve a motor control goal, and are instead believed to improve sensory feedback and prevent perceptual

fading [12, 27, 28].

The organization of the paper is as follows. Section 3 motivates the model structure from prior work

and Section 4 generalizes the model and presents the main theorem. Section 5 has the proof of the main

theorem. The Appendix provides some background concepts for the ease of understanding the tools used

in Section 3 and 5.

3 Biological motivation and simpliﬁed system

Station keeping behavior in weakly electric ﬁsh, Eigenmannia virescens, provides an ideal system for

investigating the interplay between active sensing and task-level control [11, 12, 29, 30]. These ﬁsh rou-

tinely maintain their position relative to a moving refuge and uses both vision and electrosense to collect

the necessary sensory information from its environment [31–34]. While tracking the refuge position (i.e.,

task-level control), the ﬁsh additionally produce rapid “whisking-like” forward and backward swimming

movements (i.e., active sensing). When vision is limited (for example, in darkness), the ﬁsh increase

their active sensing movements [12,30]. This increased motion compensates the lack of visual cues [11].

Suppose xis the position of an animal and z= ˙xis its velocity as it moves in one degree of freedom.

We assume that a sensory receptor measures only the local rate of change of a stimulus, s(x) as the

animal moves relative to the sensory scene, i.e. y=d

dts(x). Deﬁning γ(x) := d

dxs(x), we arrive at a

2-dimensional, single-input, single-output normalized mass-damper system of the following form [35,36]:

˙x=z, x ∈R

˙z=−z+u, z, u ∈R

y=d

dts(x) = γ(x)z, y ∈R

(1)

where the mass and the damping constant both are assumed to be unity. Linearization of the above

system (1) around any equilibrium, (x∗,0), is given by (A, B, C) as follows:

A=0 1

0−1, B =0

1, C =0γ∗,

where γ∗=γ(x∗). Clearly (A,C) is not observable irrespective of γ∗[37]. Indeed, the output introduces

a zero at the origin that cancels a pole at the origin, rendering xunobservable. Assuming no input u,

we can write the system (1) as, ˙

ξ=f(ξ), y =h(ξ),(2)

where ξ= (x, z)>, f = (z, −z)>and h(ξ) = γ(x)z. We can construct the observation space, O(set of

all inﬁnitesimal observables) by taking y=γ(x)zwith all repeated time derivatives

y(k)=L(k)

f(γ(x)z)

as in [38, 39]. The superscript “(k)” indicates kth order derivative. Note that L(k)

f((γ(x)z)) lies in

the span of the functions γ(j)(x)zj+1, j = 0,1, . . . , k. The rank condition on the observability co-

distribution [38, 39] implies a suﬃcient condition for local observability as follows [36]:

z2(2(γ0(x))2−γ(x)γ00(x)) 6= 0.(3)

Clearly for an non-hyperbolic γ(6= 1/(αx +β), with constants of integration α, β), the non-zero veloc-

ity requirement (z6= 0) implies the need for active sensing to maintain the local observability of the

system [36, 40].

What does this simpliﬁed example say about the need for active sensing? As the proposition below

illustrates, for such a system, dynamic output feedback cannot asymptotically stabilize the origin (0,0),

indicating the need for extra inputs to perform active state estimation:

Proposition 3.1 Consider the system (1). Let

˙q=g(q, y)

u=k(y, q)(4)

be a dynamic output feedback (Fig 1). Suppose (x∗, z∗, q∗) = (0,0, q∗) is an equilibrium of the coupled

system. Then all points (ξ∗,0, q∗), ξ∗∈R, are equilibiria. 2

Proof. Since (0,0, q∗) is an equilibrium, we see from the second equation in (1) that k(0, q∗) = 0. That

means that k(γ(ξ∗)·0, q∗) = 0, i.e. (ξ∗,0, q∗) is also an equilibrium, for all ξ∗∈R.

Figure 1: The system (1) cannot be stabilized to an equilibrium point by the dynamic feedback in (4).

Remark 3.2 The impossibility of stabilizing a system with high-pass sensing to an equilibrium point,

using only dynamic output feedback, generalizes to the class of systems described below. 2

4 The class of systems and main result

Now we consider a general 2n-dimensional, single-input, n-output system of the following form:

˙x=z

˙z=F(z) + bu

y=H(x)z

where the state space variable ξ∈R2nis partitioned as ξ= (x>, z>)>into variables x∈Rnand

z∈Rn, and H(x) = diag (γ1(x1), . . . , γn(xn)) is a diagonal n×nmatrix. The entries of the column

vector function F(z) as well the functions γi(xi) are real-analytic functions of their arguments, and bis

a column vector of size nall whose entries are nonzero.

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AnobservabilityresultrelatedtoactivesensingEduardoD.Sontag1,DebojyotiBiswas2,andNoahJ.Cowan2,31DepartmentofElectricalandComputerEngineeringandDepartmentofBioengineering,NortheasternUniversity,Boston,Massachusetts,UnitedStates2LaboratoryforComputationalSensingandRobotics,JohnsHopkinsUniversity,Baltim...

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An observability result related to active sensing Eduardo D. Sontag1 Debojyoti Biswas2 and Noah J. Cowan23 1Department of Electrical and Computer Engineering and Department of

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