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# Gap Reduced Minimum Error Robust Simultaneous Estimation For Unstable Nano
Air Vehicle

Jinraj V. Pushpangathan, Harikumar Kandath, , Suresh Sundaram, , and
Narasimhan Sundararajan Research fellow @ Department of Aerospace Engineering,
Indian Institute of Science, Bangalore-560012, India, e-mail:
(jinrajaero@gmail.com).Assistant professor @ International Institute of
Information Technology, Hyderabad, India, e-mail: (
harikumar.k@iiit.ac.in)Associate professor @ Department of Aerospace
Engineering, Indian Institute of Science, Bangalore- 560012, India, e-mail:
(vssuresh@iisc.ac.in).Professor (Retd.) @ School of Electrical and Electronics
Engineering, Nanyang Technological University, Singapore, e-mail:
(ensundara@ntu.edu.sg).

###### Abstract

This paper proposes a novel Gap Reduced Minimum Error Robust Simultaneous
(GRMERS) estimator for resource-constrained Nano Aerial Vehicle (NAV) that
enables a single estimator to provide simultaneous and robust estimation for a
given $N$ unstable and uncertain NAV plant models. The estimated full state
feedback enables a stable flight for NAV. The GRMERS estimator is implemented
utilizing a Minimum Error Robust Simultaneous (MERS) estimator and Gap
Reducing (GR) compensators. The MERS estimator provides robust simultaneous
estimation with minimal largest worst-case estimation error even in the
presence of a bounded energy exogenous disturbance signal. The GR compensators
reduce the gap between the graphs of $N$ linear plant models to decrease the
estimation error generated by the MERS estimator. A sufficient condition for
the existence of a simultaneous estimator is established using LMIs and robust
estimation theory. Further, MERS estimator and GR compensator design are
formulated as non-convex tractable optimization problems and are solved using
the population-based genetic algorithms. The performance of the GRMERS
estimator consisting of MERS estimator and GR compensators from the
population-based genetic algorithms is validated through simulation studies.
The study results indicate that a single GRMERS estimator can produce state
estimates with reduced errors for all flight conditions. The results indicate
that the single GRMERS estimator is robust than the individually designed
$H_{\infty}$ filters.

###### Index Terms:

Linear matrix inequality, Nano air vehicle, Robust simultaneous estimator,
$v$-gap metric

##  I Introduction

Recent trends in Micro Air Vehicles (MAVs) point to the development of a new
class of small air vehicles called Nano Air Vehicles (NAVs) that execute
specific missions undetected with a high degree of agility. NAVs can be
broadly classified into three categories, viz., fixed-wing NAVs, rotary wing
NAVS, and flapping-wing NAVs. They are widely used for intelligence
operations, battlefield surveillance, reconnaissance, and disaster assessment
missions. These small vehicles have severe dimensional and weight constraints
as their overall dimensions and weights need to be lower than $75$ mm and $20$
g, respectively [1]. Figure 1(a) shows a typical fixed-wing NAV that weighs
$19.4$ g and has an overall dimension of $75$ mm [2].

In general, the plant models of these NAVs are multi-input-multi-output
(MIMO), unstable, uncertain, adversely coupled, and have a different number of
unstable poles [3, 5]. In this paper, for convenience, we use the terms plant
and estimator to represent the plant model and estimator model, respectively.
More details about these plant characteristics are given in [1, 2]. Due to
these plant characteristics, the NAV’s require complex feedback controllers to
accomplish a mission. For using the existing closed-form solutions of the full
state feedback for designing a feasible controller for a NAV requires the
measurements or estimates of all the system’s state variables. However, due to
the weight and dimensional constraints, the autopilot hardware of a NAV like
the one shown in Fig. 1(b) [2] has severe resource constraints, such
unavailability of lightweight sensors to measure every state variable (like
translational velocities, angle-of-attack, airspeed, side slip angle).
Moreover, these autopilots have both limited computational and memory powers.
Hence, one needs to design a computationally simple full state estimator from
the available measurements.

(a) Fixed-wing NAV  (b) Autopilot hardware

Figure 1: 75 mm wingspan NAV and autopilot hardware

Extended Kalman Filter (EKF) is the de-facto standard for UAV estimation
schemes. However, EKF is not suitable for a NAV, as its autopilot hardware has
limited computational and memory resources, specifically for Jacobian
computing. A gain-scheduled EKF reduces the computational cost of calculating
the estimator gain [6, 7], but it still requires the measurement of scheduling
variables like airspeed etc. Further, the significant model uncertainties in
NAVs can induce notable errors in the state estimates. These difficulties
clearly point out to the need for an estimation algorithm that is
computationally simpler and robust to model uncertainties. This algorithm
should also cater to both stable/unstable plant models and should not require
a computationally expensive gain-scheduling approach. To put the problem in a
sharp focus, a NAV requires a single computationally less intensive and robust
estimator to estimate the states of a finite set of MIMO LTI uncertain
unstable plants. Herein, such an estimator is referred to as a Robust
Simultaneous (RS) estimator. The overall problem definition of simultaneous
estimation is given below.

Simultaneous Estimation problem: Let us consider a finite set,
$\mathcal{P}=\\{\mathbf{P}_{i}(s)\in\mathcal{RL_{\infty}}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$, containing stabilizable
and detectable MIMO LTI plant models in a transfer function matrix form (i.e.,
$\mathbf{P}_{i}(s)\in\mathcal{RL_{\infty}}$). Here, $\mathcal{RL_{\infty}}$
symbolizes the space of proper, real-rational functions of $s\in\mathcal{C}$
which has no poles in the imaginary axis of $s$-plane. The state-space form of
$\mathbf{P}_{i}(s)$ is given by

$\displaystyle\mathbf{P}_{i}(s):\left\\{\begin{array}[]{ll}\dot{\mathbf{x}}_{i}(t)&=A_{i}\mathbf{x}_{i}(t)+B_{i}\mathbf{u}_{i}(t)\\\
\mathbf{y}_{i}(t)&=C_{i}\mathbf{x}_{i}(t)+\mathbf{v}(t)\\\
\mathbf{z}_{i}(t)&=C_{z}\mathbf{x}_{i}(t)\end{array}\right.$ (1)

where $\mathbf{x}_{i}(t)\in\mathbb{R}^{\hat{n}}$,
$\mathbf{u}_{i}(t)\in\mathbb{R}^{\hat{m}}$,
$\mathbf{y}_{i}(t)\in\mathbb{R}^{\hat{r}}$, and
$\mathbf{z}_{i}(t)\in\mathbb{R}^{\hat{q}\leq\hat{n}}$ represent the state
vector, the control input vector, the measurement vector, and the vector that
contains those states that need to be estimated, respectively. Besides this,
$\mathbf{v}(t)\in\mathbb{R}^{\hat{r}}$ is a bounded energy measurement noise
vector. Note that the noise considered in this article is zero-mean Gaussian
white. Further, $A_{i}\in\mathbb{R}^{\hat{n}\times\hat{n}}$,
$B_{i}\in\mathbb{R}^{\hat{n}\times\hat{m}}$, and
$C_{i}\in\mathbb{R}^{\hat{r}\times\hat{n}}$ are the system, input, and output
matrices, respectively. Also, $C_{z}\in\mathbb{R}^{\hat{q}\times\hat{n}}$ is a
constant matrix for all the plants. Let
$\hat{\mathbf{P}}_{l}(s)\in\mathcal{RH}_{\infty}$ be a suitable estimator for
all the plant models belonging to $\mathcal{P}$. Here, $\mathcal{RH_{\infty}}$
denotes the space of proper, real-rational functions of $s\in\mathcal{C}$
which are analytic in $\mathcal{C}_{+}$. Let the state-space form of
$\hat{\mathbf{P}}_{l}(s)$ be formed using the state-space matrices of
$\mathbf{P}_{l}(s)\in\mathcal{P}$. Following this, one of the state-space
realizations of $\hat{\mathbf{P}}_{l}(s)$ is given by

$\displaystyle\hat{\mathbf{P}}_{l}(s):\left\\{\begin{array}[]{ll}\dot{\hat{\mathbf{x}}}_{l}(t)=&A_{l}\hat{\mathbf{x}}_{l}(t)+B_{l}\mathbf{u}(t)+\mathbf{L}_{l}\left(\mathbf{y}(t)-C_{l}\hat{\mathbf{x}}_{l}(t)\right)\\\
\hat{\mathbf{z}}_{l}(t)=&C_{z}\hat{\mathbf{x}}_{l}(t)\end{array}\right.$ (4)

where $\hat{\mathbf{x}}_{l}(t)\in\mathbb{R}^{\hat{n}}$ is the state vector of
the estimator, $\hat{\mathbf{z}}_{l}(t)\in\mathbb{R}^{\hat{q}}$ is the output
vector of the estimator which provides the estimate of
$\hat{\mathbf{z}}_{i}(t)$ of $\mathbf{P}_{i}(s)\in\mathcal{P}$ for all
$i\leavevmode\nobreak\ \in\\{1,\dots,l,\dots,N\\}$, and
$\textbf{L}_{l}\in\mathbb{R}^{\hat{n}\times\hat{r}}$ is the estimator gain. In
(4), $\mathbf{u}(t)$ and $\mathbf{y}(t)$ are the inputs to the estimator from
the plant. For example, $\mathbf{u}(t)=\mathbf{u}_{l}(t)$ and
$\mathbf{y}(t)=\mathbf{y}_{l}(t)$ when $\hat{\mathbf{P}}_{l}(s)$ becomes the
estimator of $\mathbf{P}_{l}(s)$. Also, for all $i\in\\{1,\dots,l,\dots,N\\}$,
$\mathbf{e}_{\mathbf{z}_{il}}(t)=\mathbf{z}_{i}(t)-\mathbf{\hat{z}}_{l}(t)\in\mathbb{R}^{\hat{q}}$
be the estimation error vector when $\hat{\mathbf{P}}_{l}(s)$ estimates
$\mathbf{z}_{i}(t)$ of $\mathbf{P}_{i}(s)\in\mathcal{P}$, respectively. Let
$\mathbf{d}(t)$ be a bounded energy exogenous signal vector that adversely
affect the estimation error dynamics of the estimator by increasing the
estimation errors (e.g. $\mathbf{v}(t)$ given in (1)). Now, the simultaneous
estimation problem is about finding an estimator $\hat{\mathbf{P}}_{l}(s)$ for
$\mathcal{P}$ such that the following two conditions hold
$\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$.
Condition I: when $\mathbf{d}(t)=\mathbf{0}$, all the estimation error vectors
should asymptotically converge to zero, i.e.,

$\displaystyle\lim_{t\to\infty}\mathbf{e}_{\mathbf{z}_{il}}(t)$
$\displaystyle=0$ (5)

Condition II: when $\mathbf{d}(t)\neq\mathbf{0}$, the root mean square (RMS)
gain from $\mathbf{d}(t)$ should be bounded, i.e.,

$\displaystyle\underset{\underset{\mathbf{d}(t)\neq\mathbf{0}}{\mathbf{d}(t)\in\mathcal{L}_{2}[0,\infty)}}{\sup}\leavevmode\nobreak\
\frac{\parallel\mathbf{e}_{\mathbf{z}_{il}}(t)\parallel_{2}}{\parallel\textbf{d}(t)\parallel_{2}}$
$\displaystyle<\gamma$ (6)

where $\gamma$ is a constant that satisfies $1>\gamma>0$ and $\sup$ symbolizes
supremum. In this paper, only Condition II is considered as the measurements
of NAV are always affected by $\mathbf{v}(t)$. Before formulating the problem
of simultaneous estimation for a NAV, we provide first a brief review of the
existing simultaneous estimation research work in the literature below.

The problem of simultaneous observation was first studied in [8]. In this
paper, coprime factorization technique was utilized to solve the simultaneous
observation problem. For the finite set that contains at least one stable
plant, the necessary and sufficient conditions for the existence of
simultaneous observations were obtained. Using the proposed method, a
simultaneous observer for two plants was designed. In [9], a stable inverse
approach was employed to synthesize a simultaneous observer for a given set of
plants. The restrictions on these plants were that they should not have any
right half plane zeros besides satisfying the condition,
$(\hat{m}+\hat{r})>\hat{m}N$. For plants that have the same number of inputs
and outputs without common eigenvalues, the necessary and sufficient
conditions for the existence of a simultaneous functional observer were
presented in [10]. In [11], algebraic geometry tools were presented to
characterize the simultaneous observability of a set of linear single-input
single-output plants and also to design a simultaneous state observer for the
same. The methods presented in [8]-[11] are not suitable for synthesizing a
simultaneous estimator for a NAV due to the following reasons: All the plants
of the NAV may be unstable and also can have zeros on the right half of the
$s$-plane [12]. Also, the number of outputs of the NAV is more than the number
of inputs violating the condition mentioned in [10]. Furthermore, in the case
of a NAV, satisfying the condition, $(\hat{m}+\hat{r})>\hat{m}N$ mentioned in
[9] is not possible. For example, a NAV with three inputs and five outputs, a
simultaneous estimator can be synthesized only for two plants. The measured
outputs of the plants of a NAV are also affected by noise. Apart from this,
the plants of the NAV are subjected to higher model uncertainties. In [8, 11],
no method is explicitly proposed to provide the desired performance (to
achieve the condition II) by the estimator when the plants are subjected to
measurement noise and higher model uncertainties. To overcome the above-
mentioned limitations, one needs to develop a new method to synthesis a
simultaneous estimator for a NAV.

In this paper, we propose a novel Gap Reduced Minimum Error Robust
Simultaneous (GRMERS) estimator to handle unstable plants with model
uncertainties and measurement noises. The GRMERS estimator incorporates the
solution of two problems: a Minimum Error Robust Simultaneous (MERS)
estimation problem and a Gap Reduced (GR) compensator problem. The MERS
estimation problem finds a single estimator referred to as the MERS estimator
that accomplishes robust state estimation with minimal (largest) worst-case
estimation error for $N$ number of unstable uncertain MIMO linear plants of
the NAV. The estimation error of the MERS estimator is further reduced by
decreasing the gap (see [13] for definition) between the graphs (see
Definition II.1) of the plants in $\mathcal{P}$ by cascading these plants with
suitable pre/post compensators. These compensators are called the GR
compensators, and the corresponding synthesis problem is termed as the GR
compensators problem. The GR compensators are defined by first-order
differential equations which can be solved by using the limited computational
capabilities of the NAV’s autopilot hardware. Using the robust estimation
theory, the MERS estimator and GR compensator designs problems are devised as
non-convex optimization problems following the robust estimation theory,
formulated in terms of Linear Matrix Inequalities (LMI) and the properties of
$v$-gap metric, respectively. The major highlights of the proposed GRMERS
estimator in this paper are:

  1. 1.

This approach can handle the robust simultaneous (RS) estimation of more than
three ($N>$3) minimum/non-minimum phase unstable plants even having common
eigenvalues.

  2. 2.

To our best knowledge, it has been shown here (for the first time) that
cascading the plants with compensators (GR compensators) that reduce the gap
between the graphs of the plants can reduce the root mean square value of
estimation errors of a simultaneous estimator.

The effectiveness of both the MERSE and GRC algorithms are demonstrated by
generation of the MERS estimator and the GR compensators to synthesise the
GRMERS estimator for four unstable plants of the NAV mentioned in [2]. The
stability, nominal, and robust performances of the GRMERS estimator are 1)
validated through numerical simulations with Gaussian measurement noise and 2)
compared with the performances of MERS estimator and $H_{\infty}$ filter. For
this purpose, individual $H_{\infty}$ filters are designed separately for each
plant. The nominal performance analysis indicates that the best performance is
given by individual $H_{\infty}$ filters, followed by the GRMERS estimator. As
compared to the MERS estimator, the GRMERS estimator yields up to $55\%$
reduction in estimation error, which substantiates the effectiveness of
providing GR compensators. The robust performance analysis, however, shows
that the GRMERS estimator has a lower estimation error of up to
$43\leavevmode\nobreak\ \%$ compared to the $H_{\infty}$ filters.

The paper is organized as follows. The dynamics of a fixed-wing NAV and the
problem formulations are presented in Section II. In Section III, the GRMERS
estimator is discussed. The design and performance evaluation of the GRMERS
estimator are presented in Section IV. Finally, Section V summarizes the key
results of this paper.

##  II Problem Formulations of Minimum Error Robust Simultaneous Estimator and
Gap Reduced Compensators for a Fixed-wing NAV

In this section, the dynamic model of a fixed-wing NAV along with its
complexities is described first to motivate the need for robust simultaneous
estimation for a NAV . Next, the precise mathematical problem formulations for
the design of the Minimal Error Robust Simultaneous (MERS) estimator and the
Gap Reducing (GR) compensators are presented.

###  II-A Dynamics of a Single Propeller Fixed-Wing NAV

Here, a brief description of the dynamic model for a single propeller fixed-
wing NAV is provided, and more details can be found in [2, 1]. Generally, in a
fixed-wing aircraft, the actuator′s bandwidth would be much higher than that
of the plant, whereas it is not true in the case of a NAV. Hence, in the
flight controller and estimator design of a NAV, one has to explicitly include
the dynamics of the actuator along with the plant dynamics. Besides, the
dynamics of a single propeller fixed-wing NAV has significant cross-coupling
effects. Based on these considerations, a suitable linear model for a single
propeller fixed-wing NAV with both the coupling effects and actuator dynamics
is the linear coupled model [2] given by

$\dot{\mathbf{x}}_{i}(t)={A}_{i}\mathbf{x}_{i}(t)+B\mathbf{u}_{i}(t)$ (7)

where $\mathbf{x}_{i}(t)\in\mathbb{R}^{\hat{n}=11}$,
$\mathbf{u}_{i}(t)\in\mathbb{R}^{\hat{m}=3}$, $A_{i}\in\mathbb{R}^{(11\times
11)}=\left[\begin{array}[]{c;{2pt/2pt}r}A_{{Lo}_{i}}&A_{{Lo}_i}^{La}\\\
\hdashline[2pt/2pt]A_{{La}_{i}}^{Lo}&A_{{La}_i}\end{array}\right]$, and
$B_{i}\in\mathbb{R}^{(11\times
3)}=\left[\begin{array}[]{c;{2pt/2pt}r}B_{{Lo}_{i}}&B_{{Lo}_i}^{La}\\\
\hdashline[2pt/2pt]B_{{La}_{i}}^{Lo}&B_{{La}_i}\end{array}\right]$ are the
state vector, the control input vector, the system matrix, and the control
matrix, respectively. Here, $A_{{Lo}_{i}}\in\mathbb{R}^{6\times 6}$,
$A_{{La}_{i}}\in\mathbb{R}^{5\times 5}$, $B_{{Lo}_{i}}\in\mathbb{R}^{6\times
2}$, and $B_{{La}_{i}}\in\mathbb{R}^{5\times 1}$ represent the system and
control matrices of the longitudinal and lateral state-space models,
respectively. Also, $A_{{Lo}_{i}}^{La}\in\mathbb{R}^{6\times 5}$,
$A_{{La}_{i}}^{Lo}\in\mathbb{R}^{5\times 6}$,
$B_{{Lo}_{i}}^{La}\in\mathbb{R}^{6\times 1}$, and
$B_{{La}_{i}}^{Lo}\in\mathbb{R}^{5\times 2}$ are the longitudinal coupling
block of $A_{i}$, lateral coupling block of $A_{i}$, longitudinal coupling
block of $B_{i}$, and lateral coupling block of $B_{i}$, respectively.
Furthermore, $\mathbf{x}_{i}(t)$ and $\mathbf{u}_{i}(t)$ in (7) are defined as

$\displaystyle\mathbf{x}_{i}(t)=$
$\displaystyle\left[\begin{array}[]{ccccccccccc}u&w&q&\theta&\delta_{e}&\delta_{T}&v&p&r&\phi&\delta_{r}\end{array}\right]^{T}(t)$
(9) $\displaystyle\mathbf{u}_{i}(t)=$
$\displaystyle\left[\begin{array}[]{ccc}\delta_{eu}&\delta_{Tu}&\delta_{ru}\end{array}\right]^{T}(t)$
(11)

where $\left[\begin{array}[]{ccc}u(t)&v(t)&w(t)\end{array}\right]^{T}$ is the
body-fixed linearized translational velocities in m/s,
$\left[\begin{array}[]{ccc}p(t)&q(t)&r(t)\end{array}\right]^{T}$ is the body-
fixed linearized rotational velocities in rad/s, and
$\left[\begin{array}[]{cc}\theta(t)&\phi(t)\end{array}\right]^{T}$ is the
body-fixed linearized Eulers angles in rad. Also, $\delta_{e}(t)$,
$\delta_{r}(t)$, and $\delta_{T}(t)$ are the linearized elevator deflection
(in rad), rudder deflection (in rad), and propeller speed (in rps-revolution
per second), respectively. In (11), $\delta_{eu}(t)$ (rad), $\delta_{ru}(t)$
(in rad), and $\delta_{Tu}(t)$ (in rps) represent the inputs to the elevator
actuator, the input to the rudder actuator, and the input to the electric
motor that drives the propeller, respectively.

The linear dynamics of a fixed-wing NAV is adversely coupled, uncertain, and
unstable as seen from dynamics of the $75$ mm wingspan fixed-wing NAV
mentioned in [2]. Hence, the NAVs similar to the $75$ mm wingspan NAV require
flight controllers to handle all these complexities and accomplish the desired
mission. This controller can use either full state feedback or output feedback
strategy. Generally, the full state feedback strategy is preferred as various
closed-form solutions are available when compared with the output feedback
strategy. The development of a well-proven full state feedback flight
controller requires the measurement of all the state variables. If all the
state variables can not be measured, then estimates of unmeasured states are
required. Among all the state variables of the NAV, the measured state
variables are $q(t)$, $\theta(t)$, $p(t)$, $r(t)$, and $\phi(t)$ and can be
directly used for control. Hence, the measurement vector of the NAV,
$\mathbf{y}_{i}(t)\in\mathbb{R}^{\hat{r}=5}$, is given by

$\displaystyle\mathbf{y}_{i}(t)=$
$\displaystyle\left[\begin{array}[]{cccccccc}q&\theta&p&r&\phi\end{array}\right]^{T}(t)$
(13)

Following this, the measurement equation of the NAV is given by

$\displaystyle\mathbf{y}_{i}(t)=C\mathbf{x}_{i}(t)+\mathbf{v}(t)$ (14)

where $\mathbf{v}(t)\in\mathbb{R}^{5}$ and $C\in\mathbb{R}^{5\times 11}$. Due
to the absence of lightweight sensors for measurement, $u(t),v(t),$ $w(t)$,
$\delta_{e}$, $\delta_{T}$, and $\delta_{r}$ need to be estimated. Thus, the
estimation vector for a NAV, $\mathbf{z}_{i}(t)\in\mathbb{R}^{\hat{q}=6}$, is
given by

$\displaystyle\mathbf{z}_{i}(t)=$
$\displaystyle\left[\begin{array}[]{cccccc}u&w&\delta_{e}&\delta_{T}&v&\delta_{r}\end{array}\right]^{T}(t)$
(16)

Then, the equation of $\mathbf{z}_{i}(t)$ is given by

$\displaystyle\mathbf{z}_{i}(t)=$ $\displaystyle C_{z}\mathbf{x}_{i}(t)$ (17)

where $C_{z}\in\mathbb{R}^{6\times 11}$. Also, note that, in the case of a
NAV, $\hat{n}$=$\hat{r}+\hat{q}$. Consequently, the state-space model of the
NAV used for designing the estimator is given by (1) with $B_{i}$=$B$,
$C_{i}$=$C$, $\hat{n}=11$, $\hat{m}=3$, $\hat{r}=5$, and $\hat{q}=6$. The
resource constrained autopilot hardware and the uncertain and unstable nature
of NAV’s LTI plants suggest that there is a need for a MERS estimator for the
estimation of $\mathbf{z}_{i}(t)$. The description of the MERS estimator is
given in the next subsection.

###  II-B Minimum Error Robust Simultaneous Estimation Problem

To describe the MERS estimation problem, consider
$\mathcal{P}\subset\mathcal{RL_{\infty}}$ containing $N$ number of
stabilizable and detectable LTI MIMO unstable adversely coupled uncertain
plants of the NAV given in [2]. The state-space form of any plant belonging to
$\mathcal{P}$ is given by (1) with $B_{i}$=$B$, $C_{i}$=$C$, $\hat{n}=11$,
$\hat{m}=3$, $\hat{r}=5$, and $\hat{q}=6$. Let
$\hat{\mathbf{P}}_{l}(s)\in\mathcal{RH}_{\infty}$ represents an estimator. The
estimator, $\hat{\mathbf{P}}_{l}(s)$ is formed using the state-space matrices
of $\mathbf{P}_{l}(s)\in\mathcal{P}$ and a suitable estimator gain,
$\textbf{L}_{l}\in\mathbb{R}^{\hat{n}=11\times\hat{r}=5}$. Following this, the
state-space model of $\hat{\mathbf{P}}_{l}(s)$ is given by (4) with
$\mathbf{\hat{x}}_{l}\in\mathbb{R}^{\hat{n}=11}$ and
$\mathbf{\hat{z}}_{l}(t)\in\mathbb{R}^{\hat{q}=6}$. Here,
$\mathbf{\hat{z}}_{l}(t)$ provides the estimate of
$\mathbf{z}_{i}(t)\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\leavevmode\nobreak\ \in\\{1,\dots,l,\dots,N\\}$. When we consider
$\mathbf{\hat{P}}_{l}(s)$ as the common estimator of
$\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$, then the state
estimation error vectors and the estimation error vectors are denoted by
$\mathbf{e}_{\mathbf{x}_{il}}(t)=\mathbf{x}_{i}(t)-\mathbf{\hat{x}}_{l}(t)$
and
$\mathbf{e}_{\mathbf{z}_{il}}(t)=\mathbf{z}_{i}(t)-\mathbf{\hat{z}}_{l}(t)$
for all $i\leavevmode\nobreak\ \in\\{1,\dots,l,\dots,N\\}$, respectively.
Similarly, for all $i\leavevmode\nobreak\ \in\leavevmode\nobreak\
\\{1,\dots,l,\dots,N\\}$,
$\mathbf{e}_{\mathbf{y}_{il}}(t)=\mathbf{y}_{i}(t)-C\mathbf{\hat{x}}_{l}(t)$.

Now, consider the case where $\mathbf{\hat{P}}_{l}(s)$ is employed to estimate
$\mathbf{z}_{i}(t)\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}$ of
$\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$, respectively. Then,
one can obtain $N$ number of estimator error models that are given by

$\displaystyle\dot{\mathbf{e}}_{\mathbf{x}_{il}}(t)={}$ $\displaystyle
A_{l}\mathbf{e}_{\mathbf{x}_{il}}(t)+\Delta
A_{il}\mathbf{x}_{i}(t)+B_{e}\mathbf{u}_{e}(t)$ (18)
$\displaystyle\mathbf{e}_{\mathbf{z}_{il}}(t)={}$ $\displaystyle
C_{z}\mathbf{e}_{\mathbf{x}_{il}}(t)$
$\displaystyle\mathbf{e}_{\mathbf{y}_{il}}(t)={}$ $\displaystyle
C\mathbf{e}_{\mathbf{x}_{il}}(t)+\mathbf{v}(t);i\in\\{1,\dots,l,\dots,N\\}$

where $\Delta A_{il}$ is the difference between the system matrices, $A_{l}$
and $A_{i}$ of $\mathbf{P}_{l}(s)$ and $\mathbf{P}_{i}(s)$, respectively,
$\mathbf{x}_{i}(t)$ is the state vector of $\mathbf{P}_{i}(s)$,
$B_{e}=\mathbf{I}\in\mathbb{R}^{11\times 11}$ is the input matrix of error
dynamics, and
$\mathbf{u}_{e}(t)=-\mathbf{L}_{l}\mathbf{e}_{\mathbf{y}_{il}}(t)$. Applying
this $\mathbf{u}_{e}(t)$ in (18) results in

$\displaystyle\dot{\mathbf{e}}_{\mathbf{x}_{il}}(t)=(A_{l}-B_{e}C\mathbf{L}_{l})\mathbf{e}_{\mathbf{x}_{il}}(t)+\Delta
A_{il}\mathbf{x}_{i}(t)-B_{e}\mathbf{L}_{l}\mathbf{v}(t)$ (19)

Equation (19) suggests that, along with $\mathbf{v}(t)$, $\mathbf{x}_{i}(t)$
also becomes an exogenous signal vector that adversely affect the estimation
error dynamics. This is because of the difference between the system matrices
of $\mathbf{P}_{i}(s)$ and $\mathbf{\hat{P}}_{l}(s)$. Hence, when
$\mathbf{\hat{P}}_{l}(s)$ estimates $\mathbf{z}_{i}(t)$ of
$\mathbf{P}_{i}(s)\in\mathcal{P}$, then
$\mathbf{d}(t)\triangleq\left[\begin{array}[]{cc}\mathbf{x}_{i}(t)&\mathbf{v}(t)\end{array}\right]^{T}$.

Note that $\mathbf{x}_{i}(t)$ needs to be a bounded energy exogenous signal.
For that either $\mathbf{P}_{i}(s)$ or its closed-loop plant must to be
stable. Further, when $\mathbf{\hat{P}}_{l}(s)$ becomes the common estimator
of all the plants belonging to $\mathcal{P}$, then there exist $N$ number of
closed-loop transfer function matrices,
$\textbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}\in\mathcal{RH_{\infty}}$,
from
$\mathbf{d}(s)\triangleq\left[\begin{array}[]{cc}\mathbf{x}_{i}(s)&\mathbf{v}(s)\end{array}\right]^{T}$
to $\textbf{e}_{\textbf{z}_{il}}(s)$ for all $i\leavevmode\nobreak\
\in\leavevmode\nobreak\ \\{1,\dots,l,\dots,N\\}$. Now, consider $N$ number of
estimators,
$\mathbf{\hat{P}}_{1}(s),\dots,\mathbf{\hat{P}}_{i}(s),\dots,\mathbf{\hat{P}}_{l}(s),\dots,\mathbf{\hat{P}}_{N}(s)$,
each formed using state-space matrices of
$\mathbf{P}_{1}(s)\in\mathcal{P},\dots,\mathbf{P}_{i}(s)\in\mathcal{P},\dots,\mathbf{P}_{l}(s)\in\mathcal{P},\dots,\mathbf{P}_{N}(s)\in\mathcal{P}$,
respectively. Let the finite set, $\hat{\mathcal{P}}$, contains all these $N$
estimators. The state-space forms of these estimators are given as

$\mathbf{\hat{P}}_{1}(s):\begin{cases}\dot{\mathbf{\hat{\mathbf{x}}}}_{l}(t)&=A_{l}\mathbf{\hat{x}}_{l}(t)+B\mathbf{u}(t)+\mathbf{L}_{l}(\mathbf{y}(t)-C\mathbf{\hat{x}}_{l}(t))\\\
\mathbf{\hat{z}}_{l}(t)&=C_{z}\mathbf{\hat{x}}_{l}(t);\forall\leavevmode\nobreak\
l\in\\{1,\dots,i,\dots,N\\}\end{cases}$ (20)

where $\mathbf{u}(t)$
$\in\\{\mathbf{u}_{1}(t),\dots,\mathbf{u}_{i}(t),\dots,\mathbf{u}_{N}(t)\\}$
and
$\mathbf{y}(t)\in\\{\mathbf{y}_{1}(t),\dots,\mathbf{y}_{i}(t),\dots,\mathbf{y}_{N}(t)\\}$.
Here, $\mathbf{u}_{i}(t)$ and $\mathbf{y}_{i}(t)$ are the control input and
measurement vectors of $\mathbf{P}_{i}(s)\in\mathcal{P}$, respectively.
Further, when each estimator belonging to $\hat{\mathcal{P}}$ is utilized as
the common estimator for all the $N$ number of plants of $\mathcal{P}$, then
$N\times N$ closed-loop transfer function matrices,
$\textbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}$ from $\textbf{d}(s)$ to
$\textbf{e}_{\textbf{z}_{il}}(s)\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i,l\in\\{1,\dots,N\\}$ are obtained. Let the
$H_{\infty}$ norm of $\textbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}$ that
provides the worst-case gain from $\mathbf{d}(t)$ is defined as

$\displaystyle||\textbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}=\underset{\underset{\mathbf{d}(t)\neq\mathbf{0}}{\mathbf{d}(t)\in\mathcal{L}_{2}[0,\infty)}}{\sup}\leavevmode\nobreak\
\frac{\parallel\mathbf{e}_{\mathbf{z}_{il}}(t)\parallel_{2}}{\parallel\textbf{d}(t)\parallel_{2}}$
(21)

Now, if we consider $\mathbf{\hat{P}}_{l}(s)\in\hat{\mathcal{P}}$ as the
simultaneous estimator of all the plants belonging to $\mathcal{P}$, then
$\parallel\textbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}\parallel_{\infty}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$ are the worst-case
gains from $\mathbf{d}(s)$ associated with $\mathbf{\hat{P}}_{l}(s)$. Also,
the largest worst-case gain from $\mathbf{d}(s)$ associated with
$\mathbf{\hat{P}}_{l}(s)$ is $\max\leavevmode\nobreak\
\\{\parallel\textbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}\parallel_{\infty}\\}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$. The largest worst-
case gain from $\mathbf{d}(s)$ associated with an estimator belongs to
$\hat{\mathcal{P}}$ occurs while estimating the desired state variables of a
plant referred to as the worst plant from the perspective of the simultaneous
estimation process. This worst plant is indicated through the subscript $k$
along with a superscript that shows its association with the corresponding
estimator. Following this, the largest worst-case gain from $\mathbf{d}(s)$
associated with $\hat{\mathbf{P}}_{l}(s)\in\hat{\mathcal{P}}$,
$\parallel\mathbf{e}(s)_{\mathbf{P}_{k}^{l}\mathbf{\hat{P}}_{l}}\parallel_{\infty}$,
is defined as

$\displaystyle\parallel\mathbf{e}(s)_{\mathbf{P}_{k}^{l}\mathbf{\hat{P}}_{l}}\parallel_{\infty}=\max\\{||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}\\}$ (22)

From (6) and (21), the sufficient conditions for considering
$\mathbf{\hat{P}}_{l}(s)\in\hat{\mathcal{P}}$ as a simultaneous estimator of
all the plants belonging to $\mathcal{P}$ with respect to Condition II are
$\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}<\gamma\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$. An equivalent single
condition that satisfy the above conditions can be obtained using (22), and is
given by

$\parallel\mathbf{e}(s)_{\mathbf{P}_{k}^{l}\mathbf{\hat{P}}_{l}}\parallel_{\infty}<\gamma$
(23)

Now, the MERS estimation problem can be defined precisely as: Given
$\mathcal{P}$. Find $\mathbf{\hat{P}}_{j}(s)\in\hat{\mathcal{P}}$ along with
$\mathbf{L}_{j}$ such that 1) $\mathbf{\hat{P}}_{j}(s)$ simultaneously
estimates $\mathbf{z}_{i}(t)$ of
$\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\leavevmode\nobreak\ \in\leavevmode\nobreak\
\\{1,\dots,j,\dots,N\\}$ even when these plants have model uncertainties and
2) the condition given by

$||\mathbf{e}(s)_{\mathbf{P}_{k}^{j}\mathbf{\hat{P}}_{j}}||_{\infty}=\min\left\\{||\mathbf{e}(s)_{\mathbf{P}_{k}^{l}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
|\leavevmode\nobreak\ l\in\\{1,...,N\\}\right\\}<\gamma$ (24)

is satisfied.

The solution to this problem is a RS estimator referred to as the MERS
estimator whose
$\parallel\mathbf{e}(s)_{\mathbf{P}_{k}^{j}\mathbf{\hat{P}}_{j}}\parallel_{\infty}$
is the smallest among
$\big{\\{}||\mathbf{e}(s)_{\mathbf{P}_{k}^{l}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
|\leavevmode\nobreak\ l\in\\{1,\dots,j,\dots,N\\}\big{\\}}$ with
$\parallel\mathbf{e}(s)_{\mathbf{P}_{k}^{l}\mathbf{\hat{P}}_{l}}\parallel_{\infty}<\gamma$.
This suggests that suggest that the largest worst-case estimation error of
MERS estimator is the minimal among the largest worst-case estimation errors
of simultaneous estimators. In this paper, we consider only parametric
uncertainties in the form of bounded perturbations in the system matrices. The
performance of the MERS estimator can be further improved by appending
suitable compensators to the plant dynamics. This is discussed in the next
section.

###  II-C Gap Reducing Compensator Problem

The Gap Reducing (GR) compensator problem is about finding those compensators
that modify the input-output characteristics of all the plants belonging to
$\mathcal{P}$ for reducing further the estimation errors arising from the
differences between the system matrices of the MERS estimator and
$\mathbf{P}_{i}(s)$ $\forall\leavevmode\nobreak\ i\in\\{1,\dots,j,\dots,N\\}$
($\Delta A_{ij}$). The GR compensator problem can be stated as: assume that
there exists a MERS estimator, $\mathbf{\hat{P}}_{j}(s)\in\mathcal{\hat{P}}$,
for the plants belonging to $\mathcal{P}$. Now, consider the plant,
$\mathbf{P}_{j}(s)\in\mathcal{P}$ and assume $\mathcal{P}$ as an uncertainty
model set with $\mathbf{P}_{j}(s)$ as the nominal plant. Besides this, assume
also that the plants belonging to $\mathcal{P}\setminus{\mathbf{P}_{j}(s)}$ as
the perturbed plants of ${\mathbf{P}_{j}(s)}$. Let
$\mathbf{N}_{j}(s)\in\mathcal{RH_{\infty}}$ and
$\mathbf{M}_{j}(s)\in\mathcal{RH_{\infty}}$ with det($\mathbf{M}_{j}(s)\neq
0$) are the normalized right coprime factors of $\mathbf{P}_{j}(s)$.
Subsequently, $\mathbf{P}_{j}(s)$ is given by

$\displaystyle\mathbf{P}_{j}(s)=$
$\displaystyle\mathbf{N}_{j}(s)\mathbf{M^{-1}}_{j}(s)$ (25)

Let
$\mathbf{\Delta}_{N_{\mathbf{P}_{j}\mathbf{P}_{i}}}(s)\in\mathcal{RH_{\infty}}$
and
$\mathbf{\Delta}_{M_{\mathbf{P}_{j}\mathbf{P}_{i}}}(s)\in\mathcal{RH_{\infty}}$
are the right coprime factor perturbations of $\mathbf{N}_{j}(s)$ and
$\mathbf{M}_{j}(s)$ with
$\big{|}\big{|}[\begin{array}[]{cc}\mathbf{\Delta}_{N_{\mathbf{P}_{j}\mathbf{P}_{i}}}(s)&\mathbf{\Delta}_{M_{\mathbf{P}_{j}\mathbf{P}_{i}}}(s)\end{array}]^{T}\big{|}\big{|}_{\infty}\leq\epsilon_{{}_{\mathbf{P}_{j}\mathbf{P}_{i}}}$,
respectively. Here, $\epsilon_{{}_{\mathbf{P}_{j}\mathbf{P}_{i}}}$ is the
least upper bound on the right coprime factor perturbations. Then,
$\mathbf{P}_{i}(s)\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,\dots,N\\}\setminus{j}$ are defined as

$\displaystyle\mathbf{P}_{i}(s)=$
$\displaystyle\big{(}\mathbf{N}_{j}(s)+\mathbf{\Delta}_{N_{\mathbf{P}_{j}\mathbf{P}_{i}}}(s)\big{)}\big{(}\mathbf{M}_{j}(s)+\mathbf{\Delta}_{M_{\mathbf{P}_{j}\mathbf{P}_{i}}}(s)\big{)}^{-1}$
(26) $\displaystyle\forall\leavevmode\nobreak\
i\in\\{1,\dots,N\\}\setminus{j}$

Now, consider the scenario where $\mathbf{v}(t)=0$ and
$\mathbf{\hat{P}}_{j}(s)$ is employed to estimate $\mathbf{z}_{l}(t)$ of
$\mathbf{P}_{j}(s)$. To realize this scenario, the state vector of the plant
needs to be bounded. For that, a feedback controller is employed as shown in
Fig. 2. Now, assume $\mathbf{P}_{j}(s)$ is perturbed to form
$\mathbf{P}_{i}(s)\in\mathcal{P}\setminus\\{\mathbf{P}_{j}(s)\\}$ when
$\mathbf{\hat{P}}_{j}(s)$ estimates $\mathbf{z}_{l}(t)$ of
$\mathbf{P}_{j}(s)$. In that case, $\mathbf{\hat{P}}_{j}(s)$ receives
$\mathbf{u}_{i}(t)=\mathbf{u}_{j}(t)+\Delta\mathbf{u}_{i}(t)$ and
$\mathbf{y}_{i}(t)=\mathbf{y}_{j}(t)+\Delta\mathbf{y}_{i}(t)$ instead of
$\mathbf{u}_{j}(t)$ and $\mathbf{y}_{j}(t)$, respectively as shown in Fig. 2.
Here, $\Delta\mathbf{u}_{i}(t)$ and $\Delta\mathbf{y}_{i}(t)$ are the
perturbations in the inputs and outputs of $\mathbf{P}_{j}(s)$, respectively
arising from $\Delta A_{ij}$. So when we develop the estimation error dynamics
offline, the plant considered is $\mathbf{P}_{j}(s)$. But in reality, the
inputs to $\mathbf{\hat{P}}_{j}(s)$ will be $\mathbf{u}_{i}(s)$ and
$\mathbf{y}_{i}(s)$ of the perturbed plant of $\mathbf{P}_{j}(s)$. Following
this, the estimation error dynamics is given by

$\displaystyle\dot{\mathbf{e}}_{x_{jj}}(t)=$
$\displaystyle(A_{j}-B_{e}\mathbf{L}_{j}C)\mathbf{e}_{x_{jj}}(t)-B\Delta\mathbf{u}_{i}(t)-\mathbf{L}_{j}\Delta\mathbf{y}_{i}(t)$
(27) $\displaystyle=$
$\displaystyle(A_{j}-B_{e}\mathbf{L}_{j}C)\mathbf{e}_{x_{jj}}(t)-B(\mathbf{u}_{i}(t)-\mathbf{u}_{j}(t))$
$\displaystyle-\mathbf{L}_{j}(\mathbf{y}_{i}(t)-\mathbf{y}_{j}(t))$
$\displaystyle\mathbf{e}_{z_{jj}}(t)=$ $\displaystyle
C_{z}\mathbf{e}_{x_{jj}}(t)$

Let the eigenvalues of $(A_{j}-B_{e}\mathbf{L}_{j}C)$ belong to
$\mathbb{C}_{-}$. Then, (27) suggests that $\mathbf{e}_{z_{jj}}(t)$ converges
to zero when $\mathbf{u}_{i}(t)$=$\mathbf{u}_{j}(t)$ and
$\mathbf{y}_{i}(t)$=$\mathbf{y}_{j}(t)$. Following this,
$\mathbf{e}_{z_{jj}}(t)$will be closer to zero if we make $\mathbf{u}_{i}(t)$
and $\mathbf{y}_{i}(t)$ closer to $\mathbf{u}_{j}(t)$ and $\mathbf{y}_{j}(t)$,
respectively. This indicates that when
$(\mathbf{u}_{i}(t),\mathbf{y}_{i}(t))\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ become closer
to $(\mathbf{u}_{j}(t),\mathbf{y}_{j}(t))$, then the estimation error due to
$\Delta A_{ij}\mathbf{x}_{i}(t)\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ becomes close
to zero.

To state the GR compensator problem, consider the following definition.

######  Definition II.1.

Graph, $\mathcal{G}$, of an operator: Let
$\mathbf{P}_{j}(s):\mathcal{H}\rightarrow\mathcal{H}$ be any linear operator
in Hilbert space ($\mathcal{H}$) defined on the domain,
$\mathcal{D}(\mathbf{P}_{j}(s))\subseteq\mathcal{H}$. Then, the graph of
$\mathbf{P}_{j}(s)$, $\mathcal{G}(\mathbf{P}_{j})$ is defined as

$\displaystyle\begin{split}\mathcal{G}(\mathbf{P}_{j})&=\\{(\mathbf{u}_{j}(s),\mathbf{y}_{j}(s))\in\mathcal{H}\times\mathcal{H}:\mathbf{u}_{j}(s)\in\mathcal{D}(\mathbf{P}_{j}(s)),\mathbf{y}_{j}(s)=\\\
&\qquad\mathbf{P}_{j}(s)\mathbf{u}_{j}(s)\in\mathcal{H}\\}\end{split}$ (28)

This definition suggests that $\mathcal{G}(\mathbf{P}_{j})$ is the set of all
pairs of $(\mathbf{u}_{j}(s),\mathbf{y}_{j}(s))$ with
$\mathbf{u}_{j}(s)\in\mathcal{D}(\mathbf{P}_{j}(s))$. Also, this definition
indicates that making $\mathcal{G}(\mathbf{P}_{i})\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ closer to
$\mathcal{G}(\mathbf{P}_{j})$ increases the closeness between
$(\mathbf{u}_{i}(t),\mathbf{y}_{i}(t))\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ and
$(\mathbf{u}_{j}(t),\mathbf{y}_{j}(t))$. Cascading the plants with pre and
post compensators, $\mathbf{W_{in}}(s)\in\mathcal{RH_{\infty}}$ and
$\mathbf{W_{ot}}(s)\in\mathcal{RH_{\infty}}$, respectively, we modify the
input-output characteristics and thereby the graphs of the plants. Now, the GR
compensator problem can be stated as: Find
$\mathbf{W_{in}}(s)\in\mathcal{RH_{\infty}}$ and
$\mathbf{W_{ot}}(s)\in\mathcal{RH_{\infty}}$ such that

  1. 1.

$\max\\{gap(\mathcal{G}(\mathbf{\acute{P}}_{j}(s)),\mathcal{G}(\mathbf{\acute{P}}_{i}(s))\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in{1,\dots,j,\dots,N}\\}$ is lower than
$\max\\{gap(\mathcal{G}(\mathbf{P}_{j}(s)),\mathcal{G}(\mathbf{P}_{i}(s))\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in{1,\dots,j,\dots,N}\\}$.

  2. 2.

$\max\\{gap(\mathcal{G}(\mathbf{\acute{P}}_{j}(s)),\mathcal{G}(\mathbf{\acute{P}}_{i}(s))\leavevmode\nobreak\
|\leavevmode\nobreak\ i\in{1,\dots,j,\dots,N}\\}$ is closer to zero.

where
$\mathbf{\acute{P}}_{j}(s)=\mathbf{W_{ot}}(s)\mathbf{P}_{j}(s)\mathbf{W_{in}}(s)$
and
$\mathbf{\acute{P}}_{i}(s)=\mathbf{W_{ot}}(s)\mathbf{P}_{i}(s)\mathbf{W_{in}}(s)$.

Figure 2: Simultaneous estimation process with $\mathbf{\hat{P}}_{j}(s)$

In the next section, we present the synthesis of GRMERS estimator for a
typical unstable and highly coupled plants of a NAV.

##  III Synthesis of Gap Reduced Minimum Error Robust Simultaneous Estimator

The Gap Reduced Minimum Error Robust Simultaneous (GRMERS) estimator comprises
of a MERS estimator and the GR compensators. Here, the MERS estimator performs
robust simultaneous estimation with minimal largest worst-case error as it
satisfies (24). The GR compensators reduces the estimation errors of MERS
estimator by minimizing the gap between the respective graphs of the $N$
linear plants of the NAV. This minimization makes the inputs,
$\mathbf{u}_{i}(t)$ and $\mathbf{y}_{i}(t)$ (ref. Fig. (2)), to the estimator
from the $N$ plants similar to $\mathbf{u}_{j}(t)$ and $\mathbf{y}_{j}(t)$,
respectively. Hence, the GR compensators reduces the estimation errors arising
from the differences in the system matrices of plants and estimator as
indicated by (27). In this section, we first explain the procedure for
synthesising the MERS estimator model. Before carrying out this procedure, as
preliminaries, the effect of $\mathbf{v}(t)$ and $\mathbf{x}_{i}(t)$ on the
estimation error is analyzed first.

###  III-A Preliminaries: Analysis of the Effect of Measurement Noise and
State Vector on Estimation Error Dynamics

Here, the effects of $\mathbf{v}(t)$ and $\mathbf{x}_{i}(t)$ on the estimation
error are briefly analyzed. For that, consider
$\mathbf{\hat{P}}_{l}(s)\in\hat{\mathcal{P}}$ as the simultaneous estimator
for estimating $\mathbf{z}_{i}(t)$ of $\mathbf{P}_{i}(s)\in\mathcal{P}$ for
all $i\in\\{1,\dots,l,\dots,N\\}$. We now define
$\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}\forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}$ using (18) and
$\mathbf{u}_{e}(t)=-\mathbf{L}_{l}\mathbf{e}_{\mathbf{y}_{il}}(t)$ as

$\displaystyle\begin{split}\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}&=-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}\Delta
A_{il}\\\ &\quad-
C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}B_{e}\mathbf{L}_{l};\\\
&\quad\qquad\forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}\end{split}$ (29)

In (29), $-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}\Delta A_{il}$
is the transfer function matrix from $\mathbf{x}_{i}(s)$ to
$\mathbf{e}_{z_{il}}(s)$ and
$-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}B_{e}\mathbf{L}_{l}$ is
the transfer function matrix from $\mathbf{v}(s)$ to $\mathbf{e}_{z_{il}}(s)$.
Now, when $\mathbf{\hat{P}}_{l}(s)$ estimates $\mathbf{z}_{l}(t)$ of
$\mathbf{P}_{l}(s)$, then $\Delta A_{ll}$ is a null matrix and
$\mathbf{x}_{i}(t)$=0. Following this, if $\mathbf{d}(t)=\mathbf{v}(t)=$0 and
all the eigenvalues of $[A_{l}-B_{e}\mathbf{L}_{l}C]$ belong to
$\mathbb{C}_{-}$, then $\lim_{t\to\infty}\mathbf{e}_{\mathbf{z}_{ll}}(t)=0$.
This indicates that when any estimator, say
$\hat{\mathbf{P}}_{i}(s)\in\hat{\mathcal{P}}$, estimates $\mathbf{z}_{i}(t)$
of $\mathbf{P}_{i}(s)\in\mathcal{P}$, then
$\lim_{t\to\infty}\mathbf{e}_{\mathbf{z}_{ii}}(t)=0$ if $\mathbf{v}(t)$=0 and
$\hat{\mathbf{P}}_{i}(s)\in\mathcal{RH_{\infty}}$. However, when
$\mathbf{\hat{P}}_{l}(s)$ estimates $\mathbf{z}_{l}(t)$ of $\mathbf{P}_{l}(s)$
with $\mathbf{v}\neq 0$ and $\mathbf{x}_{i}(t)=0$, then
$\mathbf{e}_{\mathbf{z}_{ll}}(s)=-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}B_{e}\mathbf{L}_{l}\mathbf{v}(s)$.
Subsequently, in time domain,
$\mathbf{e}_{\mathbf{z}_{ll}}(t)=\int_{0}^{t}-C_{z}e^{((A_{l}-B_{e}\mathbf{L}_{l}C)(t-\tau))}B_{e}\mathbf{L}_{l}\mathbf{v}(\tau)d\tau$.
This integral will never be zero when $\mathbf{v}(t)\neq$0 indicating that
when $\mathbf{v}(t)\neq$0, the estimation errors do not converge to zero
during the simultaneous estimation process. The same phenomenon will be there
even when $\mathbf{x}_{i}(t)\neq$ 0\. Equation (29) indicates that
$-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}B_{e}\mathbf{L}_{l}$ are
same for all $i\in\\{1,\dots,l,\dots,N\\}$. This indicates that the effect of
$\mathbf{v}(t)$ on the estimation errors remains identical when an estimator
belonging to $\hat{\mathcal{P}}$ performs simultaneous estimation. However,
(29) shows that $-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}\Delta
A_{il}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}$ become different when $\Delta
A_{il}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}$ are distinct. This proposes that the effect of
$\mathbf{x}_{i}(t)$ on the estimation errors may be dissimilar when an
estimator belongs to $\hat{\mathcal{P}}$ executes simultaneous estimation.
Applying $H_{\infty}$ norm on both sides of (29) and then using triangle and
Cauchy-Schwarz inequalities, the
$||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}$ can be written as

$\displaystyle\begin{split}||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}&\leq||-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}||_{\infty}||\Delta
A_{il}||_{\infty}\\\
&\quad+||-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}B_{e}\mathbf{L}_{l}||_{\infty};\\\
&\qquad\text{$\forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}$}\end{split}$ (30)

Equation (30) indicates that the estimation error due to $\mathbf{x}_{i}(t)$
increases with the increase of $||\Delta A_{il}||_{\infty}$. Next, the
development of a sufficient condition for the existence of a RS estimator
based on above arguments is presented.

###  III-B Minimum Error Robust Simultaneous Estimator

In this subsection, we describe the development of the MERS estimator model
and the MERSE algorithm. At first, the sufficient condition for the existence
of a robust simultaneous estimator is derived.

####  III-B1 Sufficient Condition for the Existence of a Robust Simultaneous
Estimator

We now state the sufficient condition for the existence of a RS estimator
through the following theorem.

######  Theorem III.1.

Given $\mathcal{P}$, $\hat{\mathcal{P}}$, and $\gamma$. Consider
$\mathbf{\hat{P}}_{l}(s)\in\hat{\mathcal{P}}$ as the simultaneous estimator of
$\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,k,\dots,l,\dots,N\\}$. Let
$\mathbf{P}_{k}(s)\in\mathcal{P}$ satisfies the condition given by

$\displaystyle||\Delta A_{kl}||_{\infty}=\max\\{||\Delta
A_{il}||_{\infty}\\}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,\dots,k,\dots,l,\dots,N\\}$ (31)

Then, $\mathbf{P}^{l}_{k}(s)=\mathbf{P}_{k}(s)$ and the sufficient condition
for the existence of $\mathbf{\hat{P}}_{l}(s)$ as the RS estimator of all the
plants of $\mathcal{P}$ is given by

$\displaystyle||\mathbf{e}(s)_{\mathbf{P}^{l}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}<\gamma$
(32)

###### Proof.

Using (30), let
$||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}\setminus\\{k\\}$ are
expressed as

$\begin{split}||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}&\leq||-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}||_{\infty}||\Delta
A_{il}||_{\infty}\\\
&\quad+||-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}B_{e}\mathbf{L}_{l}||_{\infty}\\\
&\quad;\text{$\forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}\setminus\\{k\\}$}\end{split}$ (33)

Likewise, $||\mathbf{e}(s)_{\mathbf{P}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}$ is
given by

$\begin{split}||\mathbf{e}(s)_{\mathbf{P}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}&\leq||-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}||_{\infty}||\Delta
A_{kl}||_{\infty}\\\
&\quad+||-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}B_{e}\mathbf{L}_{l}||_{\infty}\end{split}$
(34)

Subtracting (34) from (33) results in

$\begin{split}||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}-||\mathbf{e}(s)_{\mathbf{P}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}&\leq\Xi\big{[}||\Delta
A_{il}||_{\infty}-||\Delta A_{kl}||_{\infty}\big{]};\\\
&\quad\text{$\forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}\setminus\\{k\\}$}\end{split}$ (35)

where
$\Xi=||-C_{z}(s\mathbf{I}-(A_{l}-B_{e}\mathbf{L}_{l}C))^{-1}||_{\infty}$. In
(35), RHS is negative because $\Xi>0$ and $\big{[}||\Delta
A_{il}||_{\infty}-||\Delta A_{kl}||_{\infty}\big{]}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}\setminus\\{k\\}$ are
negative. The later terms are negative as $\mathbf{P}_{k}(s)$ satisfies (31).
Since, RHS of (35) is negative, we can rewrite (35) as

$\begin{split}||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}-||\mathbf{e}(s)_{\mathbf{P}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}&\leq-\Lambda_{i}\\\
&\quad\text{$\forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}\setminus\\{k\\}$}\end{split}$ (36)

where $\Lambda_{i}$ is a positive constant. Note that
$||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}>0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,l,\dots,N\\}\setminus\\{k\\}$ and
$||\mathbf{e}(s)_{\mathbf{P}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}>0$.
Consequently, the condition given in (36) ensures the condition given by

$\begin{split}||\mathbf{e}(s)_{\mathbf{P}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
\forall\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}\setminus\\{k\\}<||\mathbf{e}(s)_{\mathbf{P}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}\end{split}$
(37)

Equation (37) implies $\mathbf{P}^{l}_{k}(s)=\mathbf{P}_{k}(s)$ and
$\parallel\mathbf{e}_{\mathbf{P}^{l}_{k}\mathbf{\hat{P}}_{l}}(s)\parallel_{\infty}=\parallel\mathbf{e}_{\mathbf{P}_{k}\mathbf{\hat{P}}_{l}}(s)\parallel_{\infty}$
when $\mathbf{P}_{k}(s)$ satisfies (31). Then, from (23), the sufficient
condition for the existence of $\mathbf{\hat{P}}_{l}(s)$ as the simultaneous
estimator of $N$ number of plants belonging to $\mathcal{P}$ is (32). Now
consider the case of RS estimation. For that, let
$\mathbf{\bar{P}}_{i}(s)\notin\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,k,\dots,l,\dots,N\\}$ are the
perturbed plants of $\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,k,\dots,l,\dots,N\\}$,
respectively. These perturbed plants arises due to the perturbations in the
system matrix of $\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,k,\dots,l,\dots,N\\}$. Following
this, the system matrix of $\mathbf{\bar{P}}_{i}(s)$ be
$\bar{A}_{i}=A_{l}+\Delta A_{i}$ with $||\Delta A_{i}||_{\infty}<||\Delta
A_{kl}||_{\infty}$. Here, $\Delta A_{i}$ is the bounded perturbation of
$A_{i}$. Because of $||\Delta A_{i}||_{\infty}\leq||\Delta A_{kl}||_{\infty}$,
it obvious that
$||\mathbf{e}(s)_{\mathbf{\bar{P}}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}<||\mathbf{e}(s)_{\mathbf{P}^{l}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,k,\dots,l,\dots,N\\}$. So, if
$||\mathbf{e}(s)_{\mathbf{P}^{l}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}<\gamma$,
then
$||\mathbf{e}(s)_{\mathbf{\bar{P}}_{i}\mathbf{\hat{P}}_{l}}||_{\infty}<\gamma\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,k,\dots,l,\dots,N\\}$. Hence,
concerning $\mathbf{\hat{P}}_{l}(s)$,
$||\mathbf{e}(s)_{\mathbf{P}^{l}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}<\gamma$
is the sufficient condition for the existence of $\mathbf{\hat{P}}_{l}(s)$ as
the RS estimator of all the plants of $\mathcal{P}$. This establishes the
proof of the theorem. ∎

Following Theorem III.1, the sufficient condition for the existence of the
estimators, $\hat{\mathbf{P}}_{l}(s)\in\hat{\mathcal{P}}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,i,\dots,N\\}$, as a RS estimator is
given by

$\displaystyle||\mathbf{e}(s)_{\mathbf{P}^{l}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}<\gamma;\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,i,\dots,N\\}$ (38)

Furthermore, Theorem III.1 suggests that the sufficient condition for the
existence of $\mathbf{\hat{P}}_{j}(s)$ as the MERS estimator is given in (24).
Hence, to solve MERS estimation problem, we have to solve (24).

####  III-B2 Method to Solve the Minimum Error Robust Simultaneous Estimation
Problem

We now present the method that solves the MERS estimation problem. The MERS
estimator synthesis is about finding $\mathbf{\hat{P}}_{j}(s)$ with a static
gain $\mathbf{L}j$, such that the condition given in (24) is satisfied.
Equation (24) indicates that the solution of it follows by solving (38). For
this, the inequalities given in (38) is formulated in-terms of linear matrix
inequalities (LMIs) using bounded real lemma [14]. If there exists
$Q_{l}>0\in\mathcal{S}^{\hat{n}}$ and
$Y_{l}\in\mathbb{R}^{\hat{n}\times\hat{q}}$, then from the bounded real lemma,
the LMIs corresponding to (38) for $0<\gamma<1$ are given by

$\displaystyle\begin{bmatrix}Q_{l}A_{l}+A_{l}^{T}Q_{l}-Y_{l}C-C^{T}Y_{l}^{T}&Q_{l}\grave{B}_{l}-Y_{l}\grave{D}&C_{z}^{T}\\\
*&-\gamma\mathbf{I}&0\\\ *&*&-\gamma\mathbf{I}\end{bmatrix}<0$ (39)
$\displaystyle Q_{l}>0;$ $\displaystyle\forall\leavevmode\nobreak\
l\in\\{1,\dots,j,\dots,N\\}$

where $\grave{B}_{l}$=[$\Delta A_{kl}$ $0$] and $\grave{D}$ =[$0$
$\mathbf{I}\in\mathbb{R}^{\hat{r}\times\hat{r}}$]. For a given $\gamma$, solve
(39) for all $Q_{l}$ and $Y_{l}$. Thereafter, recover all the estimator gain
using $\mathbf{L}_{l}=Q_{l}^{-1}Y_{l}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,j,\dots,N\\}$. The feasible
solution of (39) establishes that there exists $N$ number of RS estimators.
Now, using $\mathbf{L}_{l}=Q_{l}^{-1}Y_{l}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,j,\dots,N\\}$, compute
$||\mathbf{e}(s)_{\mathbf{P}^{l}_{k}\mathbf{\hat{P}}_{l}}||_{\infty}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,l,\dots,N\\}$ and find its smallest
value. Then, identify the RS estimator to which this smallest value belongs
and from (24), this estimator is the MERS estimator,
$\mathbf{\hat{P}}_{j}(s)$. If the solution of (39) fails to satisfy the
condition specified in (38), then a dynamic compensator is required.

We now describe a method that induce the characteristics of a dynamic
compensator in the error dynamics by solving LMIs similar to (39). In this
method, the dynamic compensator is in the form of pre and post compensators
that will be cascaded with the plant models belonging to $\mathcal{P}$. Now,
let us consider the pre compensator, $\mathbf{\widetilde{W}_{ei}}(s)$, and the
post compensator, $\mathbf{\widetilde{W}_{eo}}(s)$ which would be cascaded
with the plants belonging to $\mathcal{P}$. The
$\mathbf{\widetilde{W}_{ei}}(s)$ and $\mathbf{\widetilde{W}_{eo}}(s)$ are
defined by (13) and (14), respectively, of [12] with $\hat{m}=3$ and
$\hat{r}=5$ ((13) and (14) are also given in supporting material). Also, the
elements of $\mathbf{\widetilde{W}_{ei}}(s)$ and
$\mathbf{\widetilde{W}}_{eo}(s)$ are given by (15) and (16), respectively of
[12] ((15) and (16) are also given in supporting material). The cascading of
these compensators modifies the characteristics of all the plants of
$\mathcal{P}$. This in turn alters the characteristics of the estimation error
dynamics. Following this and the MERS estimation problem, the objective is to
synthesize appropriate pre/post compensators along with suitable estimator
gain that reduces the largest worst-case gain of the MERS estimator
($||\mathbf{e}(s)_{\mathbf{P}^{j}_{k}\mathbf{\hat{P}}_{j}}||_{\infty}$) below
a given $\gamma$. The plant realized after cascading,
$\widetilde{\mathbf{P}}_{i}(s)$, is defined as
$\widetilde{\mathbf{P}}_{i}(s)=\mathbf{\widetilde{W}_{eo}}(s)\mathbf{P}_{i}(s)\mathbf{\widetilde{W}_{ei}}(s)$.
As $\mathbf{\widetilde{W}_{eo}}(s)$ is the post compensator, the
$\mathbf{v}(t)$ act at its input. Then, the output of
$\mathbf{\widetilde{W}_{eo}}(s)$ from $\mathbf{v}(t)$,
$\mathbf{\widetilde{v}}(t)$, is defined as
$\mathbf{\widetilde{v}}(t)=\mathcal{L}^{-1}[\mathbf{\widetilde{W}_{eo}}(s)\mathbf{v}(s)](t)$.
Here, $\mathcal{L}^{-1}$ denotes the inverse Laplace transform. Considering
this, the state-space form of $\mathbf{\widetilde{P}}_{i}(s)$ is given by

$\mathbf{\widetilde{P}}_{i}(s):\begin{cases}\dot{\mathbf{\widetilde{x}}}_{i}(t)=&\widetilde{A}_{i}\mathbf{\widetilde{x}}_{i}(t)+\widetilde{B}\mathbf{\widetilde{u}}_{i}(t)\\\
\mathbf{\widetilde{z}}_{i}(t)=&\widetilde{C}_{z}\mathbf{\widetilde{x}}_{i}(t)\\\
\mathbf{\widetilde{y}}_{i}(t)=&\widetilde{C}\mathbf{\widetilde{x}}_{i}(t)+\mathbf{\widetilde{v}}(t)\end{cases}$
(40)

where
$\mathbf{\bar{x}}_{i}(t)$=$[\begin{smallmatrix}\mathbf{x}_{\mathbf{\widetilde{W}_{eo}}}\leavevmode\nobreak\
|&\mathbf{x}_{i}\leavevmode\nobreak\
|&\mathbf{x}_{\mathbf{\widetilde{W}_{ei}}}\end{smallmatrix}]^{T}\in\mathbb{R}^{22}$
is the state vector. Here,
$\mathbf{x}_{\mathbf{\widetilde{W}_{eo}}}(t)\in\mathbb{R}^{5}$ and
$\mathbf{x}_{\mathbf{\widetilde{W}_{ei}}}(t)\in\mathbb{R}^{3}$ are the state
vectors of post and pre compensators, respectively. Besides,
$\widetilde{A}_{i}\in\mathbb{R}^{22\times 22}$,
$\widetilde{B}\in\mathbb{R}^{22\times 3}$, and
$\widetilde{C}\in\mathbb{R}^{5\times 22}$ are the system matrix, the control
input matrix, and the measurement matrix, respectively. Likewise,
$\widetilde{C}_{z}$=$[\begin{smallmatrix}\mathbf{0}_{3,5}\leavevmode\nobreak\
|&C_{z}\leavevmode\nobreak\ |&\mathbf{0}_{3,3}\end{smallmatrix}]$ is a
constant matrix. Additionally, $\mathbf{\widetilde{z}}_{i}(t)$ is the vector
to be estimated. The characteristic of $\mathbf{\widetilde{x}}_{i}(t)$ and
$\widetilde{C}_{z}$ suggests
$\mathbf{\widetilde{z}}_{i}(t)$=$\mathbf{z}_{i}(t)$. Now, consider a finite
set,
$\widetilde{\mathcal{P}}=\\{\widetilde{\mathbf{P}}_{i}(s)\in\mathcal{RL_{\infty}}\leavevmode\nobreak\
|\leavevmode\nobreak\
\widetilde{\mathbf{P}}_{i}(s)=\mathbf{\widetilde{W}_{eo}}(s)\mathbf{P}_{i}(s)\mathbf{\widetilde{W}_{ei}}(s),\mathbf{P}_{i}(s)\in\mathcal{P},\leavevmode\nobreak\
i\in\\{1,\dots,l,\dots,N\\}\\}$. The state-space form of any plant belongs to
$\widetilde{\mathcal{P}}$ is given by (40). Now, consider $N$ number of
estimators,
$\mathbf{\check{P}}_{1}(s),\dots,\mathbf{\check{P}}_{i}(s),\dots,\mathbf{\check{P}}_{l}(s),\dots,\mathbf{\check{P}}_{N}(s)$,
each formed using state-space matrices of
$\mathbf{\widetilde{P}}_{1}(s)\in\mathcal{\widetilde{P}},\dots,\mathbf{\widetilde{P}}_{i}(s)\in\mathcal{\widetilde{P}},\dots,\mathbf{\widetilde{P}}_{l}(s)\in\mathcal{\widetilde{P}},\dots,\mathbf{\widetilde{P}}_{N}(s)\in\mathcal{\widetilde{P}}$,
respectively. Let the finite set, $\check{\mathcal{P}}$, contains all these
$N$ estimators. The state-space forms of these estimators are given as

$\mathbf{\check{P}}_{1}(s):\begin{cases}\dot{\mathbf{\check{\mathbf{x}}}}_{l}(t)={}&\widetilde{A}_{l}\mathbf{\check{x}}_{l}(t)+\widetilde{B}\mathbf{\widetilde{u}}(t)+\mathbf{\check{L}}_{l}(\mathbf{\widetilde{y}}(t)-\widetilde{C}\mathbf{\check{x}}_{l}(t))\\\
\mathbf{\check{z}}_{l}(t)={}&\widetilde{C}_{z}\mathbf{\check{x}}_{l}(t);\forall\leavevmode\nobreak\
l\in\\{1,\dots,i,\dots,j,\dots,N\\}\end{cases}$ (41)

where $\mathbf{\widetilde{u}}(t)$
$\in\\{\mathbf{\widetilde{u}}_{1}(t),\dots,\mathbf{\widetilde{u}}_{i}(t),\dots,\mathbf{\widetilde{u}}_{N}(t)\\}$
and
$\mathbf{\widetilde{y}}(t)\in\\{\mathbf{\widetilde{y}}_{1}(t),\dots,\mathbf{\widetilde{y}}_{i}(t),\dots,\mathbf{\widetilde{y}}_{N}(t)\\}$.
Here, $\mathbf{\widetilde{u}}_{i}(t)$ and $\mathbf{\widetilde{y}}_{i}(t)$ are
the control input and measurement vectors of
$\mathbf{\widetilde{P}}_{i}(s)\in\mathcal{\widetilde{P}}$, respectively. Also,
$\mathbf{\check{\mathbf{x}}}_{l}(t)\in\mathbb{R}^{22}$ and
$\mathbf{\check{\mathbf{u}}}_{l}(t)\in\mathbb{R}^{3}$ is the state vectors of
$\mathbf{\check{P}}_{i}(s)$. Additionally,
$\mathbf{\check{\mathbf{z}}}_{l}(t)\in\mathbb{R}^{3}$ is the estimate of
$\mathbf{\widetilde{\mathbf{z}}}_{l}(t)$. Using Theorem III.1, the sufficient
condition for the existence of estimators,
$\mathbf{\check{P}}_{l}(s)\in\check{\mathcal{P}}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,i,\dots,j,\dots,N\\}$, as a RS
estimator is given by

$\displaystyle||\mathbf{e}(s)_{\mathbf{\widetilde{P}}^{l}_{k}\mathbf{\check{P}}_{l}}||_{\infty}<\gamma;\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,i,\dots,j,\dots,N\\}$ (42)

We solve (42) by solving the equivalent LMIs to obtain the estimator gains,
$\mathbf{\check{L}}_{l}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
l\in\\{1,\dots,i,\dots,j,\dots,N\\}$, mentioned in (43). If there exists
$\check{Q}_{l}>0\in\mathcal{S}^{n}$ and
$\check{Y}_{l}\in\mathbb{R}^{n\times\hat{q}}$, then from the bounded real
lemma [14], the LMIs corresponding to (42) for a $0<\gamma\leq 1$ are given by

$\displaystyle\begin{bmatrix}\check{Q}_{l}\widetilde{A}_{l}+\widetilde{A}_{l}^{T}\check{Q}_{l}-\check{Y}_{l}\widetilde{C}-\widetilde{C}^{T}\check{Y}_{l}^{T}&\check{Q}_{l}\check{B}_{l}-\check{Y}_{l}\check{D}&\widetilde{C}_{z}^{T}\\\
*&-\gamma\mathbf{I}&0\\\ *&*&-\gamma\mathbf{I}\end{bmatrix}<0$ (43)
$\displaystyle\check{Q}_{l}>0$ $\displaystyle\forall\leavevmode\nobreak\
l\in\\{1,\dots,i,\dots,j,\dots,N\\}$

where $\check{B}_{l}$=[$\Delta\widetilde{A}_{kl}$ $0$] and $\check{D}$ =[$0$
$\mathbf{I}\in\mathbb{R}^{5\times 5}$]. Here, $\Delta\widetilde{A}_{kl}$ is
the difference between the system matrices of
$\widetilde{\mathbf{P}}^{l}_{k}(s)$ and $\check{\mathbf{P}}_{l}(s)$. Now, for
a given $\gamma$, solve (43) for all $\check{Q}_{l}$ and $\check{Y}_{l}$.
Thereafter, recover all the estimator gain using
$\mathbf{\check{L}}_{l}=\check{Q}_{l}^{-1}\check{Y}_{l}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,i,\dots,j,\dots,N\\}$.

Theorem III.1 suggests that the estimators belong to $\mathcal{\check{P}}$
become RS estimators with $\mathbf{\check{L}}_{l}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,i,\dots,j,\dots,N\\}$ attained by
solving (43). Among these RS estimators, let $\mathbf{\check{P}}_{j}(s)$ be
the MERS estimator. Then, following (22), $\mathbf{\check{P}}_{j}(s)$ needs to
satisfy the condition given by

$\displaystyle\begin{split}||\mathbf{e}(s)_{\mathbf{\widetilde{P}}^{j}_{k}\mathbf{\check{P}}_{j}}||_{\infty}=&\min\\{||\mathbf{e}(s)_{\mathbf{\widetilde{P}}^{l}_{k}\mathbf{\check{P}}_{l}}||_{\infty}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,j,\dots,N\\}\\}\\\
&\qquad<\gamma\end{split}$ (44)

Assume there exists suitable $\mathbf{\widetilde{W}_{ei}}(s)$,
$\mathbf{\widetilde{W}_{eo}}(s)$, $\mathbf{\check{L}}_{j}$ such that
$\mathbf{\check{P}}_{j}(s)$ satisfies (44). Then, the state-space form of
$\mathbf{\check{P}}_{j}(s)$ is given by

$\mathbf{\check{P}}_{j}(s):\begin{cases}\dot{\mathbf{\check{\mathbf{x}}}}_{j}(t)={}&\widetilde{A}_{j}\mathbf{\check{x}}_{j}(t)+\widetilde{B}\mathbf{\widetilde{u}}(t)+\mathbf{\check{L}}_{j}(\mathbf{\widetilde{y}}(t)-\widetilde{C}\mathbf{\check{x}}_{j}(t))\\\
\mathbf{\check{z}}_{j}(t)={}&\widetilde{C}_{z}\mathbf{\check{x}}_{j}(t)\end{cases}$
(45)

Equation (45) indicates that $\mathbf{\check{u}}(t)$ and
$\mathbf{\check{y}}(t)$ are required for the implementation of
$\mathbf{\check{P}}_{j}(s)$. But only the information of $\mathbf{u}(t)$ and
$\mathbf{y}(t)$ are available. In that case, we need to obtain
$\mathbf{\check{y}}_{i}(t)$ using
$\mathbf{\check{y}}_{i}(t)=\mathcal{L}^{-1}[\mathbf{\widetilde{W}_{eo}}(s)\mathbf{y}(s)](t)$
and $\mathbf{\check{u}}_{i}(t)$ using
$\mathbf{\check{u}}_{i}(t)=\mathcal{L}^{-1}[\mathbf{\widetilde{W}_{ei}}^{-1}(s)\mathbf{u}(s)](t)$.
Hence, to realize $\mathbf{\check{P}}_{j}(s)$, there should exists
$\mathbf{\widetilde{W}_{eo}}(s)\in\mathcal{RH}_{\infty}$,
$\mathbf{\widetilde{W}_{ei}}(s)\in\mathcal{RH}_{\infty}$,
$\mathbf{\widetilde{W}_{ei}}^{-1}(s)\in\mathcal{RH}_{\infty}$, and
$\mathbf{\check{L}}_{j}=\check{Q}_{l}\check{Y}_{l}$ that satisfy (44). There
is no closed-form solution that provides $\mathbf{\widetilde{W}_{ei}}(s)$,
$\mathbf{\widetilde{W}_{eo}}(s)$, and $\mathbf{\check{L}}_{j}$ that satisfies
(44). Hence, to obtain these feasible compensators and estimator gain, an
optimization problem is formulated and expressed as

$\displaystyle\underset{\mathbf{\widetilde{W}_{ei}},\mathbf{\widetilde{W}_{eo}},\mathbf{\check{L}}_{l}}{\text{minimize}}$
$\displaystyle
J=\min\\{||\mathbf{e}(s)_{\mathbf{\widetilde{P}}^{l}_{k}\mathbf{\check{P}}_{l}}||_{\infty}\leavevmode\nobreak\
|\leavevmode\nobreak\ l\in\\{1,\dots,j,\dots,N\\}\\}$ (46) subject to
$\displaystyle\mathbf{\check{L}}_{l}=\check{Q}_{l}^{-1}\check{Y}_{l}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\in\\{1,\dots,j,\dots,N\\}$
$\displaystyle\mathbf{\widetilde{W}_{ei}}(s)\in\mathcal{RH}_{\infty},\mathbf{\widetilde{W}_{eo}}(s)\in\mathcal{RH}_{\infty},$
$\displaystyle\mathbf{\widetilde{W}_{ei}}^{-1}(s)\in\mathcal{RH}_{\infty}$

where $\check{Q}_{l}$ and $\check{Y}_{l}$ are obtained by solving (43). In
(46), the performance index is the pointwise minimum of $N$ convex function.
The pointwise minimum of $N$ convex function may not be convex. Hence, a
genetic algorithm based iterative approch refered as MERSE algorithm is
developed to solve the problem given in (46). This algorithm has a population-
based GA solver where GA employs the same steps of GA-SCP and GA-RSSD solver
mentioned in [1] (these steps are also given in supporting material).

Search Variables: The search variables of GA solver are the coefficients of
$\mathbf{\widetilde{W}_{ei}}(s)$ and $\mathbf{\widetilde{W}_{eo}}(s)$. The
feasible values of these search variables are those which satisfy the
constraints, $\mathbf{\widetilde{W}_{ei}}(s)\in\mathcal{RH}_{\infty}$,
$\mathbf{\widetilde{W}_{eo}}(s)\in\mathcal{RH}_{\infty}$, and
$\mathbf{\widetilde{W}_{ei}}^{-1}(s)\in\mathcal{RH}_{\infty}$.
Fitness functions: The fitness function of GA solver is the performance index,
$J$.
Termination Conditions: The iterative terminates when the number of generation
of GA exceeds its maximum value.
The pseudocode of MERSE algorithm is given in Algorithm 1. The RS estimation
using MERS estimator is depicted in Fig. 3. This consists of the MERS
estimator (shown inside the orange box) obtained from Algorithm 1. The blocks
in Fig. 3, $\mathbf{\widetilde{W}_{eo}}(s)\stackrel{{\scriptstyle
s}}{{=}}\left[\begin{array}[]{c|c}A_{eo}&B_{eo}\\\ \hline\cr
C_{eo}&D_{eo}\end{array}\right]$ and
$\mathbf{\widetilde{W}_{ei}}^{-1}(s)\stackrel{{\scriptstyle
s}}{{=}}\left[\begin{array}[]{c|c}\underline{A}_{ei}&\underline{B}_{ei}\\\
\hline\cr\underline{C}_{ei}&\underline{D}_{e1}\end{array}\right]$ denote the
state-space representation of $\mathbf{\widetilde{W}_{eo}}(s)$ and
$\mathbf{\widetilde{W}_{ei}}^{-1}(s)$, respectively.

Algorithm 1 Pseudocode of MERSE algorithm

1:Initialize: Genetic algorithm

2:Input: $\mathcal{P}$, maximum number of generations, and $\gamma$

3:if number of generation of GA solver $\leq$ maximum value then

4: GA obtain feasible values of search variables

5: Compute: $\mathbf{\widetilde{W}_{ei}}(s)$ and
$\mathbf{\widetilde{W}_{eo}}(s)$ using (15) and (16) of [12].

6: Compute: $\mathbf{\check{L}}_{l}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ l\leavevmode\nobreak\ \in\\{1,\dots,j,\dots,N\\}$
by solving N LMIs given in (43).

7: Compute: $J$ for fitness evaluation

8: Fitness value of GA= $J$

9: go to 3

10:else

11: if Fitness value $<$ 1 then

12: Find $\mathbf{\widetilde{P}}_{j}(s)$ and $\mathbf{\check{L}}_{j}$

13: Output: feasible $\mathbf{\widetilde{W}_{ei}}(s)$ and
$\mathbf{\widetilde{W}_{eo}}(s)$, $\mathbf{\widetilde{P}}_{j}(s)$,
$\mathbf{\check{L}}_{j}$

14: Exit

15: else

16: Output: no feasible solution to (46)

17: Exit

18: end if

19:end if

Figure 3: MERS estimator for the NAV (from Algorithm 1) shown inside the
orange box

###  III-C Gap Reducing Compensators

The development of the GRC algorithm is presented in this section. The gap
reducing compensators obtained through the GRC algorithm are augmented with
the plants in $\mathcal{P}$ in the implementation of GRMERS estimator to
further reduce the estimation error obtained from the MERS estimator. The
method developed in this section depends only on the plants in $\mathcal{P}$
and is independent of the MERS compensators $\mathbf{\widetilde{W}_{ei}}(s)$,
$\mathbf{\widetilde{W}_{eo}}(s)$ and the gain $\mathbf{\check{L}}_{j}$. From
the output $\mathbf{\widetilde{P}}_{j}(s)$ of the MERSE algorithm, the maximum
value of $\nu-$gap metric of the associated plant
$\mathbf{{P}}_{j}(s)\in\mathcal{P}$ with the other plants in $\mathcal{P}$ is
reduced further by adding suitable pre and post compensators to the plants in
$\mathcal{P}$. From the definition of $\mathcal{G}(\mathbf{P}_{j}(s))$, making
$\mathcal{G}(\mathbf{P}_{i}(s))\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ closer to
$\mathcal{G}(\mathbf{P}_{j})$ increases closeness between
$(\mathbf{u}_{i},\mathbf{y}_{i})\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ and
$(\mathbf{u}_{j},\mathbf{y}_{j})$. To make
$\mathcal{G}(\mathbf{P}_{i}(s))\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ closer to
$\mathcal{G}(\mathbf{P}_{j}(s))$, the gap between
$\mathcal{G}(\mathbf{P}_{i}(s))\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ and
$\mathcal{G}(\mathbf{P}_{j}(s))$ needs to be reduced. For developing an
algorithm that minimizes the gap between
$\mathcal{G}(\mathbf{P}_{i}(s))\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,\dots,N\\}\setminus\\{j\\}$ and
$\mathcal{G}(\mathbf{P}_{j}(s))$, it is necessary to compute the gap between
the graphs. Let
$\delta_{v}(\mathbf{P}_{1}(j\omega),\mathbf{P}_{2}(j\omega))\in[0,1]$ be the
$v$-gap metric (see [12] for details) between two plants,$\mathbf{P}_{1}(s)$
and $\mathbf{P}_{2}(s)$. Then, the gap between two graphs is given by [13]

$\displaystyle
gap(\mathcal{G}(\mathbf{P}_{1}),\mathcal{G}(\mathbf{P}_{2}))=\delta_{v}(\mathbf{P}_{1}(j\omega),\mathbf{P}_{2}(j\omega))$
(47)

Using (47), gap between two graphs is computed. Let
$\epsilon_{\mathbf{P}_{j}}$ denotes the maximum $v-$gap metric of
$\mathbf{P}_{j}(s)$. Then,
$\epsilon_{\mathbf{P}_{j}}=\max\big{\\{}gap\big{(}\mathbf{P}_{j}(j\omega),\mathbf{P}_{i}(j\omega)\big{)}\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\
\mathbf{P}_{j}(s),\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,2,\dots,N\\}\big{\\}}$. Now, using (47), $\epsilon_{\mathbf{P}_{j}}$
is rewritten as

$\displaystyle\begin{split}\epsilon_{\mathbf{P}_{j}}=\max\big{\\{}&\delta_{v}\big{(}\mathbf{P}_{j}(j\omega),\mathbf{P}_{i}(j\omega)\big{)}\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\ \mathbf{P}_{j}(s),\mathbf{P}_{i}(s)\\\
&\quad\in\mathcal{P}\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,2,\dots,N\\}\big{\\}}\end{split}$ (48)

Equation (48) suggests that to make
$\mathcal{G}(\mathbf{P}_{i})\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,\dots,N\\}\setminus\\{j\\}$ closer to $\mathcal{G}(\mathbf{P}_{j})$,
we need to reduce $\epsilon_{\mathbf{P}_{j}}$ and bring it closer towards
zero. The maximum gap of $\mathbf{P}_{j}(s)$ is improved by cascading these
models with suitable pre and post compensators,
$\mathbf{W_{in}}(s)\in\mathcal{RH_{\infty}}$ and
$\mathbf{W_{ot}}(s)\in\mathcal{RH_{\infty}}$, respectively [12].
Simultaneously, if required, these compensators can be employed to improve the
frequency characteristics of the plants in $\mathcal{P}$. The basic structure
of $\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$ are the same as that of
$\mathbf{\widetilde{W}_{in}}(s)$ and $\mathbf{\widetilde{W}_{ot}}(s)$,
respectively. However, $\mathbf{W_{ot}}(s)$ needs to be strictly proper. Let
$\kappa=\big{\\{}\mathbf{\acute{P}}_{i}(s)\in\mathcal{RL_{\infty}}\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\
\mathbf{\acute{P}}_{i}(s)=\mathbf{W_{ot}}(s)\mathbf{P}_{i}(s)\mathbf{W_{in}}(s),\leavevmode\nobreak\
$ $\mathbf{P}_{i}(s)$ $\in\mathcal{P}$, $\mathbf{W_{ot}}(s)$ $\in$
$\mathcal{RH_{\infty}}$, $\mathbf{W_{in}}(s)$ $\in$ $\mathcal{RH_{\infty}}$,
$\forall\leavevmode\nobreak\ i\in\\{1,2,\dots,N\\}\big{\\}}$. Now,
$\acute{\epsilon}_{\mathbf{\acute{P}}_{j}}$ is defined as

$\displaystyle\begin{split}\acute{\mathbf{\epsilon}}_{\mathbf{\acute{P}}_{j}}=\max\big{\\{}&\delta_{v}\big{(}\mathbf{\acute{P}}_{j}(j\omega),\mathbf{\acute{P}}_{i}(j\omega)\big{)}\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\
\mathbf{\acute{P}}_{j}(s),\mathbf{\acute{P}}_{i}(s)\\\
&\quad\in\kappa,\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,2,\dots,N\\}\big{\\}}\end{split}$ (49)

Then, from the GR compensators problem statement, the feasible
$\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$ are those that achieve the
following.

  1. 1.

$\acute{\mathbf{\epsilon}}_{\mathbf{\acute{P}}_{j}}<\epsilon_{\mathbf{P}_{j}}$
and bring $\hat{\epsilon}_{\mathbf{\tilde{P}}_{j}}$ closer to zero.

  2. 2.

$\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$ induce desired frequency
characteristics on all the plants belonging to $\kappa$.

As there does not exist any closed-form solution for the feasible
$\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$, an optimization problem is
formulated and is given by

$\displaystyle\underset{\mathbf{\acute{Q}}}{\text{minimize}}$ $\displaystyle
J_{1}=\acute{\epsilon}_{\mathbf{\acute{P}}_{j}}$ (50) subject to 1) Bound
constraints on the coefficients of pre and post compensators 2) No pole-zero
cancellation between compensators and the plants of $\mathcal{P}$

In (50), $\mathbf{\acute{Q}}$ represents the set that contains the
coefficients of $\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$. The bound
constraints on the coefficients of compensators provide desired frequency
characteristics to the plants of $\mathcal{P}$. These constraints prevent the
minimization of $J_{1}$ with any $\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$
that degrade the frequency characteristics of all the augmented plants. Note
that the pre and post compensators are physically present in the closed-loop
and therefore, these compensators need to be appended to the hardware. The
performance index of (50) is non-convex and non-smooth. Hence, the
optimization problem given in (50) is solved using an iterative algorithm
referred to as the GRC algorithm that has a population-based genetic algorithm
(GA) solver. In that solver, GA employs the same steps as in MERSE algorithm.

Search Variables: The search variables of the GA solver are the coefficients
of $\mathbf{W_{ot}}(s)$ and $\mathbf{W_{in}}(s)$. The feasible values of these
search variables are those which satisfy all the constraints of the problem
given in (50).
Fitness functions: The fitness function of the GA solver is the performance
index, $J_{1}$, of the optimization problem given in (50).
Termination Conditions: The iteration terminates when the number of
generations of GA solver exceeds the set maximum value.

The pseudocode for the iterative algorithm is given in Algorithm 2. The RS
estimation using GRMERS estimator is depicted in Fig. 4. This figure consists
of the MERS estimator (shown inside the orange box) obtained from Algorithm 1
and the gap reducing compensators (shown inside the blue box) obtained from
Algorithm 2. In Fig. 4, the blocks, $\mathbf{W_{in}}(s)\stackrel{{\scriptstyle
s}}{{=}}\left[\begin{array}[]{c|c}A_{in}&B_{in}\\\ \hline\cr
C_{in}&D_{in}\end{array}\right]$ and
$\mathbf{W_{ot}}(s)\stackrel{{\scriptstyle
s}}{{=}}\left[\begin{array}[]{c|c}A_{ot}&B_{ot}\\\ \hline\cr
C_{ot}&D_{ot}\end{array}\right]$ denote the state-space representation of
$\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$ respectively. The plants of
$\mathcal{P}$ are unstable and hence they are stabilized using a full state
feedback controller, $\mathbf{K}$ that uses all the states of
$\mathbf{W_{ot}}(s)\mathbf{P}_{i}(s)\mathbf{W_{in}}(s)$ for feedback . These
states are obtained by augmenting states of $\mathbf{W_{in}}(s)$ and
$\mathbf{W_{ot}}(s)$ and the estimated states of $\mathbf{P}_{i}(s)$,
$\mathbf{\hat{x}}_{i}(t)$ as shown in in Fig. 4. Here,
$\mathbf{\hat{x}}_{i}(t)$ is obtained by augmenting $\mathbf{{y}}_{i}(t)$ with
$\mathbf{\widetilde{z}}_{i}(t)$. Note that if we employ MERS estimator only,
then the full state vector for feedback is $\mathbf{\hat{x}}_{i}(t)$.

Algorithm 2 Pseudocode of GRC algorithm

1:Initialize: Genetic algorithm

2:Input: $\mathcal{P}$, $\mathbf{{P}}_{j}(s)$ (from Algorithm 1), maximum
value of generations, and $\epsilon_{\mathbf{P}_{j}}$.

3:if number of generation of GA solver $\leq$ maximum value then

4: GA obtain feasible values of search variables

5: Compute: $\mathbf{W_{in}}(s)$ and $\mathbf{W_{ot}}(s)$

6: Compute: $J_{1}$ for fitness evaluation

7: Fitness value of GA= $J_{1}$

8: go to 3

9:else

10: if Fitness value $<$ $\epsilon_{\mathbf{P}_{j}}$ then

11: Output: feasible compensators $\mathbf{W_{in}}$ and $\mathbf{W_{ot}}$,
$J_{1}^{*}$=Fitness value

12: Exit

13: else

14: Output: no feasible compensators

15: Exit

16: end if

17:end if

Figure 4: GRMERS estimator for the NAV with the MERS estimator (from Algorithm
1) and the GR compensators (from Algorithm 2) shown inside the orange and blue
boxes, respectively

##  IV Synthesis of the GRMERS Estimator for the candidate NAV

In this section, the synthesis of a GRMERS estimator for the candidate NAV
described in Section II-III of [2] is presented. To this end, a GRMERS
estimator is designed for four unstable MIMO plants
($\mathcal{P}=\\{\mathbf{P}_{1}(s),\mathbf{P}_{2}(s),\mathbf{P}_{3}(s),\mathbf{P}_{4}(s)\\}$)
of this NAV by designing a suitable MERS estimator and the GR compensators.
The plant, $\mathbf{P}_{1}(s)$ is associated with the steady turn and climb
flight condition at $Va$ (air speed) of $9$ m/s, climb rate ($\dot{h}$) of $1$
m/s, turn radius ($R$) of $30$ m. Similarly, $\mathbf{P}_{2}(s)$,
$\mathbf{P}_{3}(s)$ and $\mathbf{P}_{3}(4)$ are associated with the flight
condition at ($Va=10$ m/s, $\dot{h}=1$ m/s, $R=30$ m), ($Va=10$ m/s,
$\dot{h}=0.5$ m/s, $R=30$ m) and ($Va=10$ m/s, $\dot{h}=0$ m/s, $R=30$ m)
respectively. The state-space matrices of all the four plants including
actuator dynamics are given in the supporting material.

###  IV-A Synthesis of MERS estimator

For synthesizing the MERS estimator, the MERSE algorithm was run with the
$\gamma=1$ and the maximum number of generations set to 200. This algorithm
provides $\mathbf{\widetilde{P}}_{j}(s)=\mathbf{\widetilde{P}}_{4}(s)$ with
$||\mathbf{e}_{\mathbf{\widetilde{P}}_{k_{4}}\mathbf{{\check{P}}}_{4}}(s)||_{\infty}=0.65$.
The corresponding compensators $\mathbf{\widetilde{W}_{ei}}(s)$ and
$\mathbf{\widetilde{W}_{eo}}(s)$ are given by

$\displaystyle\mathbf{\widetilde{W}_{ei}}(s)=$ $\displaystyle
diag\bigg{[}\frac{47.3s+13.85}{s+8.196},\frac{1.243s+0.6176}{s+2.576},\frac{0.05965s+1.118}{s+0.7056}\bigg{]}$
(51)

$\displaystyle\begin{split}\mathbf{\widetilde{W}_{eo}}(s)={}&diag\bigg{[}\frac{323.6s+517.2}{s+0.932},\frac{2447s+3710}{s+0.4487},\frac{228.1s+163}{s+1.478},\\\
&\qquad\frac{1453s+803.5}{s+0.6791},\frac{867.4s+564.4}{s+0.8048}\bigg{]}\end{split}$
(52)

Now, the MERS estimator is $\mathbf{{\check{P}}}_{4}(s)$ with estimator gain ,
$\mathbf{\check{L}}_{4}\in\mathbb{R}^{19\times 5}$. This gain is given in the
supporting material and the state-space form of $\mathbf{{\check{P}}}_{4}(s)$
is given by

$\mathbf{{\check{P}}}_{4}(s):\begin{cases}\dot{\mathbf{\check{x}}}_{4}={}&\widetilde{A}_{4}\mathbf{\check{x}}_{4}+\widetilde{B}_{4}\mathbf{\widetilde{u}}_{i}+\mathbf{\check{L}}_{4}(\mathbf{\widetilde{y}}_{i}-\widetilde{C}\mathbf{\check{x}}_{i})\\\
\mathbf{\check{z}}_{4}={}&\widetilde{C}_{z}\mathbf{\check{x}}_{4}\end{cases}$
(53)

The dynamics of $\mathbf{\check{{P}}}_{4}(s)$ is asymptotically stable as
eigenvalues of $\mathbf{\check{{P}}}_{4}(s)$ belongs to $\mathcal{C}_{-}$ as
shown in Fig. 3 of the supporting material.

###  IV-B Synthesis of GR compensators

The maximum $v-$gap metric of the plant associated with the MERS estimator,
$\mathbf{P}_{4}(s)$, is $\epsilon_{{P}_{4}}=0.2915$. The GRC algorithm was run
to find the pre and post compensators that reduce the maximum $v-$gap metric
associated with the $\mathbf{P}_{4}(s)$. These compensators are given by

$\displaystyle\begin{split}\mathbf{W_{in}}(s)={}&diag\bigg{[}\frac{6723s+6409}{s+1},\frac{0.00026s+0.06368}{s+2.333},\\\
&\qquad\quad\frac{0.00278s+0.9258}{s+0.2322}\bigg{]}\end{split}$ (54)
$\displaystyle\begin{split}\mathbf{W_{ot}}(s)={}&diag\bigg{[}\frac{0.0002834}{s+0.3444},\frac{0.0002312}{s+0.7408},\frac{0.003528}{s+57.93},\\\
&\qquad\frac{0.00005}{s+0.7941},\frac{15.27}{s+2.917}\bigg{]}\end{split}$ (55)

Also, the obtained value of $J_{1}^{*}$ is $0.02793$ and is considerably lower
than $\epsilon_{\mathbf{P}_{4}}(s)=0.2915$. This suggests that the closeness
between $\mathbf{\acute{P}}_{4}(s)$ and other plants in $\kappa$ is increased.

###  IV-C Stability, performance, and robustness of GRMERS estimator

In this subsection, a study has been conducted to evaluate the stability,
performance, and robustness of the GRMERS estimator synthesised for the plants
belonging to $\mathcal{P}$. For this, the GRMERS estimator implementation
follows the architecture shown in Fig. 4 with $j=4$. To study the
effectiveness of the GR compensators, we utilize the MERS estimator to
estimate the desired states of all the nominal plants of $\mathcal{P}$. The
MERS estimator implementation for this purpose follows the architecture shown
in Fig. 3 with $j=4$. It is necessary to compare the nominal and robust
performance of the GRMERS estimator while estimating all the plants belonging
to $\mathcal{P}$ with a benchmark estimator. If the GRMERS estimator is
designed to estimate a single plant alone, then the GR compensators are not
required to improve the performance. In that case, the GRMERS estimator can
resemble a $H_{\infty}$ filter. Following this, a single $H_{\infty}$ filter
is designed for each nominal plant belonging to $\mathcal{P}$ (refer to the
supporting material for more details). The nominal and robust performances of
these individual filters while estimating the corresponding nominal and
perturbed plants with measurement noise are then evaluated and compared with
the performance of the GRMERS estimator while estimating all the plants
belonging to $\mathcal{P}$. The measurement noise considered in the
simulations is that of the rate-gyro which is used for measuring $p$, $q$, and
$r$. For simulating the effect of this noise in MATLAB, the noise with a zero
mean, RMS (root mean squared) value of $0.06$ ${}^{\circ}/s$, and a spectral
density of $0.005$ $\frac{{}^{\circ}/s}{\sqrt{Hz}}$ [3] is added to $p$, $q$,
and $r$ (rate-gyro output). In all the simulations, the plants are excited
with a doublet thrust input. Since all the four plants are unstable, a static
full state feedback controller was implemented first to make them stable. The
controller, $\mathbf{K_{1}}\in\mathbb{R}^{3\times 11}$ associated with both
the MERS estimator and $H_{\infty}$ filter are the same. However, the
controller, $\mathbf{K}\in\mathbb{R}^{3\times 19}$ used along with the GRMERS
estimator has different dimension because of the presence of the GR
compensators. The details of the controllers are given in the supporting
material.

####  IV-C1 Stability and nominal performance analysis

The nominal performance of the estimators is obtained through the estimation
of the nominal plants, $\mathbf{P}_{i}(s)\in\mathcal{P}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\leavevmode\nobreak\ \in\leavevmode\nobreak\
\\{1,\dots,4\\}$. The estimated vectors of these plants are:
$\mathbf{z}_{i}(t)=[u(t),\,v(t),\,w(t),\,\delta_{e}(t),\,\delta_{T}(t),\,\delta_{r}(t)]^{T}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\leavevmode\nobreak\ \in\leavevmode\nobreak\
\\{1,\dots,4\\}$. The boundedness of the estimated vector and the estimation
errors indicate that the GRMERS and MERS estimator are stable (refer to the
supporting material for the plots of the estimated vector and the estimation
errors). In this paper, the Normalized-Root-Mean-Squared Error (NRMSE),
$x^{NE}$ is used as a quantitative measure for the estimator’s performance in
estimating any scalar variable, $x(t)$. The NRMSE, $x^{NE}$, of $x(t)$ is
defined as

$x^{NE}=\frac{\sqrt{\frac{1}{T}\sum_{\acute{t}=1}^{T}(x^{\acute{t}}-\hat{x}^{\acute{t}})}}{max(x)-min(x)}$
(56)

where $T$ is the number of observation, $x^{\acute{t}}$ is the $\acute{t}$-th
observation of $x$, $\hat{x}^{\acute{t}}$ is the $\acute{t}$-th estimation of
$x$, $max(x)$ is the maximum value of $x$, and $min(x)$ is the minimum value
of $x$. Now, the normalized error vector, $\mathbf{z}_{i}^{e}$, of
$\mathbf{z}_{i}(t)$ is defined as

$\mathbf{z}_{i}^{e}=[u_{i}^{NE},v_{i}^{NE},w_{i}^{NE},\delta_{e_{i}}^{NE},\delta_{T_{i}}^{NE},\delta_{r_{i}}^{NE}]^{T}$
(57)

where $u_{i}^{NE}$, $v_{i}^{NE}$, $w_{i}^{NE}$, $\delta_{e_{i}}^{NE}$,
$\delta_{T_{i}}^{NE}$, and $\delta_{r_{i}}^{NE}$ are the NRMSE of $u(t)$,
$v(t)$, $w(t)$, $\delta_{e}(t)$, $\delta_{T}(t)$, and $\delta_{r}(t)$ of $i$th
plant, respectively. The nominal and robust performances of the robust
simultaneous estimator are acceptable if
$||\mathbf{z}_{i}^{e}||_{2}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\leavevmode\nobreak\ \in\leavevmode\nobreak\ \\{1,\dots,4\\}$ are closer to
zero. The $||\mathbf{z}_{i}^{e}||_{2}$ of the estimators are given in Table I.
The values shown in this table suggest that the performance of the individual
$H_{\infty}$ filters is the best followed by the GRMERS estimator. Moreover,
the values of $||\mathbf{z}_{i}^{e}||_{2}$ associated with the GRMERS
estimator is smaller than the values of the MERS estimator as indicated by
Table I. Following this, the reduction in the estimation error caused by the
GR compensators with reference to the MERS estimator expressed as the
percentage when the GRMERS estimator estimates $\mathbf{P}_{1}(s)$,
$\mathbf{P}_{2}(s)$, and $\mathbf{P}_{3}(s)$ are 41.13 $\%$, 55.8344 $\%$, and
47.5410 $\%$, respectively. This suggests that a GRMERS estimator, formed by
integrating GR compensators and a MERS estimator, outperforms a sole MERS
estimator. Note that the estimation error reduction by reducing the gap
between the plants using GR compensators for $\mathbf{P}_{4}(s)$ is not
required as the design of GRMERS and MERS estimators are based on same plant,
$\mathbf{P}_{4}(s)$.

Table I: Nominal performance comparison between GRMERS, MERS and individual $H_{\infty}$ filters Estimator | $\mathbf{P}_{1}(s)$ | $\mathbf{P}_{2}(s)$ | $\mathbf{P}_{3}(s)$ | $\mathbf{P}_{4}(s)$
---|---|---|---|---
| $||\mathbf{z}_{1}^{e}||_{2}$ | $||\mathbf{z}_{2}^{e}||_{2}$ | $||\mathbf{z}_{3}^{e}||_{2}$ | $||\mathbf{z}_{4}^{e}||_{2}$
GRMERS | 3.85e-2 | 3.52e-2 | 1.92e-2 | 4.9e-3
MERS | 6.54e-2 | 7.97e-2 | 3.66e-2 | 6.6e-3
$H_{\infty}$ filter | 3.80e-4 | 1.40e-4 | 1.14e-4 | 3.82e-4

####  IV-C2 Robust performance analysis

Here, the robust performance of the GRMERS estimator is presented. For this
purpose, $\mathbf{P}_{1}(s)$, $\mathbf{P}_{2}(s)$, $\mathbf{P}_{3}(s)$, and
$\mathbf{P}_{4}(s)$ are perturbed into $\mathbf{\bar{P}}_{1}(s)$,
$\mathbf{\bar{P}}_{2}(s)$, $\mathbf{\bar{P}}_{3}(s)$, and
$\mathbf{\bar{P}}_{4}(s)$, respectively, by inducing $8.5\leavevmode\nobreak\
\%$, $13\leavevmode\nobreak\ \%$, $10\leavevmode\nobreak\ \%$, and
$5\leavevmode\nobreak\ \%$ parametric uncertainties into the system matrices
of $\mathbf{P}_{1}(s)$, $\mathbf{P}_{2}(s)$, $\mathbf{P}_{3}(s)$, and
$\mathbf{P}_{4}(s)$ such that $||\Delta A_{i}||_{\infty}<||\Delta
A_{k4}||_{\infty}\forall\leavevmode\nobreak\ i\in\\{1,\dots,4\\}$. The state-
space matrices of the perturbed plants are given the supporting material. The
robust performance of the GRMERS estimator is compared with the individual
$H_{\infty}$ filters designed around each plant belonging to $\mathcal{P}$. A
simulation setup similar to the one explained earlier is used to study the
robustness of the GRMERS and the individual $H_{\infty}$ filters for each
plant. Table II shows the robust estimation performances of all the estimators
and filters considered in this paper. This table indicates that the robust
estimation performance of GRMERS estimator is better than the individual
$H_{\infty}$ filters as the $||\mathbf{z_{i}^{e}}||_{2}$ of the GRMERS
estimator is lower than the $H_{\infty}$ filters. The GRMERS estimator’s
estimation errors are $17.86\leavevmode\nobreak\ \%$,
$40.35\leavevmode\nobreak\ \%$, $41.13\leavevmode\nobreak\ \%$, and
$43.00\leavevmode\nobreak\ \%$ smaller than $H_{\infty}$ filters while
estimating $\mathbf{\bar{P}}_{1}(s)$, $\mathbf{\bar{P}}_{2}(s)$ ,
$\mathbf{\bar{P}}_{3}(s)$, and $\mathbf{\bar{P}}_{4}(s)$, respectively.

Table II: Robust performance comparison between GRMERS, and individual $H_{\infty}$ filters Estimator | $\mathbf{\bar{P}}_{1}(s)$ | $\mathbf{\bar{P}}_{2}(s)$ | $\mathbf{\bar{P}}_{3}(s)$ | $\mathbf{\bar{P}}_{4}(s)$
---|---|---|---|---
| $||\mathbf{z}_{1}^{e}||_{2}$ | $||\mathbf{z}_{2}^{e}||_{2}$ | $||\mathbf{z}_{3}^{e}||_{2}$ | $||\mathbf{z}_{4}^{e}||_{2}$
GRMERS | 5.52e-2 | 5.38e-2 | 4.39e-2 | 2.73e-2
$H_{\infty}$ filter | 6.76e-2 | 9.02e-2 | 8.10e-2 | 4.80e-2

##  V Conclusion

In this paper, a novel robust simultaneous estimator referred to as the GRMERS
estimator has been developed to estimate the states of a finite set of
unstable MIMO plants of a NAV. This GRMERS estimator comprises of a MERS
estimator and GR compensators where the former provides robust simultaneous
estimation with minimal largest worst-case estimation error and the latter
reduces this estimation error further by decreasing the gap between the graphs
of $N$ linear plants. For a given set of stable/unstable plants, a sufficient
condition for the existence of a MERS estimator has been presented using LMIs
and robust estimation theory. Two separate non-convex tractable optimization
problems, one for the solution of the sufficient conditions and the other to
obtain the GR compensators, are formulated in terms of LMIs using robust
estimation theory and $\nu$-gap metric, respectively. The solutions for these
optimization problems are obtained using two GA-based iterative algorithms.
The tractability of these algorithms is successfully demonstrated by the
generation of a feasible MERS estimator and GR compensators for four unstable
plants of a typical fixed-wing NAV. The simulation results highlight that the
GRMERS estimator is easily implementable in a typical NAV, and its performance
is within the acceptable limit. Further, the 2-norm of normalized error of the
GRMERS estimator is lower than that of the MERS estimator, which indicates
that the GR compensators are effective in reducing the simultaneous estimation
errors. The nominal and robust performance of the GRMERS estimator is compared
with the individually designed $H_{\infty}$ filters. The performance of GRMERS
and individual $H_{\infty}$ filters are evaluated for both nominal and
perturbed plants, and the results indicate that the GRMERS estimator is robust
under perturbation than $H_{\infty}$ filters and GRMERS estimator provides
satisfactory performance for all nominal plants. The novel GRMERS estimator is
ideal for implementation in computational-resource constrained systems.

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