# On-line Estimation of Stability and Passivity Metrics

Shirantha Welikala, Hai Lin and Panos J. Antsaklis The support of the National
Science Foundation (Grant No. IIS-1724070, CNS-1830335, IIS-2007949) is
gratefully acknowledged.The authors are with the Department of Electrical
Engineering, College of Engineering, University of Notre Dame, IN 46556,
<EMAIL_ADDRESS>
###### Abstract

We consider the problem of on-line evaluation of critical characteristic
parameters such as the $\mathcal{L}_{2}$-gain (L2G), input feedforward
passivity index (IFP) and output feedback passivity index (OFP) of non-linear
systems using their input-output data. Typically, having an accurate measure
of such _system indices_ enables the application of systematic control design
techniques. Moreover, if such system indices can efficiently be evaluated on-
line, they can be exploited to device intelligent controller reconfiguration
and fault-tolerant control techniques. However, the existing estimation
methods of such system indices (i.e., L2G, IFP and OFP) are predominantly off-
line, computationally inefficient, and require a large amount of actual or
synthetically generated input-output trajectory data under some specific
initial/terminal conditions. On the other hand, the existing on-line
estimation methods take an averaging-based approach, which may be sub-optimal,
computationally inefficient and susceptible to estimate saturation. In this
paper, to overcome these challenges (in the on-line estimation of system
indices), we establish and exploit several interesting theoretical results on
a particular class of fractional function optimization problems. For
comparison purposes, the details of an existing averaging-based approach are
provided for the same on-line estimation problem. Finally, several numerical
examples are discussed to demonstrate the proposed on-line estimation approach
and to highlight our contributions.

##  I Introduction

Stability and passivity are two critical concepts that are being abundantly
used in various control systems design tasks [1, 2, 3]. In particular,
quantitative measures of stability and passivity such as the
$\mathcal{L}_{2}$-gain (L2G), input-feedforward passivity index (IFP) and
output feedback passivity index (OFP) provide convenient avenues for control
systems design [4, 5]. This is because such quantitative measures of a system
can adequately characterize the system in lieu of an accurate theoretical
model [6]. Moreover, with the increasing complexity of systems, the problem of
identifying an accurate system model becomes extremely challenging [7]. Thus,
for such instances, designing control solutions based on estimated
quantitative measures like L2G, IFP and OFP (henceforth, collectively referred
to as the “ _system indices_ ”) is more suitable than designing control
solutions based on estimated system models [8]. Along the same lines, having
an accurate and on-line estimate of such system indices paves the way to
device intelligent controller reconfiguration and fault-tolerant control
techniques. Therefore, this paper focuses on the problem of estimating system
indices on-line using input-output data of the system.

Different approaches have been used in the literature to address this
estimation problem under different settings. In [9], certain dissipation
inequalities (that provide bounds on system indices) are determined using a
finite set of input-output data samples (tuples). While this approach is valid
for non-linear systems, it is computationally inefficient, off-line and may
require many data samples to get reasonable bounds. In contrast, the work in
[10] and its extension [7] have proposed a systematic input-output data sample
generation method for the exact evaluation of system indices. However, this
approach assumes the ability to generate custom input-output data samples from
the system - making it an off-line solution. Moreover, it is limited to linear
time-invariant systems and is computationally inefficient.

The subsequent work in [11] has improved the computational efficiency of the
solution proposed in [10] by two folds via proving that only half of the
input-output data samples required in [10] is sufficient to obtain the same
final solution. The work in [12] has further enhanced the computational
efficiency by improving the used gradient-based update scheme to achieve
faster convergence. However the solution proposed in [12] suffers from the
remaining drawbacks mentioned earlier (for [10, 7]).

Focusing on non-linear systems, the work in [6] and its extension [13] has
developed a dissipativity learning and control framework inspired by machine
learning techniques such as one-class support vector machines. However, this
approach is an off-line method that is computationally inefficient and
requires many input-output data samples.

On the other hand, in [14], an averaging-based simple on-line approach for the
estimation of system indices has been proposed. This approach is applicable
for non-linear systems as well. However, its estimates are often sub-optimal,
computationally expensive and susceptible to saturation effects over time. In
this paper, we continue this line of research and develop an on-line
estimation method for system indices while addressing the said challenges of
[14].

### Contributions

Our main contributions are as follows: (i) We consider a general continuous-
time non-linear system and formulate input-output data-driven on-line
estimation problems to determine the L2G, IFP and OFP (i.e., the system
indices) - each as a fractional function optimization problem (FFOP) over a
two-dimensional space; (ii) For a generalized class of these FFOPs, we develop
an efficient solution approach that converts an FFOP to a much simpler
optimization problem over a one-dimensional space; (iii) We further prove the
validity of the proposed FFOP solution approach in the presence of “dead-
zones” (where the denominators of the fractional functions become zero); (iv)
We discuss how optimal control can be exploited to control the system to
rapidly achieve better estimates for the system indices (rather than passively
using the observed input-output data); (v) We provide theoretical results that
characterize the relative locations of the optimal system indices with respect
to the existing and proposed on-line estimates of such system indices; (vi) We
extend the proposed solution for general discrete-time non-linear systems;
(vii) Finally, we provide several numerical examples to demonstrate the
validity and the potential of the proposed approach.

### Organization

This paper is organized as follows. The preliminary concepts behind the
characteristic system indices of interest are introduced in Sec. II. The
formulation of the interested system indices estimation problems is provided
in Sec. III. In Sec. IV, a particular class of useful FFOPs is analyzed and
the details of the proposed FFOP based on-line estimation method are provided.
In Sec. V, the proposed problem formulation and the corresponding solution is
specialized for discrete-time systems. Finally, several numerical results are
summarized in Sec. VI before concluding the paper in Sec. VII.

### Notation

The sets of real and integer numbers are denoted by $\mathbb{R}$ and
$\mathbb{Z}$, respectively. We use the notations $[\cdot,\cdot]$ and
$[\cdot,\cdot)$ to represent closed and left-closed right-open intervals,
respectively. These intervals may be either continuous (in $\mathbb{R}$) or
discrete (in $\mathbb{Z}$) - which will be clear from the context. The
notation $(\cdot,\cdot)$ is used to represent $2$-tuples. Given sets $A$ and
$B$, $A\backslash B$ indicates the set subtraction operation. The notation
$|\cdot|$ represents the cardinality (or absolute) operation if the argument
is a set (or number). The transpose of a matrix $A$ is denoted by
$A^{\mathsf{T}}$. The zero and identity matrices are denoted by $\mathbf{0}$
and $\mathbf{I}$ (dimensions will be clear from the context), respectively.

##  II Preliminaries

### System

Consider the non-linear system:

$\mathcal{H}:\begin{cases}\dot{x}(t)=f(x(t),u(t)),\\\
y(t)=h(x(t),u(t)),\end{cases}$ (1)

where $x(t)\in X\subset\mathbb{R}^{n}$, $u(t)\in U\subset\mathbb{R}^{m}$ and
$y(t)\in Y\subset\mathbb{R}^{m}$ are state, input and output variables at time
$t\in\mathbb{R}_{\geq 0}$, and $X,U$ and $Y$ are state, input and output
spaces, respectively. Let us use the notation
$x(t)\triangleq\phi(t,t_{0},x_{0},u_{t_{0}:t})$ to denote the state at time
$t$ reached from the initial state $x_{0}$ at initial time $t_{0}$ under the
control profile $u_{t_{0}:t}\triangleq\\{u(\tau),\tau\in[t_{0},t]\\}$.

By definition, the following assumption holds regarding the dynamics and the
feasible spaces $X,U,Y$ given in (1).

###### Assumption 1

For any $x_{0}\in X$ and $u_{t_{0}:t}\subset U$, the system $\mathcal{H}$ (1)
results in $x(t)=\phi(t,t_{0},x_{0},u_{t:t_{0}})\in X$ and
$y(t)=h(\phi(t,t_{0},x_{0},u_{t_{0}:t}),u(t))\in Y$, for all $t\geq t_{0}\geq
0$.

### Supply Rate, Storage Function and Dissipativity

A _supply rate_ $w(t)=w(u(t),y(t))$ represents a considered rate of energy
inserted into a system. In particular, it satisfies the condition stated in
the following Def. 1 and helps to define the _dissipativity_ of a system in
the subsequent Def. 2.

###### Definition 1

[2] The _supply rate_ $w(t)=w(u(t),y(t))$ for the system $\mathcal{H}$ (1) is
such that, for any $x_{0}\in X$ and $u_{t_{0}:t}\subset U$ with
$y(t)=h(\phi(t,t_{0},x_{0},u_{t_{0}:t}),u(t))\in Y$, it satisfies
$\int_{t_{0}}^{t_{1}}|w(t)|dt\leq\infty,\mbox{ for all }t_{1}\geq t_{0}\geq
0.$

###### Definition 2

[2] The system $\mathcal{H}$ (1) under the supply rate $w(t)=w(u(t),y(t))$ is
said to be _dissipative_ if there exists a positive semidefinite function
$S(x):X\rightarrow\mathbb{R}_{\geq 0}$ called the _storage function_ , such
that for any $x_{0}\in X$ and $u_{t_{0}:t}\subset U$,

$S(x_{1})-S(x_{0})\leq\int_{t_{0}}^{t_{1}}w(t)dt,$ (2)

for all $t_{1}\geq t_{0}\geq 0$, where $x_{1}\triangleq
x(t_{1})=\phi(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})$.

Note that the _storage function_ $S(x)$ mentioned in Def. 2 represents the
energy of the system at a state $x$.

Next, we define the _available storage_ $S_{a}(x):X\rightarrow\mathbb{R}_{\geq
0}$, which is the largest amount of energy that can be extracted from a system
starting from an initial condition $x(0)=x$.

###### Definition 3

[2] The _available storage_ $S_{a}(x)$ of the system $\mathcal{H}$ (1) with
the supply rate $w(t)=w(u(t),y(t))$ is:

$S_{a}(x)\triangleq\underset{\begin{subarray}{c}x(0)=x,\,u_{0:t}\subset
U,\,t_{1}>0\end{subarray}}{\sup}\ \left\\{-\int_{0}^{t_{1}}w(t)\,dt\right\\}.$
(3)

As shown in [2]: (i) any possible storage function $S(x)$ satisfies $0\leq
S_{a}(x)\leq S(x),\forall x\in X$, (ii) if $S_{a}(x)$ is continuous,
$S_{a}(x)$ itself is a possible storage function, and (iii) if
$S_{a}(x)<\infty,\forall x\in X$, then the system is dissipative with respect
to the corresponding supply rate $w(t)$.

### System Indices

In the following Defs. 4-7, we define the concepts of: (i) finite-gain
$\mathcal{L}_{2}$ stability, (ii) input feedforward passivity and (iii) output
feedback passivity, along with their corresponding indices. Note that,
henceforth, we use the notation $x_{1}\triangleq
x(t_{1})=\phi(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})$.

###### Definition 4

[2] The system $\mathcal{H}$ (1) is _finite-gain $\mathcal{L}_{2}$ stable_
with the gain $\gamma$ (also denoted as: $\mathcal{H}$ is L2G($\gamma$)), if
there exists a storage function $S:X\rightarrow\mathbb{R}_{\geq 0}$ with
$S(\mathbf{0})=0$ such that for any $x_{0}\in X$ and $u_{t_{0}:t}\subset U$,

$S(x_{1})-S(x_{0})\leq\int_{t_{0}}^{t_{1}}\gamma^{2}u^{\mathsf{T}}(t)u(t)-y^{\mathsf{T}}(t)y(t)dt,$
(4)

for all $t_{1}\geq t_{0}\geq 0$.

Note that, if the system $\mathcal{H}$ (1) is linear and is characterized by a
transfer function $H(s)$, then, its $\mathcal{L}_{2}$-gain is given by
$\gamma=\sup_{\omega\in\mathbb{R}}\|H(j\omega)\|_{2}$, where $j$ represents
the _imaginary unit_ [1]. Therefore, if $\gamma<1$, then, $\mathcal{H}$ has a
positive gain margin and (consequently) is stable.

The knowledge of the $\mathcal{L}_{2}$-gain value of a system is not only
important to reach conclusions regarding its stability, but also can be
helpful to synthesize controllers (e.g., via small-gain theorem [2]) and
controller re-configurations [14].

###### Definition 5

[2] The system $\mathcal{H}$ (1) is _passive_ if there exists a storage
function $S:X\rightarrow\mathbb{R}_{\geq 0}$ with $S(\mathbf{0})=0$ such that
for any $x_{0}\in X$ and $u_{t_{0}:t}\subset U$,

$S(x_{1})-S(x_{0})\leq\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)y(t)dt,$ (5)

for all $t_{1}\geq t_{0}\geq 0$.

The concept of passivity strives to provide an energy-based characterization
to the input-output behavior of a dynamical system. In essence, a passive
system can be thought of as a system that stores and dissipates energy without
generating its own [3, 15]. Under some additional assumptions, passivity
implies stability. Moreover, it can be conveniently used to device controllers
[2] and controller re-configurations [14]. Motivated by these potential
usages, we next define two degrees of passivity that can be characterized
corresponding to a dynamical system.

###### Definition 6

The system $\mathcal{H}$ (1) is _input feedforward passive_ with the index
$\nu\in\mathbb{R}$ (also denoted as: $\mathcal{H}$ is IFP($\nu$)), if there
exist a storage function $S:X\rightarrow\mathbb{R}_{\geq 0}$ with
$S(\mathbf{0})=0$ such that for any $x_{0}\in X$ and $u_{t_{0}:t}\subset U$,

$S(x_{1})-S(x_{0})\leq\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)y(t)-\nu
u^{\mathsf{T}}(t)u(t)dt.$ (6)

for all $t_{1}\geq t_{0}\geq 0$. Moreover, if $\mathcal{H}$ is IFP($\nu$) with
$\nu>0$, then, $\mathcal{H}$ is said to be _input strictly passive_.

###### Definition 7

[2] The system $\mathcal{H}$ (1) is _output feedback passive_ with the index
$\rho\in\mathbb{R}$ (also denoted as: $\mathcal{H}$ is OFP($\rho$)), if there
exist a storage function $S:X\rightarrow\mathbb{R}_{\geq 0}$ with
$S(\mathbf{0})=0$ such that for any $x_{0}\in X$ and $u_{t_{0}:t}\subset U$,

$S(x_{1})-S(x_{0})\leq\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)y(t)-\rho
y^{\mathsf{T}}(t)y(t)dt,$ (7)

for all $t_{1}\geq t_{0}\geq 0$. Moreover, if $\mathcal{H}$ is OFP($\rho$)
with $\rho>0$, then, $\mathcal{H}$ is said to be _output strictly passive_.

Based on Defs. 4 and 7, it can be shown that if the system $\mathcal{H}$ (1)
is OFP($\rho$) with $\rho>0$, then it is L2G($\gamma$) with
$\gamma\leq\frac{1}{\rho}$.

It is worth noting that each of the properties defined in Defs. 4-7 can be
interpreted as the dissipativity property (Def. 2) under a specific supply
rate. For example, note that, $\mathcal{H}$ is L2G($\gamma$) if and only if
$\mathcal{H}$ is dissipative under the supply rate
$w(u(t),y(t))=\gamma^{2}u^{\mathsf{T}}(t)u(t)-y^{\mathsf{T}}(t)y(t)$.

At this point, we remind that our main focus in this paper is on estimating
the _system indices_ : (i) $\mathcal{L}_{2}$-gain (L2G) $\gamma^{2}$, (ii)
input feedforward passivity (IFP) $\nu$ and (iii) output feedback passivity
(OFP) $\rho$, of the system $\mathcal{H}$ (1).

###### Remark 1

Based on the forms of (4), (6) and (7), it is clear that if the system
$\mathcal{H}$ (1) is L2G($\gamma$), IFP($\nu$) and OFP($\rho$), then, it is
also L2G($\gamma+\epsilon$), IFP($\nu-\epsilon$) and OFP($\rho-\epsilon$),
respectively, for all $\epsilon>0$. Therefore, the smallest possible $\gamma$
value that satisfies (4) in Def. 4, is the actual/optimal $L_{2}$-gain of the
system. Similarly, the largest possible $\nu$ and $\rho$ values that
respectively satisfy (6) and (7) in Defs. 6 and 7, are the actual/optimal IFP
and OFP indices of the system.

Taking Rm. 1 into account, let us denote the _optimal system indices_ L2G, IFP
and OFP as $\gamma_{*},\nu_{*}$ and $\rho_{*}$, respectively. In the sequel,
we will show how these optimal system indices can be estimated.

##  III Problem Formulation

### Optimal System Indices

Let us first define three fractional functions as

$\displaystyle\psi_{\gamma}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\triangleq\frac{\int_{t_{0}}^{t_{1}}y^{\mathsf{T}}(t)y(t)dt+(S(x_{1})-S(x_{0}))}{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)u(t)dt},$
(8)
$\displaystyle\psi_{\nu}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\triangleq\frac{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)y(t)dt-(S(x_{1})-S(x_{0}))}{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)u(t)dt},$
$\displaystyle\psi_{\rho}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\triangleq\frac{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)y(t)dt-(S(x_{1})-S(x_{0}))}{\int_{t_{0}}^{t_{1}}y^{\mathsf{T}}(t)y(t)dt},$

respectively over the spaces $\Psi_{\gamma}\triangleq\Psi\backslash\Psi_{u}$,
$\Psi_{\nu}\triangleq\Psi\backslash\Psi_{u}$ and
$\Psi_{\rho}\triangleq\Psi\backslash\Psi_{y}$, where

$\displaystyle\Psi\ \,\triangleq$ $\displaystyle\
\\{(t_{0},t_{1},x_{0},u_{t_{0}:t_{1}}):t_{1}>t_{0}\geq 0,x_{0}\in
X,u_{t_{0}:t_{1}}\subset U\\},$ $\displaystyle\Psi_{u}\triangleq$
$\displaystyle\
\\{(t_{0},t_{1},x_{0},u_{t_{0}:t_{1}})\in\Psi:\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)u(t)dt=0\\},$
$\displaystyle\Psi_{y}\triangleq$ $\displaystyle\
\\{(t_{0},t_{1},x_{0},u_{t_{0}:t_{1}})\in\Psi:\int_{t_{0}}^{t_{1}}y^{\mathsf{T}}(t)y(t)dt=0\\}.$

Using (8) and Rm. 1, the following lemma expresses the optimal system indices
as optimization problems.

###### Lemma 1

The optimal system indices: L2G $\gamma_{*}$, IFP $\nu_{*}$ and OFP $\rho_{*}$
of the system $\mathcal{H}$ (1) are given respectively by

$\displaystyle\gamma^{2}_{*}=\max_{(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\gamma}}\
\psi_{\gamma}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}}),$ (9)
$\displaystyle\nu_{*}=\min_{(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\nu}}\
\psi_{\nu}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}}),$
$\displaystyle\rho_{*}=\min_{(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\rho}}\
\psi_{\rho}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}}).$

###### Proof:

Based Rm. 1, by re-arranging (4) and substituting $\psi_{\gamma}$ from (8),
the optimal L2G value $\gamma_{*}^{2}$ can be expressed as

$\displaystyle\gamma_{*}^{2}=$ $\displaystyle\ \underset{\gamma}{\max}\ $
$\displaystyle\gamma^{2}$ (10) sub. to:
$\displaystyle\psi_{\gamma}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\leq\gamma^{2},$
$\displaystyle\ \ \ \ \ \ \
\forall(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\gamma}.$

Note that the optimization problem in (10) can be equivalently represented by
the first problem (i.e., the $\gamma_{*}^{2}$ expression) in (9). Following
the same steps, the $\nu_{*}$ and $\rho_{*}$ expressions in (9) can also be
proved. ∎

### Prior Work

The proposed optimization problems in (9) are functional optimization problems
(due to the program variable: $u_{t_{0}:t_{1}}\subset U$). Therefore, they are
hard to solve without considerably simplifying the problem setup via
introducing various limitations and assumptions.

For example, the work in [7]: (i) assumes the system dynamics to be linear,
(ii) fixes $t_{0}=0$, $t_{1}=N$ (where $N$ is a predefined value) and
$x_{0}=\mathbf{0}$, (iii) omits the effect of storage function
$(S(x_{1})-S(x_{0}))$, and only then develops an off-line gradient-based
iterative process to update the control input profile $u_{t_{0}:t_{1}}$ so as
to solve the corresponding (significantly simplified) version of (9). It is
also worth noting that, apart from the said limitations/assumptions used in
[7], it also assumes the ability to control/simulate the system dynamics under
arbitrary control input profiles - which may not be possible in many real-
world complex systems of interest.

In contrast to the off-line solution proposed in [7], the work in [14]
proposes an on-line solution (for (9)). In particular, it: (i) assumes the
system dynamics to be non-linear, (ii) considers the control input profile
$u_{t_{0}:t_{1}}$ and the initial state $x_{0}$ as given, (iii) fixes
$t_{0}=t_{o}$ and $t_{1}=t_{f}$ (where $t_{o}$ and $t_{f}$ are the initial and
current time values, respectively), (iv) omits the effect of storage function
$(S(x_{1})-S(x_{0}))$, and then provides estimates for the optimal system
indices $\gamma_{*}$, $\nu_{*}$ and $\rho_{*}$ (9) respectively as
$\hat{\gamma}$, $\hat{\nu}$ and $\hat{\rho}$, where

$\begin{gathered}\hat{\gamma}^{2}=\frac{\int_{t_{o}}^{t_{f}}y^{\mathsf{T}}(t)y(t)dt}{\int_{t_{o}}^{t_{f}}u^{\mathsf{T}}(t)u(t)dt},\\\
\hat{\nu}=\frac{\int_{t_{o}}^{t_{f}}u^{\mathsf{T}}(t)y(t)dt}{\int_{t_{o}}^{t_{f}}u^{\mathsf{T}}(t)u(t)dt},\\\
\hat{\rho}=\frac{\int_{t_{o}}^{t_{f}}u^{\mathsf{T}}(t)y(t)dt}{\int_{t_{o}}^{t_{f}}y^{\mathsf{T}}(t)y(t)dt}.\end{gathered}$
(11)

Note that the optimal L2G estimate $\hat{\gamma}^{2}$ in (11) is identical to
the objective function value
$\psi_{\gamma}(t_{0},t_{1},x_{0},u_{t_{0}:t_{1}})$ in (9) evaluated directly
at $t_{0}=t_{o}$, $t_{1}=t_{f}$, $x_{0}=x(t_{o})$,
$u_{t_{0}:t_{1}}=u_{t_{o}:t_{f}}$ while assuming $(S(x_{1})-S(x_{0}))=0$ (note
also that, when input-output profiles of the system are known over
$t\in[t_{o},t_{f}]$, the state information $x(t),\forall t\in[t_{o},t_{f}]$ is
not explicitly required to evaluate $\hat{\gamma}$ in (11)). The same
arguments are valid for the optimal IFP and OFP estimates $\hat{\nu}$ and
$\hat{\rho}$ in (11).

In essence, $\hat{\gamma}$, $\hat{\nu}$ and $\hat{\rho}$ in (11) can
respectively be seen as the values of the objective functions $\psi_{\gamma}$,
$\psi_{\nu}$ and $\psi_{\rho}$ in (9) at a particular point in
$\Psi_{\gamma}\cap\Psi_{\nu}\cap\Psi_{\rho}$. Therefore, it is clear that the
approach proposed in [14] may suffer from inaccuracies due to: (i) the lack of
any optimization stage (like in (9) and [7]), (ii) the omission of the storage
function, and (iii) the susceptibility to the saturation effects (as
$t_{f}\rightarrow\infty$). In the sequel, we propose an alternative approach
to [14] that addresses the above challenges when solving (9) on-line.

### On-Line Estimates for Optimal System Indices

The proposed on-line solution for (9) in this paper requires the following
assumption.

###### Assumption 2

The rate of change of the storage function $S:X\rightarrow\mathbb{R}_{\geq 0}$
is bounded such that $|\frac{dS(x(t))}{dt}|\leq K_{s}$ over the period
$t\in[t_{o},t_{f})$, where $K_{s}$ is a known constant.

Similar to [14], we assume the system dynamics to be non-linear and consider
the control input profile $u_{t_{0}:t_{1}}$ as a given. However, unlike in
[14], we do not fix $t_{0}$ and $t_{1}$ values (to $t_{o}$ and $t_{f}$,
respectively). Instead, we treat $t_{0}$ and $t_{1}$ as _program variables_
over which the we optimize the objective functions:

$\displaystyle\hat{\psi}_{\gamma}(t_{0},t_{1})\triangleq\frac{\int_{t_{0}}^{t_{1}}y^{\mathsf{T}}(t)y(t)dt-
K_{s}(t_{1}-t_{0})}{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)u(t)dt},$ (12)
$\displaystyle\hat{\psi}_{\nu}(t_{0},t_{1})\triangleq\frac{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)y(t)dt+K_{s}(t_{1}-t_{0})}{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)u(t)dt},$
$\displaystyle\hat{\psi}_{\rho}(t_{0},t_{1})\triangleq\frac{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)y(t)dt+K_{s}(t_{1}-t_{0})}{\int_{t_{0}}^{t_{1}}y^{\mathsf{T}}(t)y(t)dt},$

on-line using past input-output information seen over a time period
$[t_{o},t_{f})\supseteq[t_{0},t_{1})$. Note that $K_{s}$ in (12) is from As. 2
(see also (8)). Moreover, in parallel to (8), the objective functions in (12)
are respectively defined over the spaces

$\displaystyle\hat{\Psi}_{\gamma}\triangleq$
$\displaystyle\\{(t_{0},t_{1}):[t_{0},t_{1})\in[t_{o},t_{f}),(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\gamma}\\},$
$\displaystyle\hat{\Psi}_{\nu}\triangleq$
$\displaystyle\\{(t_{0},t_{1}):[t_{0},t_{1})\in[t_{o},t_{f}),(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\nu}\\},$
$\displaystyle\hat{\Psi}_{\rho}\triangleq$
$\displaystyle\\{(t_{0},t_{1}):[t_{0},t_{1})\in[t_{o},t_{f}),(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\rho}\\}.$

The following lemma defines the proposed estimates of the optimal system
indices and establishes their relationships with the corresponding optimal
system indices given in (9).

###### Lemma 2

Using a given input-output profile of the system $\mathcal{H}$ (1), the
optimal system indices (9): L2G $\gamma_{*}$, IFP $\nu_{*}$ and OFP $\rho_{*}$
can respectively be estimated by

$\begin{gathered}\hat{\gamma}^{2}_{*}=\max_{(t_{0},t_{1})\in\hat{\Psi}_{\gamma}}\
\hat{\psi}_{\gamma}(t_{0},t_{1}),\\\
\hat{\nu}_{*}=\min_{(t_{0},t_{1})\in\hat{\Psi}_{\nu}}\
\hat{\psi}_{\nu}(t_{0},t_{1}),\\\
\hat{\rho}_{*}=\min_{(t_{0},t_{1})\in\hat{\Psi}_{\rho}}\
\hat{\psi}_{\rho}(t_{0},t_{1}),\end{gathered}$ (13)

so that $\hat{\gamma}^{2}_{*}\leq\gamma^{2}_{*}$, $\hat{\nu}_{*}\geq\nu_{*}$
and $\hat{\rho}_{*}\geq\rho_{*}$ holds true.

###### Proof:

According to As. 2,

$-K_{s}(t_{1}-t_{0})\leq S(x_{1})-S(x_{0})\leq K_{s}(t_{1}-t_{0}).$

Hence, under a given input-output profile, the objective functions
$\hat{\psi}_{\gamma}(t_{1},t_{0})$ in (13) and
$\psi_{\gamma}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})$ in (9) (defined in (12) and
(8), respectively) satisfy:
$\hat{\psi}_{\gamma}(t_{1},t_{0})\leq\psi_{\gamma}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})$.
Moreover, the feasible spaces of the optimization problems in (13) are subsets
of those in (9) (e.g., $\hat{\Psi}_{\gamma}\subset\Psi_{\gamma}$). Starting
from the $\hat{\gamma}^{2}_{*}$ expression in (13) and then applying the above
two facts, we get

$\displaystyle\hat{\gamma}^{2}_{*}=\max_{(t_{0},t_{1})\in\hat{\Psi}_{\gamma}}\
\hat{\psi}_{\gamma}(t_{0},t_{1})\leq\max_{(t_{0},t_{1})\in\hat{\Psi}_{\gamma}}\
\psi_{\gamma}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})$
$\displaystyle\leq\max_{(t_{0},t_{1},x_{0},u_{t_{0}:t_{1}})\in\Psi_{\gamma}}\
\psi_{\gamma}(t_{1},t_{0},x_{0},u_{t_{0}:t_{1}})=\gamma_{*}^{2}.$

Thus, $\hat{\gamma}^{2}_{*}\leq\gamma_{*}^{2}$. Following the same steps, we
can also prove that $\hat{\nu}_{*}\geq\nu_{*}$ and
$\hat{\rho}_{*}\geq\rho_{*}$. This completes the proof. ∎

It is worth noting that the inequality relationships such as
$\hat{\rho}_{*}\geq\rho_{*}$ proven in Lm. 2 cannot be established between the
estimates proposed in (11) [14] and the optimal values stated in (9) (e.g.,
between $\hat{\rho}$ and $\rho_{*}$) due to the omission of the storage
function in (11) [14]. However, we will revisit this point at the end of the
next section.

In the next section, we propose an efficient solution for the fractional
function optimization problems of the form (13).

##  IV Fractional Function Optimization

In this section, we study a class of fractional function optimization problems
(FFOPs) that includes all three problem forms in (13). The goal here is to
develop an efficient solution for this class of FFOPs so as to systematically
solve (13).

### Preliminaries

We first establish a minor algebraic result that will be used later on.

###### Lemma 3

For any $a,c\in\mathbb{R}$ and $b,d\in\mathbb{R}_{>0}$,

$\frac{a}{b}\lesseqqgtr\frac{a+c}{b+d}\lesseqqgtr\frac{c}{d}.$ (14)

###### Proof:

First, we prove $\frac{a}{b}<\frac{a+c}{b+d}\iff\frac{a+c}{b+d}<\frac{c}{d}$.
Note that, since $b>0$ and $b+d>0$, we can write

$\frac{a}{b}<\frac{a+c}{b+d}\iff ab+ad<ab+bc\iff ad<bc.$ (15)

Now, adding a $cd$ term to the both sides of the last inequality and using the
fact that $d>0$ and $b+d>0$, we can obtain

$ad<bc\iff ad+cd<bc+cd\iff\frac{a+c}{b+d}<\frac{c}{d}.$ (16)

Combining (15) and (16), we get
$\frac{a}{b}<\frac{a+c}{b+d}\iff\frac{a+c}{b+d}<\frac{c}{d}$.

Now, note that these same steps can be used to prove
$\frac{a}{b}=\frac{a+c}{b+d}\iff\frac{a+c}{b+d}=\frac{c}{d}$ and
$\frac{a}{b}>\frac{a+c}{b+d}\iff\frac{a+c}{b+d}>\frac{c}{d}$. Therefore,
$\frac{a}{b}\lesseqqgtr\frac{a+c}{b+d}\iff\frac{a+c}{b+d}\lesseqqgtr\frac{c}{d}$.
∎

### The Fractional Function

Let $t_{o},t_{f}\in\mathbb{R}_{\geq 0}$ where $t_{o}<t_{f}$ be two continuous-
time instants and $p(t)$,$q(t)$ where $p:[t_{o},t_{f})\rightarrow\mathbb{R}$,
$q:[t_{o},t_{f})\rightarrow\mathbb{R}_{\geq 0}$ be two continuous-time
signals. Moreover, let $r(\tau)$, $s(\tau)$ where
$r:\mathbb{R}_{>0}\rightarrow\mathbb{R}$,
$s:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{\geq 0}$ be two continuous linear
mappings and $\dot{r}(0)\triangleq\lim_{\tau\rightarrow
0}\frac{dr(\tau)}{d\tau}$, $\dot{s}(0)\triangleq\lim_{\tau\rightarrow
0}\frac{ds(\tau)}{d\tau}$ where $\dot{r}(0)\in\mathbb{R}$,
$\dot{s}(0)\in\mathbb{R}_{\geq 0}$ be two known constants.

Now, for any continuous-time interval $[t_{0},t_{1})\subseteq[t_{o},t_{f})$,
let us define a corresponding _fractional function_ of the form:

$\psi(t_{0},t_{1})\triangleq\frac{\int_{t_{0}}^{t_{1}}p(t)dt+r(t_{1}-t_{0})}{\int_{t_{0}}^{t_{1}}q(t)dt+s(t_{1}-t_{0})}.$
(17)

To ensure that (17) is well-defined over any
$[t_{0},t_{1})\subseteq[t_{o},t_{f})$, we require the following assumption.

###### Assumption 3

(No-Dead-Zones) The continuous-time signal $q(t)$ involved in the fractional
function (17) is such that $q:[t_{o},t_{f})\rightarrow\mathbb{R}_{>0}$, i.e.,
$q(t)>0,\forall t\in[t_{o},t_{f})$.

Due to obvious reasons, we call this assumptions as the “no-dead-zone” (in
$q(t)$) assumption. However, we also point out that we will relax this
assumption later on in this section.

Under As. 3, note that the denominator of the fractional function
$\psi(t_{0},t_{1})$ defined in (17) will always be positive definite. Next, we
establish the following lemma.

###### Lemma 4

For any fractional function $\psi(t_{0},t_{1})$ of the form (17) under As. 3:

$\psi(t_{0},t_{m})\lesseqqgtr\psi(t_{0},t_{1})\lesseqqgtr\psi(t_{m},t_{1}),$
(18)

for any $t_{m}\in\mathbb{R}_{>0}$ such that $t_{0}<t_{m}<t_{1}$.

###### Proof:

Since $t_{0}<t_{1}$ (by definition), there always exists
$t_{m}\in\mathbb{R}_{>0}$ such that $t_{0}<t_{m}<t_{1}$. Using this fact and
the nature of the fractional function $\psi(\cdot,\cdot)$ defined in (17), we
can re-state the terms $\psi(t_{0},t_{m})$, $\psi(t_{0},t_{1})$ and
$\psi(t_{m},t_{1})$ as

$\psi(t_{0},t_{m})=\frac{a}{b},\ \ \psi(t_{0},t_{1})=\frac{a+c}{b+d}\ \ \mbox{
and }\ \ \psi(t_{m},t_{1})=\frac{c}{d},$ (19)

respectively, where $a\triangleq\int_{t_{0}}^{t_{m}}p(t)dt+r(t_{m}-t_{0})$,
$b\triangleq\int_{t_{0}}^{t_{m}}q(t)dt+s(t_{m}-t_{0})$,
$c\triangleq\int_{t_{m}}^{t_{1}}p(t)dt+r(t_{1}-t_{m})$ and
$d\triangleq\int_{t_{m}}^{t_{1}}q(t)dt+s(t_{1}-t_{m})$. Note that,
$a,c\in\mathbb{R}$, and under As. 3, $b,d\in\mathbb{R}_{>0}$. Therefore, Lm. 3
can now be applied to compare the terms involved in (19) as

$\frac{a}{b}\lesseqqgtr\frac{a+c}{b+d}\lesseqqgtr\frac{c}{d}\iff\psi(t_{0},t_{m})\lesseqqgtr\psi(t_{0},t_{1})\lesseqqgtr\psi(t_{m},t_{1}),$

which completes the proof. ∎

### Fractional Function Optimization

Under As. 3, let us denote the space over which a fractional function
$\psi(t_{0},t_{1})$ of the form (17) is well-defined as

$\Psi\triangleq\\{(t_{0},t_{1}):[t_{0},t_{1})\subseteq[t_{o},t_{f})\\}.$ (20)

In the following theorem, we establish our main theoretical result on
optimizing a fractional function $\psi(t_{0},t_{1})$ of the form (17) over all
$(t_{0},t_{1})\in\Psi$.

###### Theorem 1

Any fractional function $\psi(t_{0},t_{1})$ of the form (17) under As. 3 can
be efficiently optimized over the corresponding space $(t_{0},t_{1})\in\Psi$
exploiting the relationships:

$\Big{\\{}\underset{(t_{0},t_{1})\in\Psi}{\max}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})}{\max}\
\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)\Big{\\}},$ (21)
$\Big{\\{}\underset{(t_{0},t_{1})\in\Psi}{\min}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})}{\min}\
\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)\Big{\\}},$ (22)

where

$\lim_{\Delta\rightarrow
0}\psi(t,t+\Delta)=\frac{p(t)+\dot{r}(0)}{q(t)+\dot{s}(0)}.$

###### Proof:

We will first prove the equivalence relation in (21). Let
$t_{0}^{*},t_{1}^{*}$ be the optimal arguments corresponding to the
optimization problem in the left hand side (LHS) of (21). Note that,
irrespective of $t_{0}$,

$\displaystyle\lim_{t_{1}\rightarrow t_{0}}\psi(t_{0},t_{1})=$
$\displaystyle\lim_{t_{1}\rightarrow
t_{0}}\frac{\int_{t_{0}}^{t_{1}}p(t)dt+r(t_{1}-t_{0})}{\int_{t_{0}}^{t_{1}}q(t)dt+s(t_{1}-t_{0})}$
$\displaystyle=$ $\displaystyle\lim_{t_{1}\rightarrow
t_{0}}\frac{\frac{d}{dt_{1}}\int_{t_{0}}^{t_{1}}p(t)dt+\frac{d}{dt_{1}}r(t_{1}-t_{0})}{\frac{d}{dt_{1}}\int_{t_{0}}^{t_{1}}q(t)dt+\frac{d}{dt_{1}}s(t_{1}-t_{0})}$
$\displaystyle=$ $\displaystyle\lim_{t_{1}\rightarrow
t_{0}}\frac{p(t_{1})+\dot{r}(t_{1}-t_{0})}{q(t_{1})+\dot{s}(t_{1}-t_{0})}=\frac{p(t_{0})+\dot{r}(0)}{q(t_{0})+\dot{s}(0)}.$

In the above simplification, we have used the L’Hospital’s rule as $r(0)=0$,
$s(0)=0$ (recall that $r(\tau),s(\tau)$ are linear mappings). From this
result, it is clear that the equivalence relation in (21) is implied if
$t_{0}^{*},t_{1}^{*}$ are such that $t_{1}^{*}=t_{0}^{*}+\epsilon$ where
$\epsilon$ is an infinitesimally small positive number. In other words, (21)
is proved if we can prove that $t_{1}^{*}$ is infinitesimally close to
$t_{0}^{*}$. In the sequel, we prove this using a contradiction.

Let us assume $t_{0}^{*}$ and $t_{1}^{*}$ are not infinitesimally close. Then,
there should exist $t_{m}\in\mathbb{R}_{>0}$ such that
$t_{0}^{*}<t_{m}<t_{1}^{*}$. According to Lm. 4, for any such $t_{m}$, either:

$\begin{cases}\psi(t_{0}^{*},t_{1}^{*})<\psi(t_{m},t_{1}^{*}),\\\
\psi(t_{0}^{*},t_{1}^{*})=\psi(t_{0}^{*},t_{m})=\psi(t_{m},t_{1}^{*}),\ \
\mbox{ or }\ \ \\\
\psi(t_{0}^{*},t_{1}^{*})<\psi(t_{0}^{*},t_{m}).\end{cases}$ (23)

In other words,
$\psi(t_{0}^{*},t_{1}^{*})\leq\min\\{\psi(t_{0}^{*},t_{m}),\psi(t_{m},t_{1}^{*})\\}$.
This implies that $\psi(t_{0}^{*},t_{1}^{*})$ is sub-optimal - which is a
contradiction. Therefore, it is clear that the optimal arguments
$t_{0}^{*},t_{1}^{*}$ should be infinitesimally close. As stated earlier, this
completes the proof of (21). The same steps can be used to prove the
equivalence condition in (22). ∎

It is worth noting that the application of Th. 1 can significantly reduce the
computational cost associated with optimizing fractional functions of the form
(17). In particular, it can reduce the search space by one dimension (see (21)
and (22)). Note also that the resulting simplified optimization problems take
a particular form that can easily be implemented in an on-line setting without
using much memory.

However, note that, if we relaxed As. 3, we no longer can consider optimizing
fractional functions of the form (17) over the space $\Psi$ (20). Instead, we
will have to optimize such functions over a confined space
$\Psi\backslash\Psi_{q}$ (where $\Psi_{q}$ will be defined in the sequel).
Hence, Th. 1 needs to be generalized to address situations where As. 3 does
not hold.

### Relaxing As. 3

When As. 3 is relaxed, the fractional function $\psi(t_{0},t_{1})$ in (17)
becomes ill-defined over intervals $[t_{0},t_{1})\subseteq[t_{o},t_{f})$ where
$\int_{t_{0}}^{t_{1}}q(t)dt+s(t_{1}-t_{0})=0$. Recall that, by definition,
$q:[t_{o},t_{f})\rightarrow\mathbb{R}_{\geq 0}$ is an arbitrary signal and
$s:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{\geq 0}$ is a linear mapping.
Therefore, $\int_{t_{0}}^{t_{1}}q(t)dt+s(t_{1}-t_{0})=0\iff q(t)=0,\forall
t\in[t_{0},t_{1})$ and $s(\tau)=0,\forall\tau\in\mathbb{R}_{>0}$. This
inspires the following assumption.

###### Assumption 4

The linear mapping $s(\tau)$ involved in the fractional function (17) is such
that $s(\tau)=0,\forall\tau\in\mathbb{R}_{>0}$.

Recalling $\Psi$ (20), we define

$\Psi_{q}\triangleq\\{(t_{0},t_{1})\in\Psi:q(t)=0,\forall
t\in[t_{0},t_{1})\\}$

as the space of “sub-dead-zones” (i.e., intervals
$[t_{0},t_{1})\subseteq[t_{o},t_{f})$ where $q(t)=0,\forall
t\in[t_{0},t_{1})$). Note that any subset of a sub-dead-zone will also be a
sub-dead-zone, and thus, $\Psi_{q}$ may contain infinitely many
$(t_{0},t_{1})$ tuples. We also define

$\bar{\Psi}_{q}\triangleq\\{(t_{0},t_{1})\in\Psi_{q}:q(t_{0}-\epsilon)>0,q(t_{1})>0\\},$

where $\epsilon$ is an infinitesimally small constant, as the space of “dead-
zones” (i.e., the largest possible sub-dead-zones). Consequently,
$\bar{\Psi}_{q}$ only contains a finite number of $(t_{0},t_{1})$ tuples and
$\bar{\Psi}_{q}$ is empty if there are no dead-zones inside $[t_{o},t_{f})$.
For notational convenience, let us also define

$\tilde{\Psi}_{q}\triangleq\\{t:t\in[t_{o},t_{f}),q(t)=0\\}$

as the set of time instants corresponding to the dead-zones (note that
$\tilde{\Psi}_{q}=\cup_{(t_{0},t_{1})\in\bar{\Psi}_{q}}[t_{0},t_{1})=\cup_{(t_{0},t_{1})\in\Psi_{q}}[t_{0},t_{1})$).

The following theorem generalizes Th. 1 to address the presence of dead-zones
inside $[t_{o},t_{f})$ (i.e., when $|\bar{\Psi}_{q}|>0$).

$\displaystyle\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\max}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\
\max\Big{\\{}\big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\max}\
\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\max}\ \lim_{\Delta\rightarrow
0}\psi(t_{zo}-\Delta,t)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\max}\ \lim_{\Delta\rightarrow
0}\psi(t,t_{zf}+\Delta)\big{\\}}\Big{\\}},$ (24)
$\displaystyle\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\min}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\
\min\Big{\\{}\big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\min}\
\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\min}\ \lim_{\Delta\rightarrow
0}\psi(t_{zo}-\Delta,t)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\min}\ \lim_{\Delta\rightarrow
0}\psi(t,t_{zf}+\Delta)\big{\\}}\Big{\\}},$ (25)
$\displaystyle\psi(t_{0},t_{1})\triangleq\
\frac{\int_{t_{0}}^{t_{1}}p(t)dt+r(t_{1}-t_{0})}{\int_{t_{0}}^{t_{1}}q(t)dt},\
\ \ \mbox{(i.e., \eqref{Eq:GenericRationalFunctionCont} under As.
\ref{As:NoLinearMappingInDenominator}) \ \ \ and }\ \ \
\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)=\frac{p(t)+\dot{r}(0)}{q(t)},$ (26)
$\displaystyle\lim_{\Delta\rightarrow
0}\psi(t_{zo}-\Delta,t)=\begin{cases}+\infty,\mbox{ if
}\int_{t_{zo}}^{t}p(\tau)d\tau+r(t-t_{zo})>0,\\\ -\infty,\mbox{ if
}\int_{t_{zo}}^{t}p(\tau)d\tau+r(t-t_{zo})<0,\end{cases}\mbox{ and }\ \ \
\lim_{\Delta\rightarrow 0}\psi(t,t_{zf}+\Delta)=\begin{cases}+\infty,\mbox{ if
}\int_{t}^{t_{zf}}p(\tau)d\tau+r(t_{zf}-t)>0,\\\ -\infty,\mbox{ if
}\int_{t}^{t_{zf}}p(\tau)d\tau+r(t_{zf}-t)<0.\end{cases}$ (27)

###### Theorem 2

Any fractional function $\psi(t_{0},t_{1})$ of the form (17) under As. 4 can
be efficiently optimized over the corresponding space
$(t_{0},t_{1})\in\Psi\backslash\Psi_{q}$ exploiting the relationships: (24)
and (25) (under the objective functions in (26) and (27)).

###### Proof:

The main idea behind this proof is to consider all possible intervals inside
$[t_{o},t_{f})$ that are: (i) not sub-dead-zones (i.e., have well-defined
$\psi(\cdot,\cdot)$ values) and (ii) not further breakable into two or more
non-sub-dead-zones (as they would be sub-optimal according to Lm. 4). It is
easy to see that there are three types of such intervals:

  1. 1.

$[t,t+\epsilon),\ \ \forall t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q},$

  2. 2.

$[t_{zo}-\epsilon,t),\ \ \forall
t\in[t_{zo},t_{zf}),\forall(t_{zo},t_{zf})\in\bar{\Psi}_{q},$

  3. 3.

$[t,t_{zf}+\epsilon),\ \ \forall
t\in[t_{zo},t_{zf}),\forall(t_{zo},t_{zf})\in\bar{\Psi}_{q},$

where $\epsilon$ is an infinitesimally small positive number. These three
interval types give the three different inner optimization problems in the
right hand side (RHS) of (24) and (25). Finally, the simplified objective
function expressions in (26) and (27) can be obtained using the L’Hospital’s
rule following the same steps as in the proof of Th. 1. ∎

From comparing (24), (25) in Th. 2 with (21), (22) in Th. 1, it should be
clear that the main consequence of the presence of dead-zones is the last two
inner optimization problems in the RHS of (24), (25). Note that, based on
(27), these additional inner optimization problems can be either decisive or
irrelevant (but never in-between).

The following corollary shows that, under some special conditions, when
optimizing a fractional function of the form (17), the dead-zones can be
conveniently disregarded.

###### Assumption 5

The signals $p(t)$ and $q(t)$ involved in the fractional function (17) are
such that $q(t)=0\implies p(t)=0$ for any $t\in[t_{o},t_{f})$.

###### Corollary 1

Any fractional function $\psi(t_{0},t_{1})$ of the form (17) under As. 4 and
As. 5 can be efficiently optimized over the corresponding space
$(t_{0},t_{1})\in\Psi\backslash\Psi_{q}$ exploiting the relationships: (i) If
$r(\tau)\leq 0,\forall\tau\in\mathbb{R}_{>0}$:

$\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\max}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\max}\
\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)\Big{\\}},$ (28)

(ii) If $r(\tau)\geq 0,\forall\tau\in\mathbb{R}_{>0}$:

$\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\min}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\min}\
\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)\Big{\\}},$ (29)

where

$\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)=\frac{p(t)+\dot{r}(0)}{q(t)}.$

###### Proof:

These results directly follow from applying the assumed conditions in As. 5
and Co. 1 in Th. 2. ∎

We are now ready to use the developed theories for FFOPs to solve the
optimization problems formulated in (13).

### Application to Solve (13)

First, following the same notation $\tilde{\Psi}_{q}$ used before (e.g., see
(28)), let us define

$\tilde{\Psi}_{u}\triangleq\\{t:t\in[t_{o},t_{f}),u(t)=0\\},$
$\tilde{\Psi}_{y}\triangleq\\{t:t\in[t_{o},t_{f}),y(t)=0\\}.$

The following theorem provides an efficient approach to solve the optimization
problems formulated in (13).

###### Theorem 3

Using a given input-output profile of the system $\mathcal{H}$ (1), the
optimal system indices (9): L2G $\gamma_{*}$, IFP $\nu_{*}$ and OFP $\rho_{*}$
can respectively be estimated by

$\begin{gathered}\hat{\gamma}^{2}_{*}=\max_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}}\frac{y^{\mathsf{T}}(t)y(t)-K_{s}}{u^{\mathsf{T}}(t)u(t)},\\\
\hat{\nu}_{*}=\min_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}}\frac{u^{\mathsf{T}}(t)y(t)+K_{s}}{u^{\mathsf{T}}(t)u(t)},\\\
\hat{\rho}_{*}=\min_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{y}}\frac{u^{\mathsf{T}}(t)y(t)+K_{s}}{y^{\mathsf{T}}(t)y(t)},\end{gathered}$
(30)

so that $\hat{\gamma}^{2}_{*}\leq\gamma^{2}_{*}$, $\hat{\nu}_{*}\geq\nu_{*}$
and $\hat{\rho}_{*}\geq\rho_{*}$ holds true.

###### Proof:

Due to Lm. 2, here we only need to prove the equivalence of the respective
optimization problems in (13) and (30). To this end, we use the proposed Th. 2
and Co. 1.

First, note that As. 4 holds for any fractional function defined in (12) and
As. 5 holds for the fractional functions $\hat{\psi}_{\nu}$ and
$\hat{\psi}_{\rho}$ defined in (12) (as $u(t)=\mathbf{0}$ or $y(t)=\mathbf{0}$
implies $u^{\mathsf{T}}(t)y(t)=\mathbf{0}$). Hence, Co. 1 (in particular, the
equivalence relationship in (29)) can be used to efficiently evaluate the
estimates for the optimal system indices: IFP $\hat{\nu}_{*}$ and OFP
$\hat{\rho}_{*}$ given in (13).

Therefore, by applying $p(t)=u^{\mathsf{T}}(t)y(t)$,
$q(t)=u^{\mathsf{T}}(t)u(t)$ and $r(\tau)=K_{s}(\tau)$ in (29), we can obtain
the required equivalence of $\hat{\nu}_{*}$ expressions given in (13) and
(30). Similarly, by applying $p(t)=u^{\mathsf{T}}(t)y(t)$,
$q(t)=y^{\mathsf{T}}(t)y(t)$ and $r(\tau)=K_{s}(\tau)$ in (29), we can obtain
the required equivalence of $\hat{\rho}_{*}$ expressions given in (13) and
(30).

On the other hand, note that, As. 5 does not necessarily hold for the
fractional function $\hat{\psi}_{\gamma}$ defined in (12) (as
$u(t)=\mathbf{0}$ does not imply $y^{\mathsf{T}}(t)y(t)=\mathbf{0}$). However,
since As. 4 holds for this fractional function $\hat{\psi}_{\gamma}$, we can
apply Th. 2 (in particular, the equivalence relationship in (24)) to
efficiently evaluate an estimate for the optimal system index: L2G
$\hat{\gamma}_{*}$ given in (13). Nevertheless, if we assumed that the system
$\mathcal{H}$ (1) has a finite L2G value, then, by applying (27) in (24), we
can obtain the same equivalence relationship (28) proposed in Co. 1. Hence,
Co. 1 (in particular (28)) can also be used here to efficiently evaluate
$\hat{\gamma}_{*}$ given in (13).

Therefore, by applying $p(t)=y^{\mathsf{T}}(t)y(t)$,
$q(t)=u^{\mathsf{T}}(t)u(t)$ and $r(\tau)=-K_{s}(\tau)$ in (28), we can obtain
the required equivalence of $\hat{\gamma}_{*}$ expressions given in (13) and
(30). This completes the proof. ∎

### Discussion

Comparing Lm. 2 (13) and Th. 3 (30), we can make the following set of remarks.

###### Remark 2

(Efficiency) Compared to (13), (30) is significantly computationally
inefficient due to: (i) the reduced search space size, (ii) not having to
compute/analyze continuous integrals, and (iii) the ability to implement on-
line using less amount of memory.

###### Remark 3

(Sampling) When using a digital processor to compute the estimates of the
optimal system indices (via (13) or (30)), we cannot use the complete
continuous input-output signals $\\{u(t):t\in[t_{o},t_{f})\\}$ and
$\\{y(t):t\in[t_{o},t_{f})\\}$. Instead, we may only use sampled sets of data
points from those signals $\\{u(t):t\in D_{s}\subset[t_{o},t_{f})\\}$ and
$\\{y(t):t\in D_{s}\subset[t_{0},t_{f})\\}$, where $D_{s}$ is a discrete (and
possibly non-uniform) set of sampling time instants selected from the
continuous-time interval $[t_{o},t_{f})$. In a such scenario, note that we
still can use (30) (as opposed to (13)) and the bounds stated in Th. 3 will
hold true. This is because, sampling will only reduce the feasible space size
of the optimization problems in (30) leading to increase the error between the
estimates and the optimal values. Note also that this estimation error will
decrease as the number of samples (i.e., $|D_{s}|$) increases.

###### Remark 4

(Control for Estimation) Consider a scenario where signals
$\\{p(t)\in\mathbb{R}:t\in[t_{o},t_{f})\\}$ and $\\{q(t)\in\mathbb{R}_{\geq
0}:t\in[t_{o},t_{f})\\}$ in the fractional function $\psi(\cdot,\cdot)$ (17)
are _controllable_ over time $t$ from their initial values $p(t_{o})$ and
$q(t_{o})$ via selecting $\dot{p}(t)\triangleq\frac{dp(t)}{dt}$ and
$\dot{q}(t)\triangleq\frac{dq(t)}{dt}$ values, respectively. Note that, under
As. 3 and As. 4, the RHS of (21) (equally applies for (22)) can be written as

$\max_{t\in[t_{o},t_{f})}\psi(t)\triangleq\frac{p(t)+\dot{r}(0)}{q(t)}.$ (31)

The dynamics of $\psi(t)$ defined above can be derived as

$\dot{\psi}(t)=-\frac{1}{q(t)}\left(\psi(t)\dot{q}(t)-\dot{p}(t)\right).$ (32)

Now, taking
$x=[x_{1},x_{2},x_{3}]^{\mathsf{T}}\triangleq[\psi,p,q]^{\mathsf{T}}$ as the
state variable and
$u=[u_{1},u_{2}]^{\mathsf{T}}\triangleq[\dot{p},\dot{q}]^{\mathsf{T}}$ as the
control input, a non-linear state space representation $\dot{x}=g(x)u$, i.e.,

$\begin{bmatrix}\dot{x}_{1}\\\ \dot{x}_{2}\\\
\dot{x}_{3}\end{bmatrix}=\begin{bmatrix}\frac{1}{x_{3}}&-\frac{x_{1}}{x_{3}}\\\
1&0\\\ 0&1\end{bmatrix}\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix},$ (33)

can be obtained to represent the underlying dynamics. Using this setup, an
optimal control problem can be formulated as

$\displaystyle\underset{{t_{f},u_{t_{o}:t_{f}}}}{\arg\min}$ $\displaystyle\
\int_{t_{o}}^{t_{f}}L(x(t),u(t),t)dt+\phi(x(t_{f}),t_{f}),$ (34) such that
$\displaystyle\ \dot{x}(t)=g(x(t))u(t),\ \ \forall t\in[t_{o},t_{f})$

(e.g., with $L=u^{\mathsf{T}}(t)u(t)$ and
$\phi=-x_{1}(t_{f})+t_{f}=-\psi(t_{f})+t_{f}$) to determine the optimal
control input profile that rapidly leads to maximize $\psi(t)$ in (31).

###### Remark 5

(Realizations) In reality, we might not have a single long input-output
profile of the system to carry out the on-line estimation process (using (13)
or (30)). Instead, we might only have a collection of past realizations of the
system (with arbitrary initial/terminal conditions) apart from a short recent
input-output profile. In this kind of a setting, it is not clear how (13) can
be used across all the realizations. However, in such a setting, (30) can
easily be used by treating all the realizations as a single long input-output
profile (sequenced in an arbitrary order).

### Selection of $K_{s}$

For the system indices, as opposed to the proposed estimates in (11) [14], the
proposed estimates in this paper (30) are dependent on the rate bound
$K_{s}\in\mathbb{R}_{\geq 0}$ assumed for the storage function
$S:X\rightarrow\mathbb{R}_{\geq 0}$ in As. 2. To compute this $K_{s}$
parameter, theoretically, under some assumptions, one can exploit the concept
of available storage function $S_{a}:X\rightarrow\mathbb{R}_{\geq 0}$ defined
in Def. 3. However, here we take a more practical approach as given in the
sequel.

First, by applying $K_{s}=0$ in (30), we obtain parameter-free estimates for
the optimal system indices (9): L2G $\gamma_{*}$, IFP $\nu_{*}$ and OFP
$\rho_{*}$ respectively as

$\begin{gathered}\tilde{\gamma}^{2}=\max_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}}\frac{y^{\mathsf{T}}(t)y(t)}{u^{\mathsf{T}}(t)u(t)},\\\
\tilde{\nu}=\min_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}}\frac{u^{\mathsf{T}}(t)y(t)}{u^{\mathsf{T}}(t)u(t)},\\\
\tilde{\rho}=\min_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{y}}\frac{u^{\mathsf{T}}(t)y(t)}{y^{\mathsf{T}}(t)y(t)}.\end{gathered}$
(35)

The following lemma compares (9), (11), (30) and (35).

###### Lemma 5

The optimal system indices (9) and their on-line estimates given in (11), (30)
and (35) satisfy:

$\displaystyle\hat{\gamma}^{2}_{*}$ $\displaystyle\leq\gamma^{2}_{*},\ \ \ \ $
$\displaystyle\hat{\nu}_{*}$ $\displaystyle\geq\nu_{*},\ \ \ \ $
$\displaystyle\hat{\rho}_{*}$ $\displaystyle\geq\rho_{*},$ (36)
$\displaystyle\hat{\gamma}^{2}_{*}$ $\displaystyle\leq\tilde{\gamma}^{2},$
$\displaystyle\hat{\nu}_{*}$ $\displaystyle\geq\tilde{\nu},$
$\displaystyle\hat{\rho}_{*}$ $\displaystyle\geq\tilde{\rho},$
$\displaystyle\hat{\gamma}^{2}$ $\displaystyle\leq\tilde{\gamma}^{2},$
$\displaystyle\hat{\nu}$ $\displaystyle\geq\tilde{\nu},$
$\displaystyle\hat{\rho}$ $\displaystyle\geq\tilde{\rho}.$

###### Proof:

Consider the first set of inequalities in (36). In there, note that, the
inequality $\hat{\gamma}^{2}_{*}\leq\gamma^{2}_{*}$ has already been
established in Th. 3. Further, due to the fact that $K_{s}\geq 0$ and
$u^{\mathsf{T}}(t)u(t)>0$, from comparing (30) with (35), it is easy to see
that $\hat{\gamma}^{2}_{*}\leq\tilde{\gamma}^{2}$. Furthermore, using: (11),
$(t_{o},t_{f})\in\hat{\Psi}_{\gamma}$, Co. 1 and (35), we can simplify

$\displaystyle\hat{\gamma}^{2}\triangleq\frac{\int_{t_{o}}^{t_{f}}y^{\mathsf{T}}(t)y(t)dt}{\int_{t_{o}}^{t_{f}}u^{\mathsf{T}}(t)u(t)dt}\leq$
$\displaystyle\underset{(t_{0},t_{1})\in\hat{\Psi}_{\gamma}}{\max}\frac{\int_{t_{0}}^{t_{1}}y^{\mathsf{T}}(t)y(t)dt}{\int_{t_{0}}^{t_{1}}u^{\mathsf{T}}(t)u(t)dt}$
$\displaystyle=$
$\displaystyle\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}}{\max}\frac{y^{\mathsf{T}}(t)y(t)}{u^{\mathsf{T}}(t)u(t)}\triangleq\tilde{\gamma}^{2},$

and thus, $\hat{\gamma}^{2}\leq\tilde{\gamma}^{2}$. Following the same steps,
the last two sets of inequalities in (36) can also be established. ∎

A critical consequence of Lm. 5 is that it can be used to establish bounds on
the optimal system indices (9) with respect to their estimates proposed in
(11) [14] as:

$\displaystyle\hat{\gamma}^{2}-2(\tilde{\gamma}^{2}-\hat{\gamma}_{*}^{2})\leq\gamma_{*}^{2}$
$\displaystyle\mbox{ with }\ \ (\tilde{\gamma}^{2}-\hat{\gamma}_{*}^{2})\geq
0,$ (37) $\displaystyle\hat{\nu}+2(\hat{\nu}_{*}-\tilde{\nu})\geq\nu_{*}$
$\displaystyle\mbox{ with }\ \ (\hat{\nu}_{*}-\tilde{\nu})\geq 0,$
$\displaystyle\hat{\rho}+2(\hat{\rho}_{*}-\tilde{\rho})\geq\rho_{*}$
$\displaystyle\mbox{ with }\ \ (\hat{\rho}_{*}-\tilde{\rho})\geq 0.$

On the other hand, note that, as $K_{s}\rightarrow 0$, the optimal estimates
given in (30) may become: (i) inaccurate (due to violation of As. 2), (ii)
close to the optimal values (9), (iii) close to the estimates given in (35)
and (iv) distant from the estimates given in (11). On the other hand, as
$K_{s}$ increases, the optimal estimates given in (30) may become: (i)
accurate (due to satisfaction of As. 2), (ii) distant from the optimal values
(9), (iii) distant from the estimates given in (35), and (iv) close to the
estimates given in (11). Therefore, the importance of selecting a moderate
$K_{s}$ value is clear.

For this purpose, we assume the availability of a test input-output profile of
the system (1) (i.e., $\\{(u(t),y(t)):t\in[t_{o},t_{f})\\}$) and the
corresponding evaluated estimates given in (11) (i.e.,
$\hat{\gamma}^{2},\hat{\nu},\hat{\rho}$) and (35) (i.e.,
$\tilde{\gamma}^{2},\tilde{\nu},\tilde{\rho}$). Let us define

$\bar{\gamma}^{2}\triangleq\frac{1}{2}(\hat{\gamma}^{2}+\tilde{\gamma}^{2}),\
\ \ \ \bar{\nu}\triangleq\frac{1}{2}(\hat{\nu}+\tilde{\nu}),\ \ \ \
\bar{\rho}\triangleq\frac{1}{2}(\hat{\rho}+\tilde{\rho}),$ (38)

and three candidate $K_{s}$ parameter values:

$\displaystyle K_{s}\triangleq$ $\displaystyle\
\max_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}}y^{\mathsf{T}}(t)y(t)-\bar{\gamma}^{2}u^{\mathsf{T}}(t)u(t),$
(39) $\displaystyle K_{s}\triangleq$ $\displaystyle\
\max_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}}\bar{\nu}u^{\mathsf{T}}(t)u(t)-u^{\mathsf{T}}(t)y(t),$
$\displaystyle K_{s}\triangleq$ $\displaystyle\
\max_{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{y}}\bar{\rho}y^{\mathsf{T}}(t)y(t)-u^{\mathsf{T}}(t)y(t).$

###### Theorem 4

When the three optimization problems in (30) are solved using the respective
$K_{s}$ parameters given in (39), the resulting estimates of the optimal
system indices (30) will lie within the bounds (37) found for the estimates
given by (11) [14] (implying an increased accuracy).

###### Proof:

We first consider the estimation of the L2G value (using the first equations
in (30) and (39)). From (39),

$\displaystyle K_{s}$ $\displaystyle\geq
y^{\mathsf{T}}(t)y(t)-\bar{\gamma}^{2}u^{\mathsf{T}}(t)u(t),\ \ \forall
t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}$
$\displaystyle\iff\bar{\gamma}^{2}\geq\frac{y^{\mathsf{T}}(t)y(t)-K_{s}}{u^{\mathsf{T}}(t)u(t)},\
\ \forall t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{u}$
$\displaystyle\iff\bar{\gamma}^{2}\geq\hat{\gamma}_{*}^{2},\ \ \mbox{(sub.
from \eqref{Eq:Th:EstimatedOptimalSystemIndices})}$
$\displaystyle\iff\frac{1}{2}(\hat{\gamma}^{2}+\tilde{\gamma}^{2})\geq\hat{\gamma}_{*}^{2}.\
\ \mbox{(sub. from \eqref{Eq:MeanEstimates})}$ (40)

According to Lm. 5, $\tilde{\gamma}^{2}\geq\hat{\gamma}^{2}$. It is easy see
that

$\tilde{\gamma}^{2}\geq\hat{\gamma}^{2}\iff
2\tilde{\gamma}^{2}-\hat{\gamma}^{2}\geq\frac{1}{2}(\hat{\gamma}^{2}+\tilde{\gamma}^{2}).$

Applying this result in (40), we get

$\displaystyle 2\tilde{\gamma}^{2}-\hat{\gamma}^{2}\geq\hat{\gamma}_{*}^{2}$
$\displaystyle\iff
2\tilde{\gamma}^{2}-2\hat{\gamma}_{*}^{2}-\hat{\gamma}^{2}\geq-\hat{\gamma}_{*}^{2}$
$\displaystyle\iff\hat{\gamma}^{2}-2(\tilde{\gamma}^{2}-\hat{\gamma}_{*}^{2})\leq\hat{\gamma}_{*}^{2}$
$\displaystyle\iff\hat{\gamma}^{2}-2(\tilde{\gamma}^{2}-\hat{\gamma}_{*}^{2})\leq\hat{\gamma}_{*}^{2}\leq\gamma_{*}^{2}\
\ \mbox{(Lm. \ref{Lm:Bounds})}$

Comparing this result with (37) leads to the conclusion that the estimate
$\hat{\gamma}_{*}^{2}$ (evaluated using (30) under (39)) will fall within the
bound given in (37) (for the estimate (11) [14]) ∎

##  V Generalization for Discrete-Time Systems

So far, we have focused only on continuous-time non-linear dynamical systems
(1). In contrast, this section will focus on discrete-time non-linear
dynamical systems and briefly specialize the previously presented results.

### Discrete-Time System

Consider the discrete-time non-linear system:

$\mathcal{H}:\begin{cases}x(t+1)&=f(x(t),u(t)),\\\
y(t)&=h(x(t),u(t)),\end{cases}$ (41)

where $t\in\mathbb{Z}_{\geq 0}$. It is easy to see that As. 1, Defs. 1-7, Rm.
1, Lm. 1, (11) and Lm. 2 are also valid with regard to the discrete-time
system (41) upon some obvious minor modifications (e.g., replacing continuous
integrals with discrete summations). Hence, we omit re-stating them here in
the interest of brevity. Note however that As. 2 now requires a special
modification, and thus, we re-state it as follows.

###### Assumption 6

The magnitude change in the storage function $S:X\rightarrow\mathbb{R}_{\geq
0}$ over a single time step is bounded such that $|S(x(t+1))-S(x(t))|\leq
K_{s}$ over the discrete-time period $t\in[t_{o},t_{f})$, where $K_{s}$ is a
known constant.

### Fractional Function Optimization

Considering the discrete-time case, we will now specialize some key results
established in Sec. IV. First, let us define the discrete-time version of the
generic fractional function form given in (8). Let
$t_{o},t_{f}\in\mathbb{Z}_{\geq 0}$ where $t_{o}<t_{f}$ be two discrete-time
instants and $p(t)$,$q(t)$ where $p:[t_{o},t_{f})\rightarrow\mathbb{R}$,
$q:[t_{o},t_{f})\rightarrow\mathbb{R}_{\geq 0}$ be two discrete-time signals.
Moreover, let $r(\tau)$, $s(\tau)$ where
$r:\mathbb{Z}_{>0}\rightarrow\mathbb{R}$,
$s:\mathbb{Z}_{>0}\rightarrow\mathbb{R}_{\geq 0}$ be two discrete linear
mappings.

Now, for any discrete-time interval $[t_{0},t_{1})\subseteq[t_{o},t_{f})$, let
us define a corresponding _fractional function_ of the form:

$\psi(t_{0},t_{1})\triangleq\frac{\sum_{t\in[t_{0},t_{1})}p(t)+r(t_{1}-t_{0})}{\sum_{t\in[t_{0},t_{1})}q(t)+s(t_{1}-t_{0})}.$
(42)

Note that As. 3, Lm. 4 and the set $\Psi$ (20) are also valid for any
fractional function $\psi(t_{0},t_{1})$ of the form (42) as well. Thus,
parallel to Th. 1, the following theorem can be proved.

$\displaystyle\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\max}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\
\max\Big{\\{}\big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\max}\
\psi(t,t+1)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\max}\
\psi(t_{zo}-1,t)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\max}\
\psi(t,t_{zf}+1)\big{\\}}\Big{\\}},$ (43)
$\displaystyle\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\min}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\
\min\Big{\\{}\big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\min}\
\psi(t,t+1)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\min}\
\psi(t_{zo}-1,t)\big{\\}},\
\big{\\{}\underset{\begin{subarray}{c}t\in[t_{zo},t_{zf})\\\
(t_{zo},t_{zf})\in\bar{\Psi}_{q}\end{subarray}}{\min}\
\psi(t,t_{zf}+1)\big{\\}}\Big{\\}},$ (44)
$\displaystyle\psi(t_{0},t_{1})\triangleq\
\frac{\sum_{t\in[t_{0},t_{1})}p(t)+r(t_{1}-t_{0})}{\sum_{t\in[t_{0},t_{1})}q(t)},\
\ \ \mbox{(i.e., \eqref{Eq:GenericRationalFunctionDisc} under As.
\ref{As:NoLinearMappingInDenominator})\ \ \ and }\ \ \
\psi(t,t+1)=\frac{p(t)+r(1)}{q(t)},$ (45)
$\displaystyle\psi(t_{zo}-1,t)=\frac{\sum_{\tau\in[t_{zo}-1,t)}p(\tau)+r(t-t_{zo}+1)}{q(t_{zo}-1)},\
\ \ \mbox{ and }\ \ \
\psi(t,t_{zf}+1)=\frac{\sum_{\tau\in[t,t_{zf}+1)}p(\tau)+r(t_{zf}+1-t)}{q(t_{zf})}.$
(46)

###### Theorem 5

Any fractional function $\psi(t_{0},t_{1})$ of the form (42) under As. 3 can
be efficiently optimized over the corresponding space $(t_{0},t_{1})\in\Psi$
exploiting the relationships:

$\Big{\\{}\underset{(t_{0},t_{1})\in\Psi}{\max}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})}{\max}\
\psi(t,t+1)\Big{\\}},$ (47) $\Big{\\{}\underset{(t_{0},t_{1})\in\Psi}{\min}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})}{\min}\
\psi(t,t+1)\Big{\\}},$ (48)

where

$\psi(t,t+1)=\frac{p(t)+r(1)}{q(t)+s(1)}.$

###### Proof:

The proof follows the same steps as that of Th. 1 and is, therefore, omitted.
∎

Note that unlike in Th. 1, Th. 5 does not involve $\dot{r}(0)$ and
$\dot{s}(0)$ values and limits $\lim_{\Delta\rightarrow 0}\psi(t,t+\Delta)$.
Instead, it now involves $r(1)$ and $s(1)$ values and unit steps
$\psi(t,t+1)$.

We next relax As. 3 in Th. 5 by following the same steps as before, i.e., by
making a simplifying assumption (As. 4) and defining sets $\Psi_{q}$,
$\bar{\Psi}_{q}$ and $\tilde{\Psi}_{q}$ appropriately. Note that As. 4 and the
sets $\Psi_{q}$, $\bar{\Psi}_{q}$ (with $\epsilon=1$) and $\tilde{\Psi}_{q}$
defined earlier are also valid for any fractional function $\psi(t_{0},t_{1})$
of the form (42) as well. Therefore, parallel to Th. 2, the following theorem
now can be established.

###### Theorem 6

Any fractional function $\psi(t_{0},t_{1})$ of the form (42) under As. 4 can
be efficiently optimized over the corresponding space
$(t_{0},t_{1})\in\Psi\backslash\Psi_{q}$ exploiting the relationships: (43)
and (44) (under the objective functions in (45) and (46)).

###### Proof:

The proof follows the same steps as that of Th. 2 and is, therefore, omitted.
∎

In parallel to Co. 1, the following corollary shows that under some special
conditions, when optimizing a fractional function of the form (42), the dead-
zones can be conveniently disregarded. Before proceeding, we point out that
As. 5 is also valid (upon some obvious minor modifications) for any fractional
function of the form (42) as well.

###### Corollary 2

Any fractional function $\psi(t_{0},t_{1})$ of the form (42) under As. 4 and
As. 5 can be efficiently optimized over the corresponding space
$(t_{0},t_{1})\in\Psi\backslash\Psi_{q}$ exploiting the relationships: (i) If
$r(\tau)\leq 0,\forall\tau\in\mathbb{Z}_{>0}$:

$\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\max}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\max}\
\psi(t,t+1)\Big{\\}},$ (49)

(ii) If $r(\tau)\geq 0,\forall\tau\in\mathbb{Z}_{>0}$:

$\Big{\\{}\underset{(t_{0},t_{1})\in\Psi\backslash\Psi_{q}}{\min}\
\psi(t_{0},t_{1})\Big{\\}}\equiv\Big{\\{}\underset{t\in[t_{o},t_{f})\backslash\tilde{\Psi}_{q}}{\min}\
\psi(t,t+1)\Big{\\}},$ (50)

where

$\psi(t,t+1)=\frac{p(t)+r(1)}{q(t)}.$

###### Proof:

These results directly follow from applying the assumed conditions in As. 5
and Co. 2 in Th. 6. ∎

### Application to Solve the Discrete Version of (13)

For the discrete-time system (41), recall that we had to specialize several
theoretical results such as Th. 1, Th. 2 and Co. 1 as Th. 5, Th. 6 and Co. 2,
respectively. However, when it comes to the application, i.e., solving the
optimization problems in (13) considering the discrete-time system (41), it is
easy to see that Th. 3 can still be used (i.e., valid - in particular, in the
light of Co. 2).

In essence, Th. 3 can be used to estimate the optimal system indices of both
continuous-time (1) as well as discrete-time (41) systems. Moreover, the same
can be said regarding the Remarks 2-5 and the theoretical results reported in
Lm. 5 and Th. 3. Therefore, we omit re-stating these results here in the
interest of brevity.

##  VI Numerical Results

In our numerical experiments, we considered two linear systems
$\mathcal{H}_{1},\mathcal{H}_{2}$ and two non-linear systems
$\mathcal{H}_{3},\mathcal{H}_{4}$:

$\displaystyle\mathcal{H}_{1}:$
$\displaystyle\begin{cases}\dot{x}=\begin{bmatrix}-0.261&1.027&-0.074\\\
-0.8786&-0.184&-0.644\\\ 0.537&0.364&-0.576\end{bmatrix}x+\begin{bmatrix}0\\\
0.936\\\ 0\end{bmatrix}u,\\\
y=\begin{bmatrix}-3.020&-1.103&-1.032\end{bmatrix}x,\end{cases}$ (51)
$\displaystyle\mathcal{H}_{2}:$
$\displaystyle\begin{cases}\dot{x}=\begin{bmatrix}-0.681&0.495&-0.651\\\
0.107&-0.815&-0.686\\\
0.811&0.487&-0.466\end{bmatrix}x+\begin{bmatrix}0.133\\\ 0\\\
-1.399\end{bmatrix}u,\\\
y=\begin{bmatrix}0&0&1.025\end{bmatrix}x+\begin{bmatrix}-0.380\end{bmatrix}u,\end{cases}$
$\displaystyle\mathcal{H}_{3}:$
$\displaystyle\begin{cases}\dot{x}=-x^{3}-x-u,\\\ y=2x+u,\end{cases}$
$\displaystyle\mathcal{H}_{4}:$
$\displaystyle\begin{cases}\dot{x}=-x^{3}-\frac{1}{3}x-\frac{2}{3}u,\\\
y=\frac{2}{3}x+\frac{4}{3}u.\end{cases}$

The corresponding optimal system indices (9) (evaluated analytically in off-
line using MATLAB® and [16]) are given in Tab. I (Col. 3). In Tab. I, note
that we have omitted system index types with infinite or unknown (not
established theoretically) optimal index values (e.g., optimal OFP index of
$\mathcal{H}_{1}$ is $-\infty$ and optimal IFP index of $\mathcal{H}_{3}$ is
unknown).

TABLE I: The summary of the observed numerical results.

System |  System Index |  Optimal Index Val. |  Learned $K_{s}$ Value | Estimate At $t=100$ | Abs. Est. Error |  $\%$ AEE Improvement | Mean Abs. Est. Error |  $\%$ MAEE Improvement
---|---|---|---|---|---|---|---|---
AVG [14] | FFO | AVG [14] | FFO | AVG [14] | FFO
$\mathcal{H}_{1}$ | L2G | 17.575 | 2165.0 | 7.933 | 16.931 | 9.642 | 0.644 | 93.32 | 9.344 | 1.003 | 89.27
IFP | -8.067 | 322.5 | -2.629 | -3.850 | 5.438 | 4.217 | 22.45 | 5.455 | 4.274 | 21.65
$\mathcal{H}_{2}$ | IFP | -2.017 | 35.06 | -0.955 | -1.276 | 1.062 | 0.741 | 30.23 | 1.040 | 0.748 | 28.08
OFP | -2.630 | 6.634 | -0.983 | -2.231 | 1.647 | 0.399 | 75.77 | 1.672 | 0.399 | 76.14
$\mathcal{H}_{3}$ | L2G | 1.000 | 12.37 | 0.249 | 0.627 | 0.751 | 0.373 | 50.33 | 0.742 | 0.373 | 49.73
$\mathcal{H}_{4}$ | OFP | 0.750 | 2.876 | 0.850 | 0.799 | 0.100 | 0.049 | 50.95 | 0.098 | 0.049 | 50.20

To synthetically generate input-output data (on-line), we used Simulink®. In
particular, we set each system initial state as $x(0)=\mathbf{0}$ and
generated the output $y(t)$ for the input $u(t)$:

$u(t)=a+cos(bt)+pul(2t)+0.01v(t),\ \forall t\in[0,100),$ (52)

where $v(t)$ represents a normal random process and $pul(t)$ represents a
pulse wave [17] with $50\%$ duty-cycle, unit period and unit amplitude. For
the considered four systems in (51), the parameters $a$ and $b$ in (52) were
selected as $16.71,9.71,4.71,4.71$ and $1.02,0.96,0.1,0.1$, respectively. The
observed input-output profiles are shown in Fig. 1.

(a) System: $\mathcal{H}_{1}$

(b) System: $\mathcal{H}_{2}$

(c) System: $\mathcal{H}_{3}$

(d) System: $\mathcal{H}_{4}$

Figure 1: Observed input-output profiles.

In each experiment, inspired by Th. 4, the input-output data seen over the
period $t\in[0,10)$ (sampled at a rate $1\,kHz$) was used to compute/learn the
$K_{s}$ parameter value via (39). The learned $K_{s}$ values are given in Tab.
I (Col. 4).

In Simulink®, parallel to the systems in (51), we also implemented the on-line
system indices _estimators_ described by: (i) the averaging-based approach
(11) proposed in [14] (labeled as “AVG”) and (ii) the FFOP-based approach (30)
proposed in this paper (labeled as “FFO”). Figure 2 shows the variations of
these observed on-line estimates of different interested system indices over
the period $t\in[0,100)$.

(a) System: $\mathcal{H}_{1}$, Index: L2G

(b) System: $\mathcal{H}_{1}$, Index: IFP

(c) System: $\mathcal{H}_{2}$, Index: IFP

(d) System: $\mathcal{H}_{2}$, Index: OFP

(e) System: $\mathcal{H}_{3}$, Index: L2G

(f) System: $\mathcal{H}_{4}$, Index: OFP

Figure 2: Observed on-line estimates of different interested system indices.
The on-line AVG and FFO estimators respectively use (11) and (30) (with
$t_{o}=0$ and $t_{f}=t$).

We define the absolute estimation error (AEE) as the absolute difference
between an observed terminal (at $t=100$) estimate value of an interested
system index and the corresponding optimal system index value. For example,
with regard to the L2G index of a system,
$|\gamma^{2}_{*}-\hat{\gamma}^{2}(100)|$ and
$|\gamma_{*}^{2}-\hat{\gamma}_{*}^{2}(100)|$ are the respective AEE values of
AVG [14] and FFO estimators. The terminal estimate values, the corresponding
AEE values and the percentage AEE improvement achieved by the proposed on-line
estimator in this paper (i.e., FFO) are shown in Tab. I (Cols. 5-9). From
these results, it is clear that the proposed FFO estimator is significantly
more accurate than the AVG estimator proposed in [14].

Finally, we define the mean AEE (MAEE) as the mean estimation error of an
interested system index with respect to the corresponding optimal system index
value over the period $t\in[0,100)$. For example, with regard to the OFP index
of a system, $\frac{1}{100}\int_{0}^{100}|\rho_{*}-\hat{\rho}(t)|dt$ and
$\frac{1}{100}\int_{0}^{100}|\rho_{*}-\hat{\rho}_{*}(t)|dt$ are the respective
MAEE values of AVG [14] and FFO estimators. The observed MAEE values and the
percentage MAEE improvement achieved by the proposed on-line estimator in this
paper (i.e., FFO) are shown in Tab. I (Cols. 10-12). These results again
confirm the significant accuracy improvement achieved by the proposed FFO
estimator compared to the AVG estimator proposed in [14].

We conclude this paper by re-emphasizing several critical advantages of the
proposed on-line estimator: (i) It is computationally efficient and flexible
to implement (see Rm. 2 and Rm. 3); (ii) It provides highly convergent yet
reactive/robust estimates compared those provided by the AVG estimator [14]
(see Fig. 2); (iii) It requires significantly less amount of data compared to
the existing off-line approaches such as [7]; (iv) It is applicable for
general non-linear systems; (v) It requires no control over the control input
fed into the system (hence can be implemented parallel to existing control
systems). The ongoing research focuses on trading off the latter two
features/strengths of the proposed estimator to increase the estimator
accuracy further.

##  VII Conclusion

We proposed an on-line estimation method to estimate a few critical stability
and passivity metrics (system indices) of a non-linear system using input-
output data. First, we exploited fundamental definitions of such system
indices to formulate the corresponding estimation problems as fractional
function optimization problems of a specific common form. Next, we established
several interesting theoretical results regarding this class of FFOPs. Then,
these results were exploited to derive on-line estimates for the interested
system indices. Compared to an existing on-line estimation method (state of
the art, to the best of our knowledge), the proposed approach is more
theoretically precise, computationally efficient, and enjoys several other
desirable qualities. The validity and potential of the proposed approach were
demonstrated using numerical examples. Future work aims to derive specific
results for linear systems and optimal controls to favor on-line estimation.

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