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# Disturbance sensitivity analysis and experimental evaluation of continuous
sliding mode control

Michael Ruderman, Benjamin Voß, Leonid Fridman, Johann Reger M. Ruderman is
with University of Agder, 4879-Grimstad, Norway. E-mail:
<EMAIL_ADDRESS>Voß is with Technische Universität Ilmenau,
98684-Ilmenau, Germany. E-mail<EMAIL_ADDRESS>Fridman is with
Universidad Nacional Autonoma de Mexico, 04510 Mexico City, Mexico. E-mail:
<EMAIL_ADDRESS>Reger is with Technische Universität Ilmenau,
98684-Ilmenau, Germany. E-mail<EMAIL_ADDRESS>work has received funding
from the EUs H2020-MSCA-RISE research and innovation programme under grant
agreement No 734832. Third author is also acknowledging support by CONACyT
(Consejo Nacional de Ciencia y Tecnologia) project 282013, PAPIIT-UNAM
(Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica) IN
115419

###### Abstract

Continuous higher order sliding mode (CHOSM) controllers are new efficient
tool for disturbance rejection. For the systems with relative degree $r,$
CHOSM provide theoretically exact compensation of Lipschitz matched
perturbation ensuring the finite-time convergence to the $(r+1)$-th sliding-
mode set using only information on the sliding output and its derivatives up
to the order $(r-1)$. In this paper, we investigate the properties of
disturbance rejection for a PID-like controller, as the simplest and
intuitively clear example of CHOSM one, allowing a harmonic balance-based
consideration of the residual steady-state oscillations, i.e. chattering. We
provide additional analysis of the PID-like controller based on its analytic
describing function and the disturbance sensitivity function, allowing certain
comparability with the standard linear PID control in frequency domain. For
comparison reason a simple and straightforward design procedure for robust PID
controller, which is served by the assumed upper bound of the disturbance
sensitivity function, is developed. A detailed experimental case study,
accomplished on an electro-mechanical actuator in the laboratory setting,
highlight and make the pros and cons of both PID and CHOSM controllers well
comparable for a broadband disturbance rejection.

###### Index Terms:

Sliding mode control, harmonic balance, disturbance rejection, sensitivity
analysis, control design

##  I Introduction

Sliding-mode controllers (SMC) are efficient tools for disturbance
rejection[1]. The main disadvantage of SMC is the presence of the so-called
chattering effect due to the discontinuity used in the controller, see e.g.
[2],[3],[1]. Continuous higher order sliding mode (CHOSM) algorithms (see
[4],[5],[6], [7],[8],[9],[10], [11],[12]) are a class of homogeneous sliding
mode controllers capable to compensate Lipschitz uncertainties and/or
perturbations theoretically exactly, and generating a continuous control
signal, instead of a discontinuous one. These algorithms consist of a static
homogeneous finite-time controller for the nominal model of the system and a
discontinuous integral action, aimed at estimating and compensating the
uncertainties and perturbations. CHOSM controllers are an extension of the
(classical) super-twisting algorithm [4], ensuring for the systems with
relative degree $r,$ the finite-time convergence to the $(r+1)$-th sliding-
mode set, by using only information on the sliding output and its derivatives
up to the order $(r-1).$ The homogeneity weight of output variables in the
CHOSM controllers is $r+1$, cf. [6],[9],[10],[11], and it ensures,
correspondingly, the chattering amplitude of the order $r+1$ [13].

PID-like continuous sliding mode control introduced in [5] and [12] is the
simplest and intuitively clear CHOSM algorithm. For the systems with relative
degree $r=2$, it ensures the finite-time convergence to the third sliding-mode
set using only information on the sliding output and its derivative. Moreover,
when the actuator is fast (see [14], [15]), the chattering effect caused by
the discontinuity and discretization is, therefore, strongly attenuated. To
generate the synthesis rules for PID-like controllers the authors of [12] used
the harmonic balance method and made the PID-like controller gains adjusting
the parameters of chattering, occurring in the system due to the presence of
parasitic dynamics. Finally, an approach proposed in [12] allows to minimize
the amplitude of chattering or the energy needed to maintain the system with
relative degree two in a real third-order sliding mode. The harmonic balance-
based analysis of PID-like controllers opened the door for investigation of
the propagation properties of CHOSM algorithms that will be used in this
paper. The corresponding contribution of the paper is summarized as:

  * •

an analysis of the PID-like CHOSM control based on its analytic describing
function, allowing to estimate the disturbance sensitivity function in
frequency domain;

  * •

a comparison of the properties of continuous higher order sliding mode
controllers with a simple and straightforward design procedure for a robust
linear PID controller, which is served by the assumed upper bound of the
disturbance sensitivity function;

  * •

a detailed experimental case study, accomplished on an electro-mechanical
actuator in the laboratory setting, highlighting and making the pros and cons
of both PID and PID-like CHOSM controllers well comparable.

The rest of the paper is as follows. In section II, we recall the PID-like
CHOSM control [12] for convenience of the reader and provide its disturbance
sensitivity function in a combined frequency-amplitude manner. Section III is
devoted to formulation of the control problem and description of the system we
use for evaluation. In section IV, we describe the robust PID control design
by developing an analytic two-step procedure for synthesis. A comparative
experimental evaluation of linear PID and PID-like CHOSM controllers are given
in section V. Brief conclusions are drawn in section VI.

##  II PID-like Continuous Sliding Mode Control

In the following, we first briefly introduce the class of the system plants
under consideration, before summarizing the continues sliding mode control we
are interested in.

###  II-A Class of System Plants

Consider the class of perturbed second-order systems which can be written, in
time domain, as

$\ddot{y}(t)=u(t)+d(t).$ (1)

The single measurable system state is the output $y(t)$, while $u(t)$ is the
control input. The disturbance $d(t)$ is assumed to be Lipschitz continuous,
meaning $|\mathrm{d}/\mathrm{dt}\,d(t)|\leq L$ while the upper bound $L>0$ is
known. Moreover, the control channel $u$ can be subject to an additional (here
first-order) actuator dynamics with a unity gain and not negligible time
constant $\mu$, thus transforming the original system (1) into
$y(s)=s^{-2}\,\bigl{(}d(s)+(\mu s+1)^{-1}u(s)\bigr{)}$, when written in
Laplace domain.

###  II-B Control Law and Harmonic Balance Analysis

The continuous higher order sliding mode control, as a simplified version of
the so-called Discontinuous Integral Controller [9, 11] under investigation,
is given by

$u=-k_{1}\lceil
y\rfloor^{1/3}-k_{2}\lceil\dot{y}\rfloor^{1/2}-k_{3}\int\mathrm{sign}(y)dt,$
(2)

cf. [12, 5], and denoted further as CSMC. Here $k_{1},k_{2},k_{3}>0$ are the
properly designed control gain parameters. It can be recognized that the
corresponding three control terms are equivalent to the proportional,
derivative, and integral feedback actions, respectively. The used notation
$\lceil x\rfloor^{p}\equiv|x|^{p}\mathrm{sign}(x)$, for a system variable
$x\in\mathbb{R}$ and a number $p\in\mathbb{R}$, is a commonly used expression
in the high-order-sliding-mode literature, so that for example $\lceil
x\rfloor^{0}=\mathrm{sign}(x)$. The used sign-function is defined in the
Filippov sense [16], meaning

$\mathrm{sign}(x)=\left\\{\begin{array}[]{ll}1,&\hbox{ if }x>0;\\\
\left[-1,1\right],&\hbox{ if }x=0;\\\ -1,&\hbox{ if }x<0.\end{array}\right.$
(3)

For the perturbed second-order system, with a Lipschitz constant $L>0$, the
assignment of the gains $k_{1}=\lambda_{1}L^{2/3}$,
$k_{2}=\lambda_{2}L^{1/2}$, and $k_{3}=\lambda_{3}L$, ensures the finite-time
convergence and insensibility with respect to the matched disturbances $d(t)$,
cf. with Fig. 2. This is however under assumption of no additional actuator
dynamics prior to the plant $G$.

One of the remarkable features used for analysis of the nonlinear CSMC control
(2), which in fact enables for the following comparison with linear feedback
controllers $C(s)$, is the closed analytic form of the describing function,
cf. [15]

$N(A,\omega)=\frac{2\alpha_{1}k_{1}}{\pi
A^{2/3}}+j\,\frac{2\alpha_{2}k_{2}\omega^{1/2}}{\pi
A^{1/2}}+\Bigl{(}\frac{4k_{3}}{\pi
A}\Bigr{)}\Bigl{(}\frac{1}{j\omega}\Bigr{)},$ (4)

where $\alpha_{1}\approx 1.821$, $\alpha_{2}\approx 1.748$ are the
coefficients of the first harmonic approximation [12]. Based on the Harmonic
Balance (HB) equation

$N(A,\omega)W(j\omega)=-1,$

where the linear system plant $W(j\omega)=D(j\omega)G(j\omega)$ includes an
additional parasitic actuator dynamics $D(j\omega)$, the harmonic oscillations
can be predicted. The developed methodology (see e.g. [2, 3, 17]) allows
analyzing the steady-state behavior of a control loop with sliding-mode
controllers, provided the corresponding $N(\cdot)$ is known. Studying the
steady-state oscillations of $y(t)$, different sets of optimal CSMC control
parameters, see [12] and related references therein, were proposed for such
criteria as minimizing the amplitude or average power of the steady-state
oscillations. For small time constants of the actuator (cf. $\mu$ in section
III), the scaling constants that minimize the amplitude $A$ of the $y(t)$
harmonic oscillations were determined in [12] as $\lambda_{1}=2.7$,
$\lambda_{2}=5.345$, $\lambda_{3}=1.1$. With this set of the gaining
coefficients, the Lipschitz constant (as a scaling factor) remains the single
free design parameter for computing the control gains of (2).

In analogy to the disturbance sensitivity function of linear feedback loops,
cf. further section IV-A, the $d$ to $y$ transfer characteristics of CSMC
control loop at steady-state can be analyzed by using the describing function
(4), that leads to

$S_{yd,CSMC}(j\omega)=\frac{y_{A}(j\omega)}{d(j\omega)}=\frac{W(j\omega)}{1+N(A,\omega)W(j\omega)},$
(5)

where $y_{A}(\cdot)$ is the dominant harmonic oscillation (with amplitude $A$)
of the output, as a result of disturbance acting at the angular frequency
$\omega$. Obviously, the resulted approximation of the disturbance sensitivity
function (5) constitutes a two-dimensional frequency-amplitude map, for which
the magnitude response $(\omega,A)\mapsto|S_{yd,CSMC}|$ can be directly
computed. For the tuned $L=0.4$, the substitution of (4) and $G$ and $D$ (cf.
further with section III) into (5), and evaluation of $|S_{yd,CSMC}|$ result
in the frequency-amplitude response depicted in Fig. 1.

Figure 1: Frequency-amplitude-dependent magnitude characteristics of the
disturbance sensitivity function (5) of the CSMC control loop

Note that the otherwise unknown upper bound $L$ was tuned based on the
numerical simulations and experiments so that the residual oscillations of the
output $y(t)$ at steady-state remain acceptably low, cf. further with section
V. From Fig. 1, one can recognize that the amplitude dependency of the
disturbance affecting the output is remarkable for both, lower frequency range
and peak of the $S_{yd,CSMC}$ sensitivity function. At the same time, the
disturbance sensitivity becomes less $A$-dependent for higher frequency range,
thus making CSMC-control rather invariant to amplitude of the residual
harmonic oscillations when the disturbance frequency grows. Here we note that
the residual oscillations of the CSMC-controlled output can be estimated,
according to [12], as

$A=L\Bigl{(}\frac{2\alpha_{1}\lambda_{1}}{\pi\mathbb{K}^{2}(1-\mathbb{K}^{2})}\Bigr{)}^{3/2}\mu^{3}\quad\hbox{
and }\quad\omega=\frac{\mathbb{K}}{\mu}.$ (6)

Here $A$ is the amplitude of steady-state harmonic of $y$ at the frequency
$\omega$, after finite-time convergence of the CSMC. Therefore, the steady-
state accuracy of CSMC control loop is bounded by (6) for each solution of HB
equation. Recall that the amplitude of the residual output oscillations is
predictable, provided the upper bound of the perturbation’s Lipschitz constant
and actuator’s time-constant are known.

##  III Control Problem and System Description

We consider the problem of a standard output feedback control loop, see Fig.
2, with the measurable dynamic state $y(t)$ of interest and matched
disturbance value $d(t)$. Note that all signals in the block diagram of Fig. 2
are purposefully denoted in Laplace domain, where $s$ is the complex Laplace
variable, this for the sake of compatibility with the transfer functions in
use. Since our prime focus is on the disturbance rejection, we assume zero
reference i.e. $w(t)=0$ so that the control error yields $e=-y$. The system
plant can be sufficiently approximated by the second-order transfer function
$G(s)$, from the input force $u$ to the output relative displacement $y$. Note
that the input-output characteristics of a motion system can be additionally
subject to a fast parasitic dynamics of the actuator $D(s)$. Worth noting is
that the latter belongs to the main application-related problems of the
sliding mode controllers, causing the steady-state oscillations, known also as
chattering.

$C\circ D(s)$$G(s)$$d(s)$$w(s)$$e(s)$$u(s)$$y(s)$$-$

Figure 2: Standard output feedback control loop.

The applied feedback controller is $C$, while for linear feedback controls,
like the PID one which is benchmarking in this work, the corresponding
transfer function $C(s)$ is in place. For the nonlinear PID-like sliding mode
control (CSMC), $C$ constitutes a continuous mapping $e\in\mathbb{R}\mapsto
u\in\mathbb{R}$.

The main focus of our investigations is on the comparative analysis of the
control error suppression, i.e. by means of a controller $C$, when the dynamic
disturbance $d(s)$ of certain frequency-band affects the closed-loop system as
in Fig. 2. Note that due to a finite-time convergence of CSMC and, hence, its
transient behavior that depends directly on the admissible control gains and
the initial conditions, only steady-state behavior can be adequately compared
between CSCM and PID. Furthermore, it is worth recalling that the CSMC is
efficient to deal with systems affected by non-vanishing Lipschitz
perturbations [12]. Therefore, the disturbance of a particular frequency, like
a one defined later in (11), will have upper bound of the Lipschitz constant
$L=A_{d}\omega_{d}$. Accordingly, for the known upper bound $\max A_{d}$ of
the disturbance value, the admissible bandwidth of the closed-loop with CSMC
is $\max\omega_{d}\leq L(\max A_{d})^{-1}$. In case of a linear feedback
control $C(s)$, the same admissible bandwidth $\max\omega_{d}$ can be assumed,
for the sake of comparison, while the superposition principle yields the
linear closed-loop control system invariant to the amplitude $A_{d}$ of
disturbances. Note that this requires, however, an unsaturated control value
for the system plant $G(s)$.

Figure 3: Experimental setup of the motion system

The second-order system under investigation in this study is an electro-
mechanical drive depicted in Fig. 3. The induced relative displacement $y(t)$
is indirectly measured by the contactless inductive displacement sensor with a
nominal repeatability of $\pm 12$ micrometers. While all mechanical elements
are rigid, and the system dynamics is inherently of the second-order, with one
free integrator, the nominal electrical time constant of the actuator is
$\mu=1.2$ msec. Note that this knowledge of a parasitic actuator dynamics will
be explicitly used when analyzing fast oscillations of the continuous sliding
mode control, cf. section II. Furthermore, it is known that the system
discloses a relatively high level of the sensor and process noise. The former
is due to a contactless sensing, while the latter is due to additional
parasitic by-effects which are not captured by the simplified linear modeling,
cf. sections II-A and III-A. The real-time control board operates the system
with the set sampling rate of 5 kHz. More details on the experimental system
can be found in [18], with principal difference that in the present work an
oscillating payload is purposefully detached from the drive. The available and
the identified system parameters are given below.

###  III-A Modeled and Identified System Plant

The principal dynamics of the plant under consideration are assumed to be
represented by

$m\ddot{y}(t)+\sigma\dot{y}(t)=\frac{\Psi}{R}\tilde{u}(t)-mg,$ (7)

with input $\tilde{u}(t)$ (in volts) and measurement $y(t)$ (in meters). The
values of the moving mass $m$, back electromotive force constant $\Psi$, and
resistance $R$ are given by

$m=0.538\>\mathrm{kg},\quad\Psi=17.16\,\mathrm{\frac{Vs}{m}},\quad
R=5.32\,\mathrm{\frac{V}{A}}$ (8)

and originate from the weight measurement and the actuator data sheet [19].
Note that the gravitational force $F_{\mathrm{g}}=-mg$ with the gravity
constant $g=9.81\,\mathrm{ms^{-2}}$ can be directly compensated with
$u_{\mathrm{g}}=mgR\Psi^{-1}$ and is, therefore, not a part of further
discussions. In order to estimate the unknown damping parameter $\sigma>0$, we
apply $\tilde{u}=u+u_{\mathrm{g}}$ leading to

$\frac{m}{\sigma}\ddot{y}(t)+\dot{y}(t)=\frac{\Psi}{R\sigma}u(t).$ (9)

Using the Laplace transform, with zero initial conditions, the plant’s
transfer function reads

$G(s)=\frac{K}{s(\tau
s+1)}~{}\mathrm{with}~{}K=\frac{\Psi}{R\sigma},~{}\tau=\frac{m}{\sigma}.$ (10)

Due to the free integrative behavior and bounded position measurement
$y\in[0\,\mathrm{mm},18\,\mathrm{mm}]$, the identification experiments are
conducted in a closed-loop, as in Fig. 2. In order to minimize the noise
amplification and without exciting further dynamics, a pure proportional
controller with a relatively low gain $C=100$ is utilized here for the
parameter identification experiments. Starting from the middle of the actuator
range at steady-state $y_{0}=9\,\mathrm{mm}$, a sinusoidal input disturbance

$d(t)=A_{\mathrm{d}}\sin(\omega_{\mathrm{d}}t)$ (11)

is applied for exciting the system.

Figure 4: Bode plot of the experimental data and estimated transfer function.

The amplitude $A_{\mathrm{d}}\in[0.7\,\mathrm{V},1.5\,\mathrm{V}]$ is chosen
to ensure sufficient excitation without input- and state-saturation, in accord
with the frequency range
$\omega_{\mathrm{d}}\in[1\,\mathrm{rad/s},\,1000\,\mathrm{rad/s}]$. A
measurement of 20 periods at steady-state is utilized for the frequency
response estimation at each frequency. With use of the auto-correlation
$R_{\mathrm{uu}}(\cdot)$ and cross-correlation functions
$R_{\mathrm{uy}}(\cdot)$, the magnitude and phase at $\omega_{\mathrm{d}}$ can
be calculated as, cf. with e.g. [20],

$\left|G(\mathrm{j}\omega_{\mathrm{d}})\right|=\frac{R_{\mathrm{uy,max}}}{R_{\mathrm{uu}}(0)}\quad\text{and}\quad\angle
G(\mathrm{j}\omega_{\mathrm{d}})=-\omega_{\mathrm{d}}\Delta\tau,$ (12)

where $\Delta\tau$ is the time lag between the two correlation functions. The
resulting Bode plot of the experimentally collected frequency-response data is
depicted in Fig. 4.

The phase for low frequencies corresponds to the integrating behavior.
However, high frequencies show an increasing time delay in terms of a phase
lag which is not covered by the second-order model (10). Only the magnitude
plot is therefore used for the transfer function estimation. The least-squares
optimal fit for $G$, using $\left|G(\mathrm{j}\omega_{\mathrm{d}})\right|$
with
$\omega_{\mathrm{d}}\in[4\,\mathrm{\frac{rad}{s}},380\,\mathrm{\frac{rad}{s}}]$,
results in $\sigma=80.4884$. This yields the

$K=0.0408\quad\text{and}\quad\tau=6.684\,\mathrm{ms}$ (13)

parameter values. The Bode plot of the estimated $G(j\omega)$ is equally
depicted in Fig. 4 over the measurements.

##  IV Robust Linear PID Control

In this section, a derivation of the two-step procedure for a robust PID
controller design, resting on an underlying PD control with stable pole-zero
cancelation and upper bound of the disturbance sensitivity function, is
provided.

###  IV-A Controller Design

In order to reduce the design complexity of the linear PID controller and,
correspondingly, limit the tuning degrees-of-freedom to only the overall
control gain and integration time constant, a stable pole-zero cancelation is
first performed. To this end, applying the underlying PD controller

$C_{\mathrm{PD}}(s)=\gamma(\tau s+1)$ (14)

with the proportional gain $\gamma$ and derivative time constant $\tau$, the
pole of the plant $G(s)$ at $p=-\frac{1}{\tau}$ is canceled. Therefore, only
the $\gamma$-gain can be used for further control tuning. Since disturbance
rejection is of our major interest, the input disturbance sensitivity function

$S_{\mathrm{yd}}(s)=\frac{G(s)}{1+G(s)C(s)}$ (15)

is considered. Utilizing the PD controller (14) results in

$S_{\mathrm{yd,PD}}(\mathrm{j}\omega)=\frac{-K}{(\omega\tau-\mathrm{j})(\omega-\mathrm{j}K\gamma)},$
(16)

with the corresponding magnitude response

$\left|S_{\mathrm{yd,PD}}(\mathrm{j}\omega)\right|=\frac{K}{\sqrt{\omega^{2}\tau^{2}+1}\sqrt{\omega^{2}+K^{2}\gamma^{2}}}.$
(17)

Note that this approaches its maximum at steady-state, i.e.

$\left|S_{\mathrm{yd,PD}}(\mathrm{j}\omega)\right|\Big{|}_{\omega=0}=\frac{1}{\gamma}>\left|S_{\mathrm{yd,PD}}(\mathrm{j}\omega)\right|\quad\forall\omega>0,$
(18)

which is equivalent to

$\left\|S_{\mathrm{yd,PD}}\right\|_{\infty}=\frac{1}{\gamma}.$ (19)

Therefore, for a given worst case amplification of the matched disturbance,
the proportional gain $\gamma$ is chosen as its inverse.

However, in order to ensure the steady-state accuracy, we make use of an
(improper) PID controller

$C_{\mathrm{PID}}(s)=\frac{\gamma(\tau
s+1)(T_{\mathrm{I}}s+1)}{T_{\mathrm{I}}s}$ (20)

with the integration time constant $T_{\mathrm{I}}$ as a further tuning
parameter. This leads to the corresponding sensitivity function
$S_{\mathrm{yd,PID}}$ with the magnitude characteristics

$\left|S_{\mathrm{yd,PID}}(\mathrm{j}\omega)\right|=\frac{KT_{\mathrm{I}}\omega}{\sqrt{(K\gamma-
T_{\mathrm{I}}\omega)^{2}+(KT_{\mathrm{I}}\gamma\omega)^{2}}\sqrt{\tau^{2}\omega^{2}+1}}.$
(21)

In order to utilize the previous considerations, we show that (21) is bounded
by $\left\|S_{\mathrm{yd,PD}}\right\|_{\infty}$, i.e.

$\left|S_{\mathrm{yd,PID}}(\mathrm{j}\omega)\right|\leq\frac{1}{\gamma}\quad\forall\omega>0$
(22)

holds true. This can be seen by considering

$\displaystyle\left|S_{\mathrm{yd,PID}}(\mathrm{j}\omega)\right|^{2}$
$\displaystyle=\frac{(KT_{\mathrm{I}}\omega)^{2}}{\left[(K\gamma-
T_{\mathrm{I}}\omega)^{2}+(KT_{\mathrm{I}}\gamma\omega)^{2}\right]\underbrace{(\tau^{2}\omega^{2}+1)}_{>1\,\forall\omega,\tau>0}}$
$\displaystyle\leq\frac{(KT_{\mathrm{I}}\omega)^{2}}{(K\gamma-
T_{\mathrm{I}}\omega)^{2}+(KT_{\mathrm{I}}\gamma\omega)^{2}}\equiv\Xi(\mathrm{j}\omega)$

and knowing that $K,T_{\mathrm{I}},\omega>0$, which yields

$\Xi(\mathrm{j}\omega)=\frac{1}{\left(\frac{K\gamma-
T_{\mathrm{I}}\omega}{KT_{\mathrm{I}}\omega}\right)^{2}+\gamma^{2}}\leq\frac{1}{\gamma^{2}}.$

Thus, the relation (22) becomes evident and can be used to determine an
appropriate proportional gain for a given minimal input disturbance
attenuation. The conservativeness of such design is discussed in section IV-B,
for the specific case of the given plant and value for $\gamma$.

The free tuning parameter $T_{\mathrm{I}}$ is chosen based on the frequency
characteristics of the open-loop and a predefined phase margin $\Phi$. The
open-loop transfer function for zero initial conditions is given by

$L(s)=C_{\mathrm{PID}}(s)\,G(s)=\frac{K\gamma(T_{\mathrm{I}}s+1)}{T_{\mathrm{I}}s^{2}}$
(23)

and consists of a double-integrator and one stable zero. Therefore, the
corresponding phase never crosses $-180^{\circ}$, leading to a (theoretically)
infinite gain margin. However, the phase margin $\Phi$ can be used as a
further robustness measure, especially with respect to additional phase lags.
For this purpose, consider the amplitude and phase at open-loop crossover
frequency $\omega_{\mathrm{s}}$, given by

$\displaystyle\left|L(\mathrm{j}\omega_{\mathrm{s}})\right|$ $\displaystyle=1$
(24a) $\displaystyle\pi+\arg(L(\mathrm{j}\omega_{\mathrm{s}}))$
$\displaystyle=\Phi.$ (24b)

From (23) it can be seen that

$L(\mathrm{j}\omega)=\frac{-K\gamma(1+\mathrm{j}T_{\mathrm{I}}\omega)}{T_{\mathrm{I}}\omega^{2}}=-\frac{K\gamma}{T_{\mathrm{I}}\omega^{2}}-\mathrm{j}\frac{K\gamma}{\omega}$

and, hence, at $\omega_{\mathrm{s}}$ one obtains

$\displaystyle|L(\mathrm{j}\omega_{\mathrm{s}})|$
$\displaystyle=\frac{K\gamma\sqrt{1+\omega_{\mathrm{s}}^{2}T_{\mathrm{I}}^{2}}}{\omega_{\mathrm{s}}^{2}T_{\mathrm{I}}}\quad\text{and}$
(25a) $\displaystyle\arg\left(L(\mathrm{j}\omega_{\mathrm{s}})\right)$
$\displaystyle=\arctan\left(T_{\mathrm{I}}\omega_{\mathrm{s}}\right)-\pi,\quad
T_{\mathrm{I}},\omega_{\mathrm{s}}>0.$ (25b)

The combination of (24) and (25) yields

$\displaystyle\omega_{\mathrm{s}}$
$\displaystyle=\frac{K\gamma\sqrt{1+\tan^{2}(\Phi)}}{\tan(\Phi)}\quad\text{and}$
(26a) $\displaystyle T_{\mathrm{I}}$
$\displaystyle=\frac{\tan(\Phi)}{\omega_{\mathrm{s}}}.$ (26b)

The above developments result in the two-step procedure for a dedicated PID
controller design:

  1. 1.

Set worst case disturbance amplification $S_{\mathrm{max}}$ leading to

$\gamma=\frac{1}{S_{\mathrm{max}}}.$ (27)

  2. 2.

Set the desired phase margin $\Phi$ and calculate

$T_{\mathrm{I}}=\frac{\tan^{2}(\Phi)}{K\gamma\sqrt{1+\tan^{2}(\Phi)}}.$ (28)

With the parameters $\tau$, $\gamma$ and $T_{\mathrm{I}}$, the PID controller
(20) is entirely determined.

###  IV-B Application of PID Controller

The design approach developed above is now utilized to determine the PID
controller for the identified plant model $G$. For a good disturbance
rejection and robustness, we choose

$S_{\mathrm{max}}=-52\,\mathrm{dB}\quad\text{and}\quad\Phi=60^{\circ}$

leading to

$\gamma=400\quad\text{and}\quad T_{\mathrm{I}}=0.092\,\mathrm{s}.$ (29)

The magnitude plot of $S_{\mathrm{yd}}$, using the underlying PD controller,
is depicted in Fig. 5 opposite to the PID controller for a large variation of
the integration time constant $T_{\mathrm{I}}$. Obviously, the maximum
disturbance amplification is only little dependent on the value of
$T_{\mathrm{I}}$, while a clear asymptotic reduction for lower frequencies is
evident. The PID controller with $T_{\mathrm{I}}=0.1$ s, cf. with the
determined parameters (29), provides certain optimality in shaping the
disturbance sensitivity function. That means without sharp peak of
$|S_{\mathrm{yd}}|$ and without a flat plateau of
$|S_{\mathrm{yd}}|\rightarrow S_{\max}$ upper bound dictated by the underlying
PD control.

Figure 5: Magnitude plot of the input disturbance sensitivity function using a
PD controller opposite to PID controllers with the different integration time
constants $T_{\mathrm{I}}\in[0.01\,\mathrm{rad/s},~{}10\,\mathrm{rad/s}]$.

A closer look at the $\mathcal{H}_{\infty}$ norms for a variation of
$T_{\mathrm{I}}$ in the relevant range, see Fig. 6, illustrates (19) and (22).
Worth noting here is that for this set of parameters, the above mentioned
conservativeness of (22) is limited to $0.31\,\mathrm{dB}$.

Figure 6: $\mathcal{H}_{\infty}$ norms of the disturbance sensitivity
functions from Fig. 5.

For the ease of implementation and better comparison with the PID-like sliding
mode control, the parallel form of the PID controller $C_{\mathrm{PID}}$ (20)
reads as

$\bar{C}_{\mathrm{PID}}(s)=K_{\mathrm{p}}+K_{\mathrm{I}}\frac{1}{s}+K_{\mathrm{D}}s,$
(30)

with the proportional, integral and derivative gains as

$\displaystyle K_{\mathrm{p}}$
$\displaystyle=\gamma\frac{T_{\mathrm{I}}+\tau}{T_{\mathrm{I}}}=429.064,$ (31)
$\displaystyle K_{\mathrm{I}}$
$\displaystyle=\frac{\gamma}{T_{\mathrm{I}}}=4348.267,$ (32) $\displaystyle
K_{\mathrm{D}}$ $\displaystyle=\gamma\tau=2.674.$ (33)

Here worth mentioning is that the parameters result directly from a comparison
of coefficients between $C_{\mathrm{PID}}$ and $\bar{C}_{\mathrm{PID}}$.

##  V Experimental Study

In this section, we provide an experimental comparison of the PID controller,
designed according to section IV, and the PID-like CSMC controller, cf.
section II. At first, we recall and tune a robust exact differentiator, which
is required for both controllers in order to use $\dot{y}(t)$ in feedback.
Then, we show an experimental evaluation of the PID controller that confirms
analysis and performance of the two-step procedure developed in section IV.
Finally, we compare both the PID and CSMC controllers in suppressing a
broadband disturbance.

Figure 7: Magnitude plot of the $d$ to $y$ transfer characteristics: the
disturbance sensitivity function $S_{yd}$ with PID control versus the
amplitude-dependent harmonic balance approximation with CSMC control, for
three exemplary amplitudes $3A$, $A$, $A/3$ of steady-state $y$ (differing by
order of magnitude).

For the designed CSMC and PID controllers, the $d(j\omega)$ to $y(j\omega)$
transfer characteristics, with the describing function-based approximation (5)
and disturbance sensitivity function $S_{yd}(j\omega)$, respectively, are
shown in Fig. 7. For the PID control loop, it shows a maximum of
$\hat{S}_{\mathrm{max}}=-52.08\,\mathrm{dB}$ at
$\omega_{\mathrm{max}}=13.31\,\mathrm{rad/s}$. Therefore, the largest impact
on the output value is expected from the disturbances with a frequency around
$\omega_{\mathrm{max}}$. The magnitude plot of the amplitude-dependent
harmonic balance approximation for CSMC control, i.e. (5), is shown for
steady-state of $y$ with the exemplary amplitudes $A=0.001$ m and its $3A$ and
$A/3$ variations, cf. with Fig. 10. Shrinking of the disturbance sensitivity
characteristics of CSMC control is clearly visible for amplitude reduction,
marked by an arrow in Fig. 7. One can recognize that starting from certain
frequencies (see rectangular mark in Fig. 7) the expected disturbance
rejection becomes better comparing to PID. It is also evident that for a
further decreasing amplitude of $y$-oscillations, the $|d/y|$ plot of CSMC
will lie entirely below the $|S_{yd}(j\omega)|$ plot of the linear feedback
control.

###  V-A Robust Exact Differentiator

In order to obtain an estimate of the output derivative $\dot{y}(t)$, to be
practically suitable for the feedback loop, we apply a robust exact
differentiator [21] (further as RED), which is based on the sliding-mode
estimation principles, see e.g. [1]. Its remarkable features are an
insensitivity to the bounded noise (provided the Lipschitz constant of n-th
time-derivative is available) and a finite-time convergence. This makes a RED,
at least theoretically, free of a phase lag. Recall that the latter is
otherwise deteriorating the loop performance if a low-pass filtering of the
differentiated measurement of $y(t)$ is used. The second-order RED, cf. [22],
with the parametrization according to [23] is given by

$\displaystyle\dot{x}_{0}$
$\displaystyle=x_{1}+3.1~{}r~{}|y-x_{0}|^{\frac{2}{3}}~{}\mathrm{sign}(y-x_{0}),$
(34a) $\displaystyle\dot{x}_{1}$
$\displaystyle=x_{2}+3.2~{}r^{2}~{}|y-x_{0}|^{\frac{1}{3}}~{}\mathrm{sign}(y-x_{0}),$
(34b) $\displaystyle\dot{x}_{2}$
$\displaystyle=1.1~{}r^{3}~{}\mathrm{sign}(y-x_{0}),$ (34c)

where the scaling factor $r>0$ is the single design parameter. It is worth
noting that $r^{n+1}$ (in our case $r^{3}$) corresponds to the Lipschitz
constant $L$ of the highest derivative ${y}^{(n)}$, cf. [23].

Figure 8: Exemplary evaluation of the 2nd order RED with $r=8$: the measured
output value in (a) and RED estimate $x_{1}$ over the low-pass (LP) filtered
discrete time derivative of $y$ in (b).

Note that the second-order (and not first-order) RED is purposefully used
here, in order to obtain a smoother estimate $x_{1}(t)$ of the output
derivative. Recall that the second-order RED provides $x_{0}(t)=y(t)$,
$x_{1}(t)=\dot{y}(t)$, $x_{2}(t)=\ddot{y}(t)$, for all $t>t_{c}$, where
$t_{c}$ is a finite convergence time. The scaling factor $r=8$ was
experimentally tuned on the collected $y(t)$ data, for which an up-chirp
signal until $\omega=80$ rad/s disclosed a still satisfying match between the
$x_{1}(t)$ estimate and theoretically calculated $\dot{y}(t)$. An exemplary
evaluation of the RED (34) with $r=8$ is shown in Fig. 8, in comparison with
the discrete time derivative of the measured signal $y$, which is then low-
pass filtered with the cutoff frequency of 100 Hz.

###  V-B Evaluation of PID Controller

In order to evaluate efficiency of the proposed PID design procedure, as well
as the accuracy of the model fit and the associated linear loop shaping, cf.
section IV, the sinusoidal input disturbance $d(t)$ (given by eq. (11)) is
used in experiments. The assigned amplitude is $A_{\mathrm{d}}=1\,\mathrm{V}$
and the frequencies selected around $\omega_{\mathrm{max}}$ are

$\omega_{\mathrm{d,1}}=5\,\mathrm{\frac{rad}{s}},\quad\omega_{\mathrm{d,2}}=10\,\mathrm{\frac{rad}{s}}\quad\text{and}\quad\omega_{\mathrm{d,3}}=20\,\mathrm{\frac{rad}{s}}.$

The experimental results are depicted in Fig. 9 by disclosing the control
errors. Evaluating the theoretically expected disturbance attenuation at
$\omega_{\mathrm{d,1,\ldots,3}}$ i.e.
$|\tilde{S}_{\mathrm{yd}}(\mathrm{j}\omega_{\mathrm{d,1,\ldots,3}})|=\\{-58.6,\,-52.9,\,-53.8\\}\,\mathrm{dB}$
and comparing these values with the corresponding numbers calculated out from
the experiments $\\{-57.5,\,-52.6,\,-50.1\\}\,\mathrm{dB}$, cf. Figs. 7 and 9,
one can recognize a good accordance between both.

Figure 9: Measured control error $e$ of $\bar{C}_{\mathrm{PID}}$ for the
disturbance attenuation.

###  V-C Controllers Comparison for Broadband Disturbances

The CSMC and PID controllers, both designed according to sections II and IV
respectively, are evaluated experimentally and compared to each other in the
disturbance rejection. The applied disturbance signal is an up-chirp

$d(t)=A_{d}\sin\bigl{(}(\omega_{0}+\alpha t)\,t\bigr{)}$

with a linearly increasing frequency, i.e. $\alpha>0$, and the start and end
frequencies of 0.06 rad/sec and 30 rad/sec, correspondingly. The chirp runtime
of 60 sec ensures that the transient, correspondingly convergence, phase of
the control response (which one is relatively long for the realizable gain
values of CSMC) is passed at lower frequencies, see Fig. 10 (a). Following to
that, a steady-state disturbance rejection behavior can be compared for the
times about $t>10$ sec.

Figure 10: Experimental results of the PID and CSMC controllers on rejection
of the up-chirp disturbance with bandwidth 0.06–30 rad/s: the control error
(around the reference position $y_{0}=9$ mm) in (a) and control signal in (b)

It can be seen that the error pattern of the PID control follows the expected
shape of the disturbance sensitivity function, cf. Fig. 7. On the contrary,
the CSMC control error is continuously decreasing with an increasing
frequency, that corresponds to the CSMC sensitivity function charts that
depend of the amplitude of the main harmonic of chattering at steady-state,
cf. section II. The period where the control error of both PID and CSCM are
approximately the same, i.e. at the time about 24 sec, corresponds to the
disturbance frequency $w_{d}\approx 9.5$ rad/sec, cf. Fig. 7. Regarding the
control values, see Fig. 10 (b), one can recognize that for the PID control,
the signal pattern is also inline with the frequency response of the closed-
loop system (cf. Fig. 2). For that, the transfer characteristics between
$d(j\omega)$ and $u(j\omega)$ are indicative. One can recognize that with an
increasing angular frequency, the amplitude of the PID control value $u(t)$ is
also growing, thus resulting in a higher power and, correspondingly, energy
consumption. On the contrary, the control value of CSMC keeps an almost
constant amplitude level, that speaks for a lower power consumption. Also
recall that for finite-time convergence of CSMC the control signal tends to
the opposite value of perturbation, that is the case with a chirp disturbance
of a constant amplitude, Fig. 10 (b).

Figure 11: Comparison between the simulated and experimental response of CSMC
control; experimental data are from higher-frequencies from Fig. 10.

The comparison between the simulated and experimental response of the closed
loop with CSMC controller is exemplary shown in Fig. 11. Here the same data of
a higher-frequency clipping from Fig. 10 is used. Note that in the numerical
simulation of the system (7), augmented by $d$ with the control (2), an
additional band-limited white noise is added to the output $y(t)$, so as to
bring the modeled control system behavior closer to the real experiments. Up
to the noise level, the simulated and experimental response of the CSMC-
controlled output coincide well with each other, cf. Fig. 11.

##  VI Conclusions

A PID-like continuous sliding mode (CSMC) controller, known to be robust for
rejection of the matched Lipschitz continuous perturbations, was analyzed in
terms of a disturbance sensitivity function. This frequency-domain approach,
which takes additionally into account the amplitude of steady-state harmonics
of the CSMC control loop (when perturbed by the actuator) allows a comparison
with standard linear feedback controllers. Due to the structural similarity,
i.e. with feedback of the output, its derivative, and integral, the CSMC was
compared with a standard linear PID control, designed also for rejection of
the matched disturbances. Both controllers were compared experimentally as the
fair competitors. The developed two-step procedure for a robust PID controller
design yields _two tunable parameters_ , the overall control gain and the
integration time constant. Both are shown as associated with the H-infinity
norm of the disturbance sensitivity function and with the desired phase
margin, respectively. The CSMC control parametrization follows exactly the
analysis and developments provided in [12], yielding only _one tunable
parameter_ , which is the scaling factor out from the Lipschitz constant of
perturbations. The comparative study of both controllers is made for the
second-order experimental motion system with additional fast parasitic
dynamics of the actuator. The two-parameters linear model of the system plant
is identified in frequency domain and shown to be sufficient for design of the
feedback controllers. For the time derivative of the output, required for both
PID and CSMC controls, the robust second-order sliding-mode differentiator is
designed and used in experiments. The PID and CSMC controllers are evaluated
experimentally on rejection of a broadband input disturbance, which is an up-
chirp between 0.06 rad/sec and 30 rad/sec. The resulted control performance is
shown to be fully inline with the theoretically expected (i) disturbance
sensitivity characteristics (i.e. sensitivity function) of the PID control
loop, and (ii) describing-function based prediction of the steady-state
harmonic oscillations (i.e. chattering) of the CSMC control loop. The CSMC
disturbance rejection performance is shown to be clearly less frequency-
sensitive in the control error pattern, and more energy efficient as for the
control signal amplitude. At the same time, the transient behavior of CSMC
control disclosed inferior, comparing to the PID one, that for the maximal
achievable $L$-scaling factor of the CSMC parametrization. It can be noted
that describing the propagation of slow signals through the oscillating
system, correspondingly transient behavior, might require the concept of an
equivalent gain, cf. [24]. This is, however, out of scope of the present work.
The $L$-scaling factor, also related to the actuator dynamics, proved
experimentally to be most sensitive for application of the CSMC control. In
summary, the demonstrated practical comparison of PID and CSMC controllers
allows for a better understanding and distinction of the application benefits
and challenges of their use in compensating for amplitude- and band-specific
disturbances.

## References

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tcb@breakable

# Revisiting Human Information Foraging: Adaptations for LLM-based Chatbots

Sruti Srinivasa Ragavan<EMAIL_ADDRESS>Indian Institute of
TechnologyKanpurIndia and  Mohammad Amin Alipour  University of
HoustonHouston, TXUSA

###### Abstract.

Information Foraging Theory’s (IFT) framing of human information seeking
choices as decision-theoretic cost-value judgments has successfully explained
how people seek information among linked patches of information (e.g., linked
webpages). However, the theory stands to be adopted and validated in non-
patchy LLM-based chatbot environments, before its postulates can be reliably
applied to the design of such chat-based information seeking environments.
This paper is a thought experiment applying IFT’s cost-value proposition to
LLM-based chatbot environments and presents a set of preliminary hypotheses to
guide future theory-building efforts for how people seek information in such
environments.

##  1\. Introduction

The abundance and centrality of information in day-to-day tasks marks the
information age we live in, and humans are dubbed ”informavores” (Miller,
1983). The new intelligent LLM-based chatbots (e.g.
ChatGPT111https://chat.openai.com/), have transformed how people search for
information: they are able to comprehend users’ query prompts for general-
purpose information and use the interaction context to produce better quality
responses. In turn, this has led to proliferation of new interactive systems
in different domains (e.g., (Chen et al., 2023; Zhou et al., 2023; Liu et al.,
2023)). These systems seem to produce “reasonable” answers with arguably vague
or incomplete prompt instructions, and seem to facilitate easy and quick
acquisition and processing of information by their user(Cambon et al., 2023).

However, it is unclear how information seekers adapt to these new interaction
paradigms, such as cognitive adaptations to effectively learn and use these
tools for information seeking (Anderson, 2013), or to deal with their unique
limitations (e.g., inaccuracies, nondeterminism) (Sarkar et al., 2022), or the
metacognitive effects (e.g., perception of acquired skills) are unclear
(Eloundou et al., 2023; Amoozadeh et al., 2023). This lack of understanding
warrants more fundamental inquiries into the nature of work with such
systems–both by building new theories (e.g., (Barke et al., 2023)) and by
revisiting relevant classical theories–for human information acquisition,
information processing, and information refinement and retention. In this
paper, we undertake one such exercise in the latter category and revisit a
well-established theory in HCI, namely the Information Foraging Theory, in the
context of these LLM-based chatbots.

Information Foraging Theory (IFT) is a theory of how people seek information.
Proposed by Pirolli and Card in 1995, IFT is an adaptation of the optimal food
foraging theories in biology (Pirolli and Card, 1995). IFT posits that
informavores forage for information in ways similar to how animals forage for
their prey. Specifically, in both cases, the forager aims to solve an
optimization problem: the foraging animal aims to optimize the rate of energy
gain (i.e., energy gained per unit of energy expended hunting and consuming
the prey), whereas the informavore attempts to maximize the rate of
information gain (amount of valuable information gained per unit foraging
cost).

The theory also describes how this cost-value optimization plays out in a
specific kind of environment – namely, where the information is organized as
“patches”, and the patches are interconnected by ”links”. In such
environments, according to IFT, the forager follows a scent-following
strategy, wherein they metaphorically ”sniff on” (or attend to) ”cues” that
signal the value-cost payoffs of a foraging choice (e.g., text on hyperlink
signals what is on the other end of the link). They then follow the path that
smells strongest of the prey, much like the scent following behavior of
animals. The theory also makes predictions about how much time a forager will
spend on one information patch – once again framing it in terms of the value
obtained, as a function of cost (including the cost of lost opportunity in
other costs).

Prior work has explored the validity of IFT in other types of information
environments than on which the theory was developed (namely, interconnected
filesystems and the web (Pirolli and Card, 1995)). Examples include those with
heterogeneous patches (text, images, animations, etc.) (Pirolli and Card,
1995), variationed environments having several variants (or versions) of the
same information patches (Srinivasa Ragavan et al., 2016), and in the case of
screen readers (Vigo and Harper, 2013) such as for the visually impaired.
Whereas in the former two cases, foragers followed scent albeit with some
special adaptations by the forager, there were no scent following mechanisms
with screen readers and perhaps even no cost-value optimization due to the
design of the environments. Thus, with every new kind of information
environment, there is a need to test the validity of the theory.

To this end, this work is a prelude to our ongoing research on validating IFT
for LLM-based chatbot environments, especially those powered by Large Language
Models (LLMs) (e.g., ChatGPT). Our key research questions are as follows:

  * •

Does IFT’s cost-value optimization proposal apply to information seeking
inLLM-based chatbots?

  * •

If so:

    * –

what specific cost-value optimization mechanisms do users adapt to effective
forage for information in LLM-based chatbots?

    * –

What do these mechanisms tell us about how to design LLM-based chatbot
environments to better suit user needs?

  * •

If not:

    * –

where does information seeking in LLM-based chatbots deviate from IFT, and
what is another possile theory for such kinds of information seeking (or
perhaps, more aggressively, is there a more unifying theory for human
information seeking!).

In the rest of this paper, we revisit the roots and assumptions of Information
Foraging Theory, discuss the nature of information seeking in QnA, in
comparison with traditional notions of foraging (particularly in the ”patchy”
web), propose a list of initial hypotheses about human information foraging in
LLM-based chatbots, and finally. We conclude with a discussion on the
challenges and consequences of applying IFT to LLM-based chatbot contexts.

##  2\. Background: Information Foraging Theory?

Information Foraging Theory (or IFT) concerns with how people seek information
as part of a larger task. With roots in evolutionary psychology, IFT assumes
that humans have exapted their food-foraging adaptations to aid their
information seeking (Pirolli, 2009); thus IFT itself is an adaptation of the
optimal foraging theory from biology (Stephens and Krebs, 1986). In fact, IFt
uses the constructs predator and prey to refer to human information seekers or
foragers, and to the information they are looking for, respectively.

IFT has three key propositions (listed below). These propositions are not only
descriptive–in that they explain how humans forage for information, but also
explanatory, in that they offer explanations for why a forager might engage in
the foraging behaviors they do. This explanatory power is then leverageable
for practical purposes, such as to predict how a person would forage in a
given environment (such as to evaluate its usability), or as a corollary to
design information environments that are better suited to human foraging
behaviors.

####  2.0.1. Value-cost optimization

: Central to IFT is the cost-value proposition which says that, analogous to
predatory animals maximizing the rate of energy gain during foraging,
informavores foraging for information adopt mechanisms to optimize the rate of
information gain (or the amount of valuable information per unit cost). It is
important to note that despite efforts, foragers might not always be
successful in their cost-value optimization attempts, and this is largely due
to incomplete information about their environment (for example, the cheapest
way to get to an information piece, or the ability to estimate what
information is in a webpage). In other words, there is a distinction between
predator’s estimates or expectations of costs and values from various foraging
choices, and the actual costs and values in the foraging; only when the
estimates and actual match do a forager actually engage in optimal user
experiences and efficient foraging.

####  2.0.2. Patch models

: The second set of IFT propositions assumes that information in the
environment is organized as “patches” (e.g., web pages, files). In turn, these
patches could form a hierarchy (e.g., Internet contains several domains
containing many subdomains containing many websites containing many pages each
containing sections, etc.). In fact, such hierarchies are a common way in
which complexity is organized in several information environments (folder
structure in computers, sections in large documents, websites, sections, and
chapters in books and libraries).

By laws of evolutionary psychology, humans adapt themselves to their
environment, and in this case, they have adaptation mechanisms to optimally
forage for information in such patchy heirarchies. The patch models of IFT
seek to explain what these adaptations are.

  * •

Between-patch foraging times: In a patchy information environment, a predator
must choose among several available patches (e.g., which page among hundreds
of search results), so as to maximize the rate of information gain. For this
between-patch foraging, IFT says, a predator relies on the scent from various
cues in the environment, and follows the path with strongest scent of prey.
For example, in web search results, a person might look at the ratings of a
webpage, or look at its preview or a snippet to guess what the page contains,
and whether to click on the link or not.

  * •

Within-patch foraging times: Having navigated to a patch, the forager must
then gather information foraging within that patch (including any sub-patches
therein). How long a predator spends doing so, before leaving the page for
another one, is dependent on the average rate of gain of information. The
longer a predator spends in a page, the more information s/he would consume,
thereby diminishing the value of a page. Thus, a predator will remain in a
patch until the average rate of information gain in the current patch is
greater than what s/he expects to get from find, navigating to and foraging in
another patch.

Indeed, both these activities are tedious when a predator must seek very
specific information in too many large patches. Therefore, another cost-value
optimization strategy is called ”enrichment”. Here, a predator aims to improve
the value or lower the cost of foraging by engaging in activities such as
search, filtering, sorting, or even customizing the environment (e.g., split
tab to constantly lookup something, filter to get only relevant information).

####  2.0.3. Diet model

: The third proposition of IFT is the diet model. This model also describes
cost-value adaptations of information foragers, but cases where there are
multiple possible prey(s) or prey kind(s) available (for example, variety of
news, emails, books to choose from). In such cases, IFT predicts what kinds of
”prey” a predator chooses to be part of the optimal diet for the forager: that
a rational forager will select high-value prey(s) and leave out low-value
prey(s), irrespective of cost, assuming similar occurrence of both high- and
low-value prey(s). For example, a rational and task-oriented person would
never read what he/she knows to be a spam email (unless the task itself has to
do with reading the spam email). In fact, there has even been recent work on
what diet is part of the foraging of AI users when seeking explanations about
the output of an AI system, as part of efforts in eXplainable AI (XAI) systems
(Dodge et al., 2018).

##  3\. Information Foraging: Web vs. LLM-based chatbots

###  3.1. Information seeking on the Web

We mentioned earlier that the World Wide Web (WWW) is a patchy structure, and
that the patches for a hierarchy: for example, web domains contain subdomains
contain websites contain web pages contain sections. response,that Webby, a
new college graduate, is foraging the WWW to learn about mutual funds and
investments after earning her first paycheck. She starts by going to an
Internet Search Engine (e.g., Google 222www.google.com) and types in ”mutual
funds”. She receives a set of links to various pages that contain information
relevant to her search query. Webby then engages in between-patch foraging via
scent-following: she skims through the search results, paying attention to the
link title, the snippets of content below them, or the reputation of the site
and her past experiences with it, and clicks on one that seems potentially
useful (high value) and/or easy to grasp (low cost).

Once on a page, Webby then reads the content skimming, scanning, and skipping
ahead as needed–all ways of reducing cost. Then, when she feels she has
gathered enough from the page, she leaves to go back to the search results
page, and then the cycle continues. Sometimes, while on a page, Webby finds a
link that seems promising, and she clicks on it, and other times, she lands on
a page and is disappointed at its content and immediately abandons the patch.

At this point, it is easy to see that all these behaviors are cost-value
optimizations–the mechanisms of which are explained by IFT’s patch and diet
models (Pirolli and Card, 1995).

###  3.2. Information seeking with LLM-based chatbots

Now, let us consider the case of another forager, Chad. Like Webby, Chad also
wants to start investing in mutual funds and first wants to learn about them.
But unlike Webby, Chad goes to ChatGPT and prompts it with a question: “What
are mutual funds?”. In response, the LLM-based chatbot provides a detailed
answer about what mutual funds are. Both Chad and the LLM-based chatbot go
back and forth until he feels he has learned about mutual funds.

Chad’s interactions with ChatGPT look as follows:

  * •

Sometimes, Chad gets lucky and gets the right answer the very first time, and
is able to grok the response quickly and move on.

  * •

Often, Chad reads the LLM-based chatbot’s response to a prompt and decides
that it is not what he wanted, and therefore reframes his prompt to better
specify his needs.

  * •

He sometimes also finds the content too hard to grasp (costly) or not quite
relevant, and so he refines the prompt, or asks a follow-up mimicking human-
human conversation (e.g., ”can you put it in simpler terms?”, ”explain”, ”what
are mutual funds in simple terms?”, ”oh, I meant apple the computer and not
the company”).

  * •

Occasionally, Chad might see something suspicious or hard to believe, and
might ask the chatbot for further clarifications, reasoning, or evidence (or
check its veracity elsewhere).

  * •

Sometimes, Nash also wants to go back to a thread he left, but repeating the
same prompt or something similar does not always guarantee the same answer,
and so he cannot exactly pick up from where he left. He tries and gives up,
and plows through the chat history to go back to where he left.

All of the above behaviors described for Chad are based on our ongoing lab
study comparing information seeking between Web and LLM-based chatbot, and
from prior studies (e.g., (Skjuve et al., 2023; Menon and Shilpa, 2023)).

###  3.3. Information seeking similarities: Web vs. LLM-based chatbots

Comparing information seeking on the Web and in LLM-based chatbots, several
similarities and differences become apparent; the similarities are described
in this section, and the differences are described in the next section.

####  3.3.1. Similarity #1: Forager specifies foraging goals.

First, in both the Web and the Q&A chatbot, the information seeking starts
with a forager specifying the foraging goal. Whereas in the search engine,
this is typically keywords in the foraging goal, the expected prompt in the
chatbot is a question, such as in human-human conversations. 333Indeed,
sometimes a web forager does not use a search and instead directly goes to a
website and follows the scent to their prey–however, that is arguably less
common that using a search engine.

####  3.3.2. Similarity #2: Foraging goals evolve.

Second, in the case of both the web and LLM-based chatbot, the foraging goal
evolves. This could be in the form of seeking clarifications about some
information the forager had just gathered, or asking follow-up questions that
in some cases can be formed as the next subtask, based on what has been found.
Both environments cater to such kinds of foraging – by way of refining the
query, or asking a follow-up question (e.g., “what does X mean?”, “why do you
say X?”), or sometimes by clicking on a link on the Web (such as hovering over
a word for meaning, or citation). In fact, one might argue that this is a
fundamental characteristic of information seeking, when the prey is ill-
defined.

####  3.3.3. Similarity #3: Value-cost judgments guide foraging.

Third, information seeking in both the Web and LLM-based chatbot is driven by
value-cost judgments. For example, on the web, participants seek pages that
seem easy to understand based on snippets of search results. Similarly, in the
chatbot environment, participants read the first few lines of the response, or
eyeball the structure and length, to estimate whether the response is
relevant, or too hard, and then seek ways to optimize it (for example,
reframing prompt to simplify something).

Similarities: Web vs. LLM-based chatbot foraging. • Foragers begin foraging by
specifying a verbal description of the prey. • Foraging goals evolve as the
forager gathers information, as well as in response to the environment. • The
fundamental cost-value optimization (i.e., maximizing value-to-cost ratio) is
central to what drives foraging behaviors in both environments.

###  3.4. Information seeking differences: Web vs. LLM-based chatbots

In terms of differences, we observe distinctions in the environment as well as
foraging behaviors, and we frame them as hypotheses (within the framework of
IFT) that we aim to empirically verify, dismiss or refine, in future work.

####  3.4.1. Difference #1: Prey specification.

Both in the preliminary results of our study, as well as in prior work
(sarkar2022), there is some evidence that new users of LLM-based chatbots
initially treat them similar to web search engines, and provide search
keywords – eventually gravitating towards asking questions that the LLM-based
chatbot environments are built for. In other words, in an LLM-based chatbot
environment, a user is _expected to know_ what they are looking for and to
specify it clearly (e.g., ”where is Paris located?”), rather than just words
about which they might be interested (e.g., ”Paris” or ”Paris location”).
Framed in terms of costs and values, it therefore appears that the former
questioning of chatbots (e.g., via prompt engineering) is harder than keyword-
based web search.

Hypothesis 1: The cost of specifying prey, or enrichment to improve overall
value of information in the environment, is higher in the LLM-based chatbot
environment (via prompt engineering) than in the web environment (via web
searches).

####  3.4.2. Difference #2: Structure of the information environment.

A second point of difference between the web and the chat environment is based
on the structure. We already saw that the web is organized as hierarchical
patches, and that the prey lives in one or more of those patches. In contrast,
the chat is one long patch, that evolves and changes with each interaction
between the user and the LLM-based chatbot environment. Notably, as each new
question is asked and responses are generated, the new information is appended
to the same patch. One could argue that each question and answer are a smaller
patch, but there are no explicit boundaries (e.g., each question-response pair
can be considered a patch, but what about follow-up questions?). Moreover,
there are also limited enrichment options (e.g., filtering out irrelevant
patches, or removing them) compared to traditional patchy environments. In
fact, even considering other human-human conversations, the resultant
transcript is perhaps considered as one single patch, rather as multiple pages
on the web, or files in a folder. However, whether users perceive chat as one
large ”blob” or as logical patches remains to be empirically understood.

Hypothesis 2: Whereas users treat the LLM-based chatbot environment has a
single constantly evolving patch, the web environment is a collection of
multiple patches of varying size

####  3.4.3. Difference #3: Transience of patches.

In WWW, the prey is nearly ”permanent”; a forager can find their way to the
same prey, any number of time times, such as by bookmarking, or leaving an
open tab, or a new similar web search – both within the same session and
another future session. In contrast, in ChatGPT or other similar LLM-based
chatbot environments, the prey is ephemeral – it is created on the fly in
response to a user prompt and context, a user might not be able to get the
same prey by typing in the same prompt. They might have to go through the
tedium of finding it in chat history (provided that it is saved, and not too
long!).

This has notable consequences for cost-value estimation. First, some prey
might be very hard or impossible to get; this can be framed as infinite cost,
or zero availability of the prey. In driving this down, a predator might still
engage in very costly forms of foraging, such as scrolling through chat
history, constantly remembering to bookmark, or spending effort in prompt
engineering or finally adapting and reorienting themselves to a new prompt’s
response. Second, repeated experiences of this nature might affect the users’
estimations of costs and values in the future, and as we shall see later in
this section, trust has a role to play in this regard.

Hypothesis 3: When prey is scattered across responses or patches, the overall
foraging costs are higher in LLM-based chatbot environments than in the web.

.

####  3.4.4. Difference #4: What information is available.

Another distinction is in what is in the information environment. In the web,
there exist several patches that a predator must forage to piece together the
prey. There might even be multiple alternative prey and prey types (e.g.,
video, text, audio) and a predator has access to all of this prey, and can
even see the presence of what else might be available (even when they are not
foraging in a patch). In other words, a predator is aware of, and makes
decisions about what opportunities to pursue, and what opportunities they lose
and at what cost.

In contrast, in the chat environment, there is a specification of the prey,
and then one instance of the prey presented in one coherent manner. This has
two consequences:

  * •

The forager does not know what other alternative prey(s) exist, and therefore
a predator has no sense of opportunity cost, nor knows whether what they
actually get as response is really the most profitable.

  * •

The lack of diversity in prey results in a lack of diversity in diet (e.g.,
multiple viewpoints).

Hypothesis 4: Foraging in LLM-based chatbots blinds foragers to other, more
profitable and diverse diets compared to the web.

####  3.4.5. Difference #5: Cost-value judgment adaptations

This also brings us to the next difference in foraging. When foraging in the
web, a forager has different patches to pick from; they consider the costs and
values of those patches before making a decision to navigate into one and
consume its content or not. This is done by scent-following: where scent is a
predator’s estimate of what is in a patch even before getting there.

Then, once they enter a patch, they do another evaluation (such as by seeing
length of page, or nature of content) to see whether the actual costs and
values are as per their prior expectations, and if not, they leave the patch,
and readjust their expectations of costs and values of that patch for future
foraging decisions.

In contrast, in LLM-based chatbot environments, there are no diverse patches
to choose from, and a predator has no way of gathering diverse prey(s) or even
knowing that diversity in prey(s) exists. Moreover, there is not even a real
patch whose cost and value a forager can estimate before deciding whether to
navigate or not. We believe that the user’s adaptation to such ecological
uncertainty is trust.

Before performing the action of what to prompt the chatbot with, and what to
expect, the predator has a certain level of trust in the LLM-based chatbot. We
hypothesize that this trust is effectively a predator’s expectation of cost
and value (e.g., based on hearsay, reputation, past experiences) about what to
expect from the LLM-based chatbot, or what prompt would generate what
response, and with what accuracy. Then, once they do generate the patch, they
then evaluate the content for both relevance and accuracy – based on the cost-
value payoffs expected (cost of errors, prior trust)– in turn recalibrating
their expectations of costs and values for future foraging.

Hypotheses 5:  • Cost-value estimations in LLM-based chatbots include
estimations of relevance and accuracy. • To cope with high levels of
uncertainty and lack of other patches in LLM-based chatbot environments,
foragers use trust to make preliminary cost-value estimates, such as for what
response a prompt will provide, or how much to trust a response. (This is
analogous to scent following adaptation for too many patches in the web!). •
Once within a patch, a forager will evaluate costs and values, both in terms
of relevance, as well as accuracy (just as they might do on the web); this
investment is based on the potential costs of mistakes. This happens by
skimming through the content, presence of cues such as such as citations,
seeking additional information (e.g., counterfactuals and explanations in
chatbot or fact checking in another environment). • Foragers’ trust (cost-
value estimates based on past experiences) in the LLM-based chatbot acts as a
moderator (and not a mediator) in how much cost the forager is willing to
expend for accuracy estimation. • Trust (similar to scent in the web) is
foragers’ ability to estimate the value and cost to be gained from that
environment (e.g., based on knowledge and prior experience). Overtrust and
undertrust, are simply misjudgments of such costs and values.

##  4\. Future directions

Indeed, the above hypotheses are preliminary and un-verified. Our intention
then with these verbal hypotheses is as a first step in theory building. Our
future steps plans include: mathematical formulation of verifiable hypotheses,
combining them to form propositions, evaluating their explanatory and
predictive abilities empirically, and finally building a theory that fits
easily into the existing information-foraging theoretic framework, or as a
standalone theory for information seeking with LLM-based chatbots.

##  5\. Concluding Remarks

In this paper, we reported our ongoing efforts to adapt the information
foraging theory for LLM-based chatbots. We compared classic foraging in the
web and in LLM-based chatbots and discussed how current theory may be
inadequate to be applied to this new domain. We proposed five sets of
hypotheses to augment or modify the current theory. In particular, we
discussed novel characteristics of patches and scent-following behaviors in
such systems, and the role of trust as an ecological adaptation of human
foragers in LLM-based chatbot environments. We are conducting studies to
rigorously validate these hypotheses.

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