# Frequency-Domain Stability Method for Reset Control Systems

Ali Ahmadi Dastjerdi<EMAIL_ADDRESS>Alessandro Astolfi
<EMAIL_ADDRESS>S. Hassan HosseinNia<EMAIL_ADDRESS>Department of Precision and Microsystem Engineering, Delft University of
Technology, 2826 CD Delft, The Netherlands  Department of Electrical and
Electronic Engineering, Imperial College London, London, SW7 2AZ, UK

###### Abstract

Reset controllers have the potential to enhance the performance of high-
precision industrial motion systems. However, similar to other non-linear
controllers, the stability analysis for these controllers is complex and often
requires parametric model of the system, which may hinder their applicability.
In this paper a frequency-domain approach for assessing stability properties
of control systems with first and second order reset elements is developed.
The proposed approach is also able to determine uniformly bounded-input
bounded-state (UBIBS) property for reset control systems in the case of
resetting to non-zero values. An illustrative example to demonstrate the
effectiveness of the proposed approach in using frequency response
measurements to assess stability properties of reset control systems is
presented.

###### keywords:

reset controllers; stability; frequency-domain; $H_{\beta}$ condition.

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author S. H. HosseinNia. Tel. +31152784248.

, ,

##  1 Introduction

High-tech precision industrial applications have control requirements which
are hard to fulfil by means of linear controllers. One way to increase the
performance of these systems is to replace linear controllers with non-linear
ones, for instance reset controllers. Owing to their simple structure, these
controllers have attracted significant attention from academia and industry
[17, 12, 2, 19, 44, 3, 42, 28, 22, 13]. In particular, reset controllers have
been utilized to improve the performance of several mechatronic systems (see,
e.g. [27, 23, 21, 41, 16, 40, 38, 9, 10, 24]).

The first reset element was introduced by Clegg [17] in 1958. The Clegg
Integrator (CI) is an integrator which resets its state to zero whenever its
input signal is zero. To provide additional design freedom and flexibility,
extensions of the CI including First Order Reset Elements (FORE) [45, 27],
Generalized First Order Reset Elements (GFORE) [38], Second Order Reset
Elements (SORE) [23], Generalized Second Order Reset Elements (GSORE) [38],
and Second Order Single State Reset Elements (SOSRE) [29] have been developed.
Moreover, to improve the performances of these controllers several methods
such as reset bands [8, 6], fixed reset instants, partial reset techniques
(resetting to a non-zero value or resetting a selection of the controller
states) [46], use of shaping filters in the reset instants line [14], and the
PI+CI approach [46] have also been investigated.

Similar to every control system, stability is one of the most essential
requirements of reset control systems [32, 12, 42, 22, 3, 33, 5, 37].
Stability properties for reset control systems have been studied using
quadratic Lyapunov functions [3, 22, 36, 43], reset instants dependent methods
[5, 4, 34], passivity, small gain, and IQC approaches [32, 20, 15, 26].
However, most of these methods are complex, require parametric models of the
system and the solution of LMI’s, and are only applicable to specific types of
systems. Thus, since industry often favors the use of frequency-domain
methods, these methods are not well matched with the current control design
requirements in industry. To overcome this challenge, some frequency-domain
approaches for assessing stability properties of reset control systems have
been proposed [11, 12, 42]. A method for determining stability properties of a
FORE in closed-loop with a mass-spring-damper system has been developed in
[11]. However, this method is only applicable to a specific type of systems.
Under the specific reset condition $e(t)u(t)<\dfrac{u^{2}(t)}{\varepsilon}$,
for some $\varepsilon>0$, in which $e$ and $u$ are the input and the output of
the reset element, respectively, the approach in [42] is applicable to reset
control systems. However, this method is not applicable to traditional reset
control systems in which the reset condition is $e(t)=0$.

The $H_{\beta}$ condition is one of the most widely-used methods for assessing
stability properties of reset control systems [12, 5, 22]. In particular, when
the base linear system of the reset element has a first order transfer
function, it gives sufficient frequency-domain conditions for uniform bounded-
input bounded-state (UBIBS) stability. However, assessing the $H_{\beta}$
condition in the frequency-domain is not intuitive, especially for high order
transfer function plants. In addition, the effect of a shaping filter in the
reset line on the $H_{\beta}$ condition has not been studied yet. Furthermore,
there is a lack of methods to assess the $H_{\beta}$ condition for GSORE using
Frequency Response Function (FRF) measurements. Finally, the $H_{\beta}$
condition is not applicable to assess UBIBS stability of reset control systems
in the case of partial reset techniques. Hence, obtaining a general easy-to-
use frequency-domain method for assessing UBIBS stability of reset control
systems is an important open question.

In this paper, on the basis of the $H_{\beta}$ condition, novel frequency-
domain stability conditions for control systems with first and second order
reset elements with a shaping filter in the reset line are proposed. This
approach allows for assessing UBIBS stability of reset control systems in the
frequency-domain. In this approach, the $H_{\beta}$ condition does not have to
be explicitly tested and stability properties are directly determined on the
basis of the FRF measurements of the base linear open-loop system. In
addition, the approach can be used in the case of partial reset techniques.

The remainder of the paper is organized as follows. In Section 2 preliminaries
about reset elements are presented and the problem is formulated. The
frequency-domain approaches for assessing stability properties of control
systems with first and second order reset elements are presented in Section 3
and Section 4, respectively. In Section 5 the effectiveness of these
approaches is demonstrated via a practical example. Finally, conclusions and
suggestions for future studies are given in Section 6.

##  2 Preliminaries

In this section the description of reset elements and the $H_{\beta}$
condition are breifly recalled and some preliminaries are given. The focus of
the paper is on the single-input single-output (SISO) control architecture
illustrated in Fig. 1. The closed-loop system consists of a linear plant with
transfer function $G(s)$ (which we assume strictly proper), linear controllers
with proper transfer functions $C_{L_{1}}(s)$ and $C_{L_{2}}(s)$, a reset
element with base transfer function $C_{R}(s)$, and a shaping filter with a
proper stable transfer function $\mathcal{C}_{s}(s)$.

The state-space representation of the reset element is

$-$$r$$y$$e_{r}$Liner ControllerPlant$+$$d$$\mathbf{C_{\mathbf{R}}}$Reset
Controller$\mathbf{C_{\mathbf{L}_{2}}}$$u_{r}$$\mathbf{G}$$\mathcal{L}$$\mathcal{C}_{s}$$\mathbf{C_{\mathbf{L}_{1}}}$$e$$u_{1}$Liner
Controller

Figure 1: The closed-loop architecture of a reset control system

$\left\\{\begin{aligned}
\dot{x}_{r}(t)&=A_{r}x_{r}(t)+B_{r}u_{1}(t),&e_{r}(t)&\neq 0,\\\
x_{r}(t^{+})&=A_{\rho}x_{r}(t),&e_{r}(t)&=0\land(I-A_{\rho})x_{r}(t)\neq 0,\\\
u_{r}(t)&=C_{r}x_{r}(t)+D_{r}u_{1}(t),\end{aligned}\right.$

(1)

in which $x_{r}(t)\in\mathbb{R}^{n_{r}}$ is the vector containing the reset
state, $A_{r}$, $B_{r}$, $C_{r}$, and $D_{r}$ are the dynamic matrices of the
reset element, $A_{\rho}$ is the reset matrix, which determines the values of
the reset state after the reset action, and $u_{1}(t)\in\mathbb{R}$ and
$u_{r}(t)\in\mathbb{R}$ are the input and output of the reset element,
respectively. The transfer function $C_{r}(sI-A_{r})^{-1}B_{r}+D_{r}$ is
called the base transfer function of the reset element. The base transfer
function in case of GFORE is (in all cases $\omega_{r}>0$)

$C_{R}(s)=\dfrac{1}{\displaystyle\frac{s}{\omega_{r}}+1},$ (2)

for CI and Proportional Clegg Integrator (PCI) one has

$\displaystyle C_{R}(s)$ $\displaystyle=\dfrac{1}{s},$ (3) $\displaystyle
C_{R}(s)$ $\displaystyle=1+\dfrac{\omega_{r}}{s},$ (4)

and for GSORE one has

$C_{R}(s)=\dfrac{1}{s^{2}+2\xi\omega_{r}s+\omega_{r}^{2}},\ \xi>0.$ (5)

Thus, for GFORE, $A_{r}=-C_{r}=-\omega_{r}$ ($\omega_{r}$ is the so-called
corner frequency), $D_{r}=0$, and $B_{r}=1$, whereas for the PCI, $A_{r}=0$,
$C_{r}=\omega_{r}$, and $B_{r}=D_{r}=1$. In the case of CI, $A_{r}=D_{r}=0$,
$B_{r}=C_{r}=1$, and if we consider the controllable canonical form
realization for GSORE, we obtain

$A_{r}=\begin{bmatrix}-2\xi\omega_{r}&-\omega_{r}^{2}\\\ 1&0\end{bmatrix},\
B_{r}=\begin{bmatrix}1\\\ 0\end{bmatrix},\
C_{r}=\begin{bmatrix}0&1\end{bmatrix},\text{ and }D_{r}=0.$ (6)

Let $\mathcal{L}$ be the linear time-invariant (LTI) part of the system, see
Fig. 1, with input $u_{r}(t)\in\mathbb{R}$, external disturbance
$w(t)=\begin{bmatrix}r(t)&d(t)\end{bmatrix}^{T}\in\mathbb{R}^{2}$, and outputs
$y(t)\in\mathbb{R}$, $e_{r}(t)\in\mathbb{R}$, and $u_{1}(t)\in\mathbb{R}$. The
state-space realization of $\mathcal{L}$ is given by equations

$\mathcal{L}:\left\\{\begin{aligned}
\dot{\zeta}(t)&=A\zeta(t)+B_{u}u_{r}(t)+Bw(t),\\\ y(t)&=C\zeta(t),\\\
e_{r}(t)&=C_{e}\zeta(t)+D_{e}r(t),\\\
u_{1}(t)&=C_{u}\zeta(t)+D_{1}r(t),\end{aligned}\right.$ (7)

where $\zeta(t)\in\mathbb{R}^{n_{p}}$ describes the states of the plant and of
the linear controllers ($n_{p}$ is the number of states of the whole linear
part), and $A$, $B$, $B_{u}$, and $C$ are the corresponding dynamic matrices.
The closed-loop state-space representation of the overall system can,
therefore, be written as

$\left\\{\begin{aligned} \dot{x}(t)&=\bar{A}x(t)+\bar{B}w(t),&e_{r}(t)&\neq
0,\\\ x(t^{+})&=\bar{A}_{\rho}x(t),&e_{r}(t)&=0\land(I-\bar{A}_{\rho})x(t)\neq
0,\\\ y(t)&=\bar{C}x(t),\\\
e_{r}(t)&=\bar{C}_{e}x(t)+D_{e}r(t),\end{aligned}\right.$ (8)

where
$x(t)=\begin{bmatrix}x_{r}(t)^{T}&\zeta(t)^{T}\end{bmatrix}^{T}\in\mathbb{R}^{n_{r}+n_{p}}$,
$\bar{C}=\begin{bmatrix}0_{1\times n_{r}}&C\end{bmatrix}$,
$\bar{B}=\begin{bmatrix}0_{n_{r}\times 2}\\\
B\end{bmatrix}+\begin{bmatrix}B_{r}D_{1}&0_{n_{r}\times 1}\\\
B_{u}D_{r}D_{1}&0_{n_{p}\times 1}\end{bmatrix}$,
$\bar{C}_{e}=\begin{bmatrix}0_{1\times n_{r}}&C_{e}\end{bmatrix}$,
$\bar{A}=\begin{bmatrix}A_{r}&B_{r}C_{u}\\\
B_{u}C_{r}&A+B_{u}D_{r}C_{u}\end{bmatrix}$, and
$\bar{A}_{\rho}=\begin{bmatrix}A_{\rho}&0_{n_{r}\times n_{p}}\\\
0_{n_{p}\times n_{r}}&I_{n_{p}\times n_{p}}\end{bmatrix}$.

######  Definition 1.

A time $\bar{T}>0$ is called a reset instant for the reset control system (8)
if $e_{R}(\bar{T})=0\land(I-\bar{A}_{\rho})x(T)\neq 0$. For any given initial
condition and input $w$ the resulting set of all reset instants defines the
reset sequence $\\{t_{k}\\}$, with $t_{k}\leq t_{k+1}$, for all
$k\in\mathbb{N}$. The reset instants $t_{k}$ have the well-posedness property
if for any initial condition $x_{0}$ and any input $w$, all the reset instants
are distinct, and there exists $\lambda>0$ such that, for all
$k\in\mathbb{N}$, $\lambda\leq t_{k+1}-t_{k}$ [7, 3].

One of the methods for determining stability properties of reset control
systems is the $H_{\beta}$ condition [12, 3, 5, 22, 25], which is briefly
recalled. Let

$C_{0}=[\Varrho\quad\beta C],\quad B_{0}=\begin{bmatrix}I_{n_{r}\times
n_{r}}\\\ 0_{n_{p}\times
n_{r}}\end{bmatrix},\quad\Varrho=\Varrho^{T}>0,\quad\Varrho\in\mathbb{R}^{n_{r}\times
n_{r}},$ (9)

and $\beta\in\mathbb{R}^{n_{r}\times 1}.$ The $H_{\beta}$ condition [12, 3, 5,
22, 25] states that the zero equilibrium of the reset control system (8) with
$C_{L_{1}}=\mathcal{C}_{s}=1$ and $w=0$ is globally uniformly asymptotically
stable if and only if there exist $\Varrho=\Varrho^{T}>0$ and $\beta$ such
that the transfer function

$H(s)=C_{0}(sI-\bar{A})^{-1}B_{0}$ (10)

is Strictly Positive Real (SPR), $(\bar{A},B_{0})$ and $(\bar{A},C_{0})$ are
controllable and observable, respectively, and

$A_{\rho}^{T}\Varrho A_{\rho}-\Varrho\ \leq 0.$ (11)

Evaluating the $H_{\beta}$ condition requires finding the parameters $\Varrho$
and $\beta$, which may be very difficult when the system has a high order
transfer function. Furthermore, in the case of GSORE there is no direct
frequency-domain method to assess this condition. Besides, the UBIBS property
of GSORE and of GFORE have not yet been studied, and the effects of the
shaping filter on the $H_{\beta}$ condition have not been considered yet. In
the current paper, frequency-domain methods to determine stability properties
without finding $\Varrho$ and $\beta$ for GFORE and of GSORE with considering
the shaping filter are proposed.

######  Assumption 1.

There are infinitely many reset instants and
$\displaystyle\lim_{k\to\infty}t_{k}=\infty$.

Assumption 1 is introduced to rule out a trivial situation. In fact, if there
are finitely many reset instants, then there exists a $T_{K}\in[0,\infty)$
such that for all $t\geq T_{K}$ the reset control system (8) is a linear
stable system provided the $H_{\beta}$ condition is satisfied. In addition to
Assumption 1, we need the following assumption, which is instrumental to study
the UBIBS property of reset control systems.

######  Assumption 2.

In the case of partial reset technique, if $A_{\rho}$ has the structure

$A_{\rho}=\begin{bmatrix}I_{\tilde{n}_{r}}&0\\\
0&A^{\prime}_{n^{\prime}_{r}}\end{bmatrix},$

then $A_{r}$ has the structure

$A_{r}=\begin{bmatrix}A_{r_{1}}&A_{r_{2}}\\\ 0_{\tilde{n}_{r}\times
n^{\prime}_{r}}&A_{r_{3}}\end{bmatrix}.$

######  Remark 1.

In the case of GFORE, GSORE, PCI, and CI in which all states of the reset
element reset, Assumption 2 holds.

Before stating the main theorem, an important technical lemma, which is
instrumental for all proofs, is formulated and proved.

######  Lemma 1.

Consider the reset control system (8). Suppose that

  * •

Assumption 1 holds;

  * •

$A_{\rho}^{T}\Varrho A_{\rho}-\Varrho<0$;

  * •

the $H_{\beta}$ condition holds;

  * •

at least one of the following conditions holds:

    1. 1.

$\mathcal{C}_{s}=1$ and Assumption 2 holds;

    2. 2.

the reset instants have the well-posedness property.

Then the reset control system (8) has a well-defined unique left-continuous
solution for any initial condition $x_{0}$ and any input $w$ which is a Bohl
function111See definition Bohl function in [7]. In addition, this solution is
UBIBS and the reset instants have the well-posedness property.

See Appendix A.

##  3 Stability analysis of reset control systems with first order reset
elements

In this section frequency-domain methods for assessing stability properties of
the reset control system (8) with GFORE (2), CI (3), and PCI (4) are proposed
on the basis of the $H_{\beta}$ condition. To this end, the Nyquist Stability
Vector (NSV=$\vv{\mathcal{N}}(\omega)\in\mathbb{R}^{2}$) in a plane with axis
$\chi-\Upsilon$ (see Fig. 2) is defined as follows.

######  Definition 2.

The Nyquist Stability Vector is, for all $\omega\in\mathbb{R}^{+}$, the vector

$\displaystyle\vv{\mathcal{N}}(\omega)$ $\displaystyle{=}$
$\displaystyle[\mathcal{N}_{\chi}\quad\mathcal{N}_{\Upsilon}]^{T}$
$\displaystyle{=}$
$\displaystyle[\Re(L(j\omega)\mathcal{C}_{s}(j\omega)\kappa(j\omega))\quad\Re(\kappa(j\omega)C_{R}(j\omega))]^{T},$

in which $L(s)=C_{L_{1}}(s)C_{R}(s)C_{L_{2}}(s)G(s)$,
$L(j\omega)=a(\omega)+b(\omega)j$, and $\kappa(j\omega)=1+L^{*}(j\omega)$
($L^{*}(j\omega)$ is the conjugate of $L(j\omega)$).

$\chi$$\Upsilon$$\theta_{\mathcal{N}}$$\vv{\mathcal{N}}$$\mathcal{N}_{\Upsilon}$$\mathcal{N}_{\chi}$

Figure 2: Representation of the NSV in the $\chi-\Upsilon$ plane

For simplicity, and without loss of generality, let
$\phase{\vv{\mathcal{N}}(\omega)}=\theta_{\mathcal{N}}\in[-\frac{\pi}{2},\
\frac{3\pi}{2})$ and define the open sets

$\mathcal{I}_{1}=\left\\{\omega\in\mathbb{R}^{+}|\
0<\phase{\vv{\mathcal{N}}(\omega)}<\frac{\pi}{2}\right\\},$
$\mathcal{I}_{2}=\left\\{\omega\in\mathbb{R}^{+}|\
\dfrac{\pi}{2}<\phase{\vv{\mathcal{N}}(\omega)}<\pi\right\\},$
$\mathcal{I}_{3}=\left\\{\omega\in\mathbb{R}^{+}|\
\pi<\phase{\vv{\mathcal{N}}(\omega)}<\dfrac{3\pi}{2}\right\\},$
$\mathcal{I}_{4}=\left\\{\omega\in\mathbb{R}^{+}|\
-\dfrac{\pi}{2}<\phase{\vv{\mathcal{N}}(\omega)}<0\right\\}.$

Let
$L(s)\mathcal{C}_{s}(s)=\dfrac{K_{m}s^{m}+K_{m-1}s^{m-1}+...+K_{0}}{s^{n}+K^{\prime}_{n-1}s^{n-1}+...+K^{\prime}_{0}}$
and
$\mathcal{C}_{s}(s)=\dfrac{K_{s_{m}}s^{m_{s}}+K_{s_{m-1}}s^{m_{s}-1}+...+K_{s_{0}}}{K^{\prime}_{s_{n}}s^{n_{s}}+K^{\prime}_{s_{n-1}}s^{n_{s}-1}+...+1}$.
On the basis of the definition of the NSV, systems of Type I and of Type II,
which are used to assess stability properties of the reset control system (8),
are defined.

######  Definition 3.

The reset control system (8) is of Type I if the following conditions hold.

  1. (1)

If $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ has at least one pole at the origin, then
$K_{s_{0}}>0$.

  2. (2)

In the case of CI (3), $K_{s_{0}}<0$.

  3. (3)

For all $\omega\in\mathcal{M}=\\{\omega\in\mathbb{R}^{+}|\
\mathcal{N}_{\chi}(\omega)=0\\}$ one has $\mathcal{N}_{\Upsilon}(\omega)>0$.

  4. (4)

For all $\omega\in\mathcal{Q}=\\{\omega\in\mathbb{R}^{+}|\
\mathcal{N}_{\Upsilon}(\omega)=0\\}$ one has $\mathcal{N}_{\chi}(\omega)>0$.

  5. (5)

At least one of the following statements is true:

    1. (a)

$\forall\ \omega\in\mathbb{R}^{+}:\ \mathcal{N}_{\Upsilon}(\omega)\geq 0.$

    2. (b)

$\forall\ \omega\in\mathbb{R}^{+}:\ \mathcal{N}_{\chi}(\omega)\geq 0.$

    3. (c)

Let
$\delta_{1}=\underset{\omega\in\mathcal{I}_{4}}{\max}\left\lvert\dfrac{\mathcal{N}_{\Upsilon}(\omega)}{\mathcal{N}_{\chi}(\omega)}\right\rvert$
and
$\Psi_{1}=\underset{\omega\in\mathcal{I}_{2}}{\min}\left\lvert\dfrac{\mathcal{N}_{\Upsilon}(\omega)}{\mathcal{N}_{\chi}(\omega)}\right\rvert$.
Then $\delta_{1}<\Psi_{1}$ and $\mathcal{I}_{3}=\varnothing$.

######  Remark 2.

Let

$\theta_{1}=\underset{\omega\in\mathbb{R}^{+}}{\min}\phase{\vv{\mathcal{N}}(\omega)}\text{
and
}\theta_{2}=\underset{\omega\in\mathbb{R}^{+}}{\max}\phase{\vv{\mathcal{N}}(\omega)}.$
(12)

Then the conditions identifying Type I systems are equivalent to the following
conditions.

  1. (1)

If $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ has at least one at the origin, then
$K_{s_{0}}>0$.

  2. (2)

In the case of CI (3), $K_{s_{0}}<0$.

  3. (3)

The condition

$\left(-\dfrac{\pi}{2}<\theta_{1}<\pi\right)\ \land\
\left(-\dfrac{\pi}{2}<\theta_{2}<\pi\right)\ \land\
(\theta_{2}-\theta_{1}<\pi)$ (13)

holds.

######  Definition 4.

The reset control system (8) is of Type II if the following conditions hold.

  1. (1)

If $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ has at least one at the origin, then
$K_{s_{0}}<0$.

  2. (2)

In the case of CI (3), $K_{s_{0}}>0$.

  3. (3)

For all $\omega\in\mathcal{M}$ one has $\mathcal{N}_{\Upsilon}(\omega)>0$.

  4. (4)

For all $\omega\in\mathcal{Q}$ one has $\mathcal{N}_{\chi}(\omega)<0$.

  5. (5)

At least, one of the following statements is true:

    1. (a)

$\forall\ \omega\in\mathbb{R}^{+}:\ \mathcal{N}_{\Upsilon}(\omega)\geq 0$;

    2. (b)

$\forall\ \omega\in\mathbb{R}^{+}:\ \mathcal{N}_{\chi}(\omega)\leq 0$;

    3. (c)

Let
$\delta_{2}=\underset{\omega\in\mathcal{I}_{3}}{\max}\left\lvert\dfrac{\mathcal{N}_{\Upsilon}(\omega)}{\mathcal{N}_{\chi}(\omega)}\right\rvert$
and
$\Psi_{2}=\underset{\omega\in\mathcal{I}_{1}}{\min}\left\lvert\dfrac{\mathcal{N}_{\Upsilon}(\omega)}{\mathcal{N}_{\chi}(\omega)}\right\rvert$.
Then, $\delta_{2}<\Psi_{2}$ and $\mathcal{I}_{4}=\varnothing$.

######  Remark 3.

The conditions identifying Type II systems are equivalent to the following
conditions.

  1. (1)

If $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ has at least one at the origin, then
$K_{s_{0}}<0$.

  2. (2)

In the case of CI (3), $K_{s_{0}}>0$.

  3. (3)

The condition

$\left(0<\theta_{1}<\dfrac{3\pi}{2}\right)\ \land\
\left(0<\theta_{2}<\dfrac{3\pi}{2}\right)\ \land\ (\theta_{2}-\theta_{1}<\pi)$
(14)

holds.

######  Theorem 1.

The zero equilibrium of the reset control system (8) with GFORE (2), or CI
(3), or PCI (4) is globally uniformly asymptotically stable when $w=0$, and
the system has the UBIBS property for any input $w$ which is a Bohl function
if all of the following conditions are satisfied.

  * •

The base linear system is stable and the open-loop transfer function does not
have any pole-zero cancellation.

  * •

In the case of CI (3), $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ does not have any pole
at the origin and $n-m=2$.

  * •

The reset control system (8) is either of Type I and/or of Type II.

  * •

$A_{\rho}=\gamma,\ -1<\gamma<1.$

  * •

$\mathcal{C}_{s}(s)=1$ and/or the reset instants have the well-posedness
property.

For $w(t)=0$, for all $t\geq 0$, reset happens when
$x(t)\in\text{ker}(\bar{C}_{e})$. Looking at the proof of the $H_{\beta}$
condition, which is given in [12, 22, 3], when there is a shaping filter in
the reset line, $C_{0}$ in the $H_{\beta}$ condition is changed to

$C_{0}=[\Varrho\quad\beta\bar{C}_{e}].$ (15)

Theorem 1 is now proved in several steps.

  * •

Step 1: It is shown that there is a $\beta$ and $\Varrho>0$ such that
$\Re(H(j\omega))>0$, for all $\omega\in\mathbb{R}^{+}$.

  * •

Step 2: For systems with poles at the origin it is shown that
$\displaystyle\lim_{\omega\to 0}\Re(H(j\omega))>0$.

  * •

Step 3: It is shown that either $\displaystyle\lim_{s\to\infty}H(s)>0$ or
$\displaystyle\lim_{\omega\to\infty}\omega^{2}\Re(H(j\omega))>0.$

  * •

Step 4: It is shown that $(A,C_{0})$ and $(A,B_{0})$ are observable and
controllable, respectively.

Step 1: For simplicity take $\beta^{\prime}=-\beta$ and
$\Varrho^{\prime}=\dfrac{\Varrho}{C_{r}}$. The transfer function (10) with the
modified $C_{0}$ as in (15) can be rewritten as (see also Fig. 3)

$H(s)=\dfrac{y}{r}=\dfrac{\beta^{\prime}L(s)\mathcal{C}_{s}(s)+\Varrho^{\prime}C_{R}(s)}{1+L(s)}.$
(16)

$-$$y$$\mathbf{C_{\mathbf{R}}}$$\mathbf{C_{\mathbf{L}_{2}}}$$\mathbf{G}$$\mathcal{C}_{s}$$\mathbf{C_{\mathbf{L}_{1}}}$$r$$\beta^{\prime}$$\Varrho^{\prime}$$+$

Figure 3: The block diagram of the $H_{\beta}$ condition for the closed-loop
architecture Fig. 1 with GFORE or PCI

Thus222Omitting arguments for simplicity

$\Re(H(j\omega))=\dfrac{\beta^{\prime}\mathcal{N}_{\chi}+\Varrho^{\prime}\mathcal{N}_{\Upsilon}}{(a+1)^{2}+b^{2}}.$
(17)

Define now the vector $\vv{\xi}$ in the $\chi-\Upsilon$ plane as
$\vv{\xi}=[\beta^{\prime}\ \ \Varrho^{\prime}]^{T}$. Using Definition 2,
equation (17) can be re-written as

$\Re(H(j\omega))=\dfrac{\vv{\xi}\cdot\vv{\mathcal{N}}}{(a+1)^{2}+b^{2}}.$ (18)

Therefore

$\begin{array}[]{*{35}{c}}\forall\omega\in\mathbb{R}^{+}:\
\Re(H(j\omega))>0\iff\vv{\xi}\cdot\vv{\mathcal{N}}>0\iff\\\
-\frac{\pi}{2}<\phase{(\vv{\xi},\vv{\mathcal{N}})}<\frac{\pi}{2}\ \land\
\left\lvert\vv{\mathcal{N}}\right\rvert\neq 0\ \land\
\left\lvert\vv{\mathcal{\xi}}\right\rvert\neq 0.\end{array}$ (19)

The rest of the proof of this step are the same as the proof of Step 1
provided in [1].
Step 2: When the open-loop system has poles at the origin and $C_{R}$ is a
GFORE, equation (16) becomes

$\displaystyle\lim_{\omega\to 0}\Re(H(j\omega))=K_{s_{0}}\beta^{\prime}>0,$
(20)

whereas in the case of PCI and CI when $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ does not
have any pole at the origin, (16) becomes

$\displaystyle\lim_{\omega\to
0}\Re(H(j\omega))=K_{s_{0}}\beta^{\prime}+\Varrho^{\prime}\dfrac{\omega_{r}}{C_{L_{1}}(0)C_{L_{2}}(0)G(0)}>0.$
(21)

Setting
$\vv{\mathcal{N}^{\prime}}=[K_{s_{0}}\quad\dfrac{\omega_{r}}{C_{L_{1}}(0)C_{L_{2}}(0)G(0)}]^{T}$,
yields

$\displaystyle\lim_{\omega\to
0}\Re(H(j\omega))=\vv{\xi}\cdot\vv{\mathcal{N}^{\prime}}.$ (22)

In addition

$\phase{\vv{\mathcal{N}^{\prime}}}=\displaystyle\lim_{\omega\to
0}\phase{\vv{\mathcal{N}}}\xRightarrow{\ (\ref{E_4-333})\
}\theta_{1}\leq\phase{\vv{\mathcal{N}^{\prime}}}\leq\theta_{2}.$ (23)

As a result, by Step 1, $\displaystyle\lim_{\omega\to
0}\Re(H(j\omega))=\vv{\xi}\cdot\vv{\mathcal{N}^{\prime}}>0$. For PCI, when
$C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ has poles at the origin,

$\displaystyle\lim_{\omega\to 0}\Re(H(j\omega))=K_{s_{0}}\beta^{\prime}>0.$
(24)

Note that for CI in equations (21)-(23), $\omega_{r}=1$. It is therefore
concluded that if $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ has poles at the origin, then
$K_{s_{0}}\beta^{\prime}>0$. If $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ does not have
any pole at the origin, $\beta$ can be either positive or negative.
Step 3: In the case of GFORE with $n-m=2$, setting
$\vv{\mathcal{N}^{\prime\prime}}=[-K_{n}\quad\omega_{r}^{2}]^{T}$ yields

$\displaystyle\lim_{\omega\to\infty}\omega^{2}\Re(H(j\omega))=-\beta^{\prime}K_{n}+\Varrho^{\prime}\omega_{r}^{2}=\vv{\xi}\cdot\vv{\mathcal{N}^{\prime\prime}}.$
(25)

In addition,

$\phase{\vv{\mathcal{N}^{\prime\prime}}}=\displaystyle\lim_{\omega\to\infty}\phase{\vv{\mathcal{N}}}\xRightarrow{\
(\ref{E_4-333})\
}\theta_{1}\leq\phase{\vv{\mathcal{N}^{\prime\prime}}}\leq\theta_{2}.$ (26)

Thus, by Step 1
$\displaystyle\lim_{\omega\to\infty}\omega^{2}\Re(H(j\omega))=\vv{\xi}\cdot\vv{\mathcal{N}^{\prime\prime}}>0$.
For GFORE with $n-m>2$,
$\displaystyle\lim_{\omega\to\infty}\omega^{2}\Re(H(j\omega))=\Varrho^{\prime}\omega_{r}^{2}>0$.
For PCI $\displaystyle\lim_{s\to\infty}H(s)=\Varrho^{\prime}>0$. Moreover, in
the case of CI when $n-m>2$,

$\displaystyle\lim_{\omega\to\infty}\omega^{2}\Re(H(j\omega))=0,$ (27)

which implies that $H(s)$ is not SPR in the case of $n-m>2$. Whereas in the
case of CI with $n-m=2$,

$\displaystyle\lim_{\omega\to\infty}\omega^{2}\Re(H(j\omega))=-K_{s_{0}}\beta^{\prime}>0,$
(28)

which means that in the case of CI, $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ must not
have any pole at the origin.
Step 4: In order to show that the pairs $(A,C_{0})$ and $(A,B_{0})$ are
observable and controllable, respectively, it is sufficient to show that the
denominator and the numerator of $H(s)$ do not have any common root. Let
$a_{0}+jb_{0}$ be a root of the denominator. Then

$1+R_{L}(a_{0},b_{0})+jI_{L}(a_{0},b_{0})=0\Rightarrow\begin{cases}R_{L}(a_{0},b_{0})=-1,\\\
I_{L}(a_{0},b_{0})=0.\end{cases}$ (29)

Now, the numerator must not have a root at $a_{0}+jb_{0}$, that is

$\begin{array}[]{*{35}{c}}\beta^{\prime}\left(R_{\mathcal{C}_{s}}(a_{0},b_{0})+jI_{\mathcal{C}_{s}}(a_{0},b_{0})\right)\neq\Varrho^{\prime}\left(R_{C_{R}}(a_{0},b_{0})+jI_{C_{R}}(a_{0},b_{0})\right)\\\
\Rightarrow\beta^{\prime}R_{\mathcal{C}_{s}}(a_{0},b_{0})\neq\Varrho^{\prime}R_{C_{R}}(a_{0},b_{0})\
\lor\
\beta^{\prime}I_{\mathcal{C}_{s}}(a_{0},b_{0})\neq\Varrho^{\prime}I_{C_{R}}(a_{0},b_{0}).\end{array}$

(30)

Therefore, using Step 1 and (30) it is possible to find a pair
$(\beta^{\prime},\Varrho^{\prime})$ such that $H(s)$ does not have any pole-
zero cancellation. According to Step 1-4, $H(s)$ is SPR [32],
$(\bar{A},C_{0})$ is observable and $(\bar{A},B_{0})$ is controllable, and the
base linear system is stable. Moreover, since $-1<\gamma<1$, one has that
$A_{\rho}^{T}\Varrho A_{\rho}-\Varrho<0$. As a result, the $H_{\beta}$
condition is satisfied for the reset control system (8) with GFORE (2), or CI
(3), or PCI (4). Hence, the zero equilibrium of the reset control system (8)
is globally uniformly asymptotically stable when $w=0$, and according to Lemma
1, it has the UBIBS property for any initial condition $x_{0}$ and any input
$w$ which is a Bohl function.

######  Corollary 3.1.

Let $\mathcal{C}_{s}(s)=1$, $\theta_{L}=\phase{L(j\omega)}$, and
$\theta_{C_{R}}=\phase{C_{R}(j\omega)}$. Suppose that the base linear system
of the reset control system (8) is stable, $A_{\rho}=\gamma,\ -1<\gamma<1$,
$L(s)$ and the open-loop system does not have any pole-zero cancellation. Then
the zero equilibrium of the reset control system (8) with GFORE (2), or CI
(3), or PCI (4) is globally uniformly asymptotically stable when $w=0$, and
the system has the UBIBS property for any input $w$ which is a Bohl function
if at least one of the following conditions hold.

  1. 1.

For all $\omega\in\mathbb{R}^{+}$, $\sin(\theta_{L})\geq 0$.

  2. 2.

For all $\omega\in\mathbb{R}^{+}$, $\cos(\theta_{L}-\theta_{C_{R}})\geq 0$ and
the reset element is not CI (3).

When $\mathcal{C}_{s}(s)=1$,
$\mathcal{N}_{\chi}(\omega)=a(\omega)^{2}+b(\omega)^{2}+b(\omega)$. By
Hypothesis 1, $b(\omega)\geq 0$, for all $\omega\in\mathbb{R}^{+}$, which
implies that $\mathcal{N}_{\chi}(\omega)>0$. Thus, the reset control system
(8) is of Type I. In addition, defining
$C_{R}(j\omega)=a_{R}(\omega)+jb_{R}(\omega)$, yields
$\mathcal{N}_{\Upsilon}(\omega)=a(\omega)a_{R}(\omega)+b(\omega)b_{R}(\omega)+a_{R}(\omega)$.
By Hypothesis 2,

$\forall\ \omega\in\mathbb{R}^{+}:\ \cos(\theta_{L}-\theta_{C_{R}})\geq
0\Rightarrow\dfrac{a(\omega)a_{R}(\omega)+b(\omega)b_{R}(\omega)}{|L(j\omega)C_{R}(j\omega)|}\geq
0,$

(31)

and since $a_{R}(\omega)>0$ in the cases of PCI and GFORE,
$\mathcal{N}_{\Upsilon}(\omega)>0$, for all $\omega\in\mathbb{R}^{+}$.
Therefore, the reset control system (8) is of Type I and/or Type II, hence the
claim. In [30] the GFORE, CI and PCI architectures have been modified to
improve the performance of reset control systems. Using the same procedure as
Theorem 1 a frequency-domain method to assess stability properties of these
reset control systems illustrated in Fig. 4 is proposed.

$-$$r$$y$$e_{r}$Liner ControllerPlant$+$$d$$\mathbf{C_{\mathbf{R}}}$Reset
Controller$\mathbf{C_{\mathbf{L}_{2}}}$$u_{r}$$\mathbf{G}$$\mathcal{C}_{s}$$\mathbf{C_{\mathbf{L}_{1}}}$$e$Liner
Controller

Figure 4: The closed-loop architecture of a modified reset element

######  Corollary 3.2.

Let the NSV vector for the reset control system shown in Fig. 4 be

$\displaystyle\vv{\mathcal{N}}_{MF}(\omega)$ $\displaystyle{=}$
$\displaystyle[\mathcal{N}_{MF_{\chi}}\quad\mathcal{N}_{MF_{\Upsilon}}]^{T}$
(32) $\displaystyle{=}$
$\displaystyle\left[\Re(\dfrac{L^{\prime}(j\omega)\kappa(j\omega)}{\mathcal{C}_{s}(j\omega)})\quad\Re(\kappa(j\omega)C_{R}(j\omega))\right]^{T},$

in which
$L^{\prime}(s)=C_{L_{1}}(s)C_{R}(s)C_{L_{2}}(s)\mathcal{C}_{s}(s)G(s)$. Then,
the zero equilibrium of the reset control system (8) in the configuration of
Fig. 4 with GFORE (2), or CI (3), or PCI (4) is globally uniformly
asymptotically stable when $w=0$, and the system has the UBIBS property for
any input $w$ which is a Bohl function if all of the following conditions are
satisfied.

  * •

The base linear system is stable and the open-loop transfer function does not
have any pole-zero cancellation.

  * •

In the case of CI (3), $C_{L_{1}}(s)C_{L_{2}}(s)G(s)$ does not have any pole
at the origin and $n-m=2$.

  * •

The reset control system (8) is either of Type I and/or of Type II.

  * •

$A_{\rho}=\gamma,\ -1<\gamma<1.$

  * •

$\mathcal{C}_{s}(s)=1$ and/or the reset instants have the well-posedness
property.

See Appendix B.

##  4 Stability analysis of reset control systems with second order reset
elements

###  4.1 Reset control systems with GSORE

In this section a frequency-domain method for assessing stability properties
of the reset control system (8) with GSORE (5), which has the canonical
controllable form state-space realization (6), is proposed. In this method the
$H_{\beta}$ condition is combined with optimization tools to provide
sufficient conditions to guarantee stability properties of the reset control
system (8). Before presenting the main result, one preliminary fact, which is
useful for assessing stability properties of the reset control system (8) with
GSORE (5), is presented.

######  Proposition 2.

Let $\vv{\mathcal{Q}}\in\mathbb{R}^{2}$ and
$\vv{\mathcal{F}}\in\mathbb{R}^{2}$ be defined as
$\vv{\mathcal{Q}}=\begin{bmatrix}Q_{1}&Q_{2}\end{bmatrix}^{T}$ and
$\vv{\mathcal{F}}(\omega)=\begin{bmatrix}\mathcal{F}_{1}(\omega)&\mathcal{F}_{2}(\omega)\end{bmatrix}^{T}$.
Let
$\phase{\vv{\mathcal{Q}},\vv{\mathcal{F}}(\omega)}=\vartheta(\omega,\dfrac{Q_{2}}{Q_{1}})$,
$\omega_{p}=\\{\omega\in\mathbb{R}^{+}|\ \mathcal{F}_{3}(\omega)\geq 0\\}$,
$\omega_{N}=\mathbb{R}^{+}-\omega_{p}$,

$g_{p}=\left\\{\dfrac{Q_{2}}{Q_{1}}\in\mathbb{R}|\
\forall\omega\in\omega_{p}:\
Q_{1}\mathcal{F}_{1}(\omega)+Q_{2}\mathcal{F}_{2}(\omega)>0\right\\},$

and

$g_{N}=\left\\{\dfrac{Q_{2}}{Q_{1}}\in\mathbb{R}|\
\forall\omega\in\omega_{N}:\
Q_{1}\mathcal{F}_{1}(\omega)+Q_{2}\mathcal{F}_{2}(\omega)>0\right\\}.$

Then the condition

$Q_{1}\mathcal{F}_{1}(\omega)+Q_{2}\mathcal{F}_{2}(\omega)>\mathcal{F}_{3}(\omega),$
(33)

holds for all $\omega\in\mathbb{R}$ if and only if

  * •

$\eta_{1}(\dfrac{Q_{2}}{Q_{1}})<\sqrt{Q_{1}^{2}+Q_{2}^{2}}<\eta_{2}(\dfrac{Q_{2}}{Q_{1}})$,

  * •

$\dfrac{Q_{2}}{Q_{1}}\in\left\\{\dfrac{Q_{2}}{Q_{1}}\in g_{p}|\
\eta_{1}(\dfrac{Q_{2}}{Q_{1}})<\eta_{2}(\dfrac{Q_{2}}{Q_{1}})\right\\}$,

where

$\displaystyle\eta_{1}(\dfrac{Q_{2}}{Q_{1}})$
$\displaystyle=\begin{cases}-\infty&\omega_{p}=\varnothing,\\\
\underset{\omega\in\omega_{p}}{\max\
}\dfrac{\mathcal{F}_{3}(\omega)}{\cos(\vartheta)\sqrt{\mathcal{F}^{2}_{1}(\omega)+\mathcal{F}^{2}_{2}(\omega)}},&\omega_{p}\neq\varnothing,\end{cases}$
$\displaystyle\eta_{2}(\dfrac{Q_{2}}{Q_{1}})$
$\displaystyle=\begin{cases}+\infty&\dfrac{Q_{2}}{Q_{1}}\in
g_{N},\lor\omega_{N}=\varnothing,\\\ \underset{\omega\in\omega_{N}}{\min\
}\dfrac{\mathcal{F}_{3}(\omega)}{\cos(\vartheta)\sqrt{\mathcal{F}^{2}_{1}(\omega)+\mathcal{F}^{2}_{2}(\omega)}},&\dfrac{Q_{2}}{Q_{1}}\notin
g_{N}.\end{cases}$ (34)

See Appendix C.

######  Remark 4.1.

The sets $g_{p}$ and $g_{N}$ can be easily obtained using the method described
in [1].

Define now
$\Gamma(\gamma_{1},\gamma_{2})=\dfrac{(\gamma_{1}\gamma_{2}-1)^{2}}{(\gamma_{1}^{2}-1)(\gamma_{2}^{2}-1)}$

$\displaystyle f_{1}(\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{X}_{3},\omega)$
$\displaystyle{=}$
$\displaystyle\mathcal{X}_{1}\Re(C_{R}(j\omega)\kappa(\omega)j\omega)$
$\displaystyle{+}\>\mathcal{X}_{2}\Re(C_{R}(j\omega)\kappa(\omega))$
$\displaystyle{+}\>\mathcal{X}_{3}\Re(\mathcal{C}_{s}(j\omega)(a^{2}+b^{2}+a)),$
$\displaystyle f_{2}(\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{X}_{3},\omega)$
$\displaystyle{=}$
$\displaystyle\mathcal{X}_{1}\Re(C_{R}(j\omega)\kappa(\omega)(j\omega+2\xi\omega_{r}))$
$\displaystyle{+}\>\mathcal{X}_{3}\Re(L(j\omega)\kappa(\omega)\mathcal{C}_{s}(j\omega)(j\omega+2\xi\omega_{r}))$
$\displaystyle{+}\>\mathcal{X}_{2}\Re(C_{R}(j\omega)\kappa(\omega)(2j\xi\omega_{r}\omega-\omega^{2})$
$\displaystyle{-}\>(a+1)^{2}-b^{2}),$ $\displaystyle
G_{1}(Q_{1},Q_{2},Q_{3},Q_{4})$ $\displaystyle{=}$
$\displaystyle\displaystyle\sup_{\omega\in(0,\infty)}\Bigg{[}\dfrac{f_{1}(Q_{1},Q_{2},1,\omega)}{f_{1}(Q_{2},\dfrac{Q_{2}Q_{3}}{Q_{4}},\dfrac{Q_{2}}{Q_{4}},\omega)+f_{2}(Q_{2},Q_{1},1,\omega)}$
$\displaystyle{\times}\dfrac{f_{2}(Q_{3},Q_{4},1,\omega)}{f_{1}(Q_{4},Q_{3},1,\omega)+f_{2}(Q_{4},\dfrac{Q_{1}Q_{4}}{Q_{2}},\dfrac{Q_{4}}{Q_{2}},\omega)}\Bigg{]},$
$\displaystyle G_{2}(Q_{1},Q_{2},Q_{3},Q_{4})$ $\displaystyle{=}$
$\displaystyle\displaystyle\sup_{\omega\in[0,\infty)}\Bigg{[}\dfrac{f_{1}(1,Q_{2},Q_{1},\omega)}{f_{1}(Q_{2},\dfrac{Q_{2}}{Q_{4}},\dfrac{Q_{2}Q_{3}}{Q_{4}},\omega)+f_{2}(Q_{2},1,Q_{1},\omega)}$
$\displaystyle{\times}\dfrac{f_{2}(1,Q_{4},Q_{3},\omega)}{f_{1}(Q_{4},1,Q_{3},\omega)+f_{2}(Q_{4},\dfrac{Q_{4}}{Q_{2}},\dfrac{Q_{1}Q_{4}}{Q_{2}},\omega)}\Bigg{]}.$

We define systems of Type III, of Type IV, and of Type V to assess stability
properties of the reset control system (8) with GSORE (6).

######  Definition 4.2.

The reset control system (8) with GSORE (6) is of Type III if the following
conditions hold.

  1. (1)

$M<4$, where $M=\underset{Q_{1},Q_{2},Q_{3},Q_{4}}{\min}\
G_{1}(Q_{1},Q_{2},Q_{3},Q_{4})$, in which $Q_{1}$, $Q_{2}$, $Q_{3}$, and
$Q_{4}$ are such that the following constraints hold:

$\displaystyle\mathcal{S}_{1}:$ $\displaystyle\forall\omega\in(0,\infty):\
K_{s_{0}}f_{1}(Q_{1},Q_{2},1,\omega)>0,$ $\displaystyle\mathcal{S}_{2}:$
$\displaystyle\forall\omega\in(0,\infty):\
K_{s_{0}}f_{2}(Q_{3},Q_{4},1,\omega)>0,$ $\displaystyle\mathcal{S}_{3}:$
$\displaystyle
K_{s_{0}}\left(\dfrac{2\xi\omega_{r}}{Q_{1}}+\dfrac{Q_{2}}{Q_{1}Q_{4}}+\dfrac{2}{Q_{1}}\sqrt{\dfrac{2Q_{2}\xi\omega_{r}}{Q_{4}}-\dfrac{Q_{2}}{K_{s_{0}}}}\right)>1,$
$\displaystyle\mathcal{S}_{4}:$ $\displaystyle
K_{s_{0}}\left(\dfrac{2\xi\omega_{r}}{Q_{1}}+\dfrac{Q_{2}}{Q_{1}Q_{4}}-\dfrac{2}{Q_{1}}\sqrt{\dfrac{2Q_{2}\xi\omega_{r}}{Q_{4}}-\dfrac{Q_{2}}{K_{s_{0}}}}\right)<1,$
$\displaystyle\mathcal{S}_{5}:$
$\displaystyle\dfrac{\omega_{r}^{2}Q_{1}}{Q_{2}}+2\omega_{r}\left(\xi+2\sqrt{\dfrac{2Q_{1}\xi\omega_{r}}{Q_{2}}-1}\right)>\dfrac{Q_{3}}{Q_{4}},$
$\displaystyle\mathcal{S}_{6}:$
$\displaystyle\dfrac{\omega_{r}^{2}Q_{1}}{Q_{2}}+2\omega_{r}\left(\xi-2\sqrt{\dfrac{2Q_{1}\xi\omega_{r}}{Q_{2}}-1}\right)<\dfrac{Q_{3}}{Q_{4}},$
(36) $\displaystyle\mathcal{S}_{7}:$ $\displaystyle K_{s_{0}}Q_{i}>0,\
2\xi\omega_{r}>\dfrac{Q_{4}}{K_{s_{0}}},\
2\xi\omega_{r}>\dfrac{Q_{2}}{Q_{1}},$ $\displaystyle\mathcal{S}_{8}:$
$\displaystyle\dfrac{Q_{1}Q_{3}}{Q_{2}Q_{4}}>\Gamma(\gamma_{1},\gamma_{2}).$

  2. (2)

The pairs $(\bar{A},C_{0})$ and $(\bar{A},B_{0})$ where
$B_{0}=\begin{bmatrix}0_{n_{p}\times 2}\\\ I_{2}\end{bmatrix}$ and
$C_{0}=\left[\begin{bmatrix}1\\\
\dfrac{Q_{2}}{Q_{4}}\end{bmatrix}\bar{C}_{e}\quad\begin{bmatrix}Q_{1}&Q_{2}\\\
Q_{2}&\dfrac{Q_{2}Q_{3}}{Q_{4}}\end{bmatrix}\right]$ are controllable and
observable, respectively.

  3. (3)

The open-loop system has at least one pole at the origin and $K_{s_{0}}\neq
0$.

######  Definition 4.3.

The reset control system (8) with GSORE (6) is of Type IV if the following
conditions hold.

  1. (1)

$M<4$, where $M=\underset{Q_{1},Q_{2},Q_{3},Q_{4}}{\min}\
G_{2}(Q_{1},Q_{2},Q_{3},Q_{4})$, in which $Q_{1}$, $Q_{2}$, $Q_{3}$, and
$Q_{4}$ are such that the following constraints hold:

$\displaystyle\mathcal{S}_{1}:$ $\displaystyle\forall\omega\in[0,\infty):\
f_{1}(1,Q_{2},Q_{1},\omega)>0,$ (37) $\displaystyle\mathcal{S}_{2}:$
$\displaystyle\forall\omega\in[0,\infty):\ f_{2}(1,Q_{4},Q_{3},\omega)>0,$
$\displaystyle\mathcal{S}_{3}:$
$\displaystyle\omega_{r}^{2}+2\omega_{r}\left(\xi
Q_{2}+2\sqrt{2Q_{2}\xi\omega_{r}-Q_{2}^{2}}\right)>\dfrac{Q_{2}}{Q_{4}},$
$\displaystyle\mathcal{S}_{4}:$
$\displaystyle\omega_{r}^{2}+2\omega_{r}\left(\xi
Q_{2}-2\sqrt{2Q_{2}\xi\omega_{r}-Q_{2}^{2}}\right)<\dfrac{Q_{2}}{Q_{4}},$
$\displaystyle\mathcal{S}_{5}:$ $\displaystyle Q_{4}>0,\
0<Q_{2}<2\xi\omega_{r},\ Q_{2}Q_{4}<\dfrac{1}{\Gamma(\gamma_{1},\gamma_{2})}.$

  2. (2)

The pairs $(\bar{A},C_{0})$ and $(\bar{A},B_{0})$ where
$B_{0}=\begin{bmatrix}0_{n_{p}\times 2}\\\ I_{2}\end{bmatrix}$ and
$C_{0}=\left[\begin{bmatrix}Q_{1}\\\
\dfrac{Q_{2}Q_{3}}{Q_{4}}\end{bmatrix}\bar{C}_{e}\quad\begin{bmatrix}1&Q_{2}\\\
Q_{2}&\dfrac{Q_{2}}{Q_{4}}\end{bmatrix}\right]$ are controllable and
observable, respectively.

  3. (3)

The open-loop system does not have any pole at the origin.

  4. (4)

$n-m>3$.

######  Definition 4.4.

The reset control system (8) with GSORE (6) is of Type V if the following
conditions hold.

  1. (1)

$M<4$, where $M=\underset{Q_{1},Q_{2},Q_{3},Q_{4}}{\min}\
G_{2}(Q_{1},Q_{2},Q_{3},Q_{4})$, in which $Q_{1}$, $Q_{2}$, $Q_{3}$, and
$Q_{4}$ are such that the following constraints hold:

$\displaystyle\mathcal{S}_{1}:$ $\displaystyle\forall\omega\in[0,\infty):\
f_{1}(1,Q_{2},Q_{1},\omega)>0,$ (38) $\displaystyle\mathcal{S}_{2}:$
$\displaystyle\forall\omega\in[0,\infty):\ f_{2}(1,Q_{4},Q_{3},\omega)>0,$
$\displaystyle\mathcal{S}_{3}:$
$\displaystyle\leavevmode\resizebox{184.28981pt}{}{$\omega_{r}^{2}-K_{n}Q_{1}+2\xi\omega_{r}Q_{2}+2\sqrt{2\xi\omega_{r}^{3}Q_{2}+\dfrac{Q_{2}^{2}Q_{3}K_{n}}{Q_{4}}-\omega_{r}^{2}Q_{2}^{2}}>\dfrac{Q_{2}}{Q_{4}},$}$
$\displaystyle\mathcal{S}_{4}:$
$\displaystyle\leavevmode\resizebox{184.28981pt}{}{$\omega_{r}^{2}-K_{n}Q_{1}+2\xi\omega_{r}Q_{2}-2\sqrt{2\xi\omega_{r}^{3}Q_{2}+\dfrac{Q_{2}^{2}Q_{3}K_{n}}{Q_{4}}-\omega_{r}^{2}Q_{2}^{2}}<\dfrac{Q_{2}}{Q_{4}},$}$
$\displaystyle\mathcal{S}_{5}:$ $\displaystyle
2\xi\omega_{r}^{3}Q_{2}+\dfrac{Q_{2}^{2}Q_{3}K_{n}}{Q_{4}}>\omega_{r}^{2}Q_{2}^{2},$
$\displaystyle\mathcal{S}_{6}:$ $\displaystyle Q_{2}<2\xi\omega_{r},\
K_{n}Q_{3}<\omega_{r}^{2}Q_{4},\
0<Q_{2}Q_{4}<\dfrac{1}{\Gamma(\gamma_{1},\gamma_{2})}.$

  2. (2)

The pairs $(\bar{A},C_{0})$ and $(\bar{A},B_{0})$ where
$B_{0}=\begin{bmatrix}0_{n_{p}\times 2}\\\ I_{2}\end{bmatrix}$ and
$C_{0}=\left[\begin{bmatrix}Q_{1}\\\
\dfrac{Q_{2}Q_{3}}{Q_{4}}\end{bmatrix}\bar{C}_{e}\quad\begin{bmatrix}1&Q_{2}\\\
Q_{2}&\dfrac{Q_{2}}{Q_{4}}\end{bmatrix}\right]$ are observable and
controllable, respectively.

  3. (3)

The open-loop system does not have any pole at the origin.

  4. (4)

$n-m=3$.

######  Theorem 3.

The zero equilibrium of the reset control system (8) with GSORE (6) is
globally uniformly asymptotically stable when $w=0$, and the system has the
UBIBS property for any input $w$ which is a Bohl function if all of the
following conditions are satisfied.

  * •

The base linear system is stable.

  * •

$A_{\rho}=\begin{bmatrix}\gamma_{1}&0\\\ 0&\gamma_{2}\end{bmatrix}$ and
$-1<\gamma_{i}<1$, for $i=1,\ 2$.

  * •

The reset control system is either of Type III, or of Type IV, or of Type V.

  * •

$\mathcal{C}_{s}(s)=1$ and/or the reset instants have the well-posedness
property.

Theorem 3 is proved in the following steps.

  * •

Step 1: The transfer function $H(s)$ in (10) for the reset control system (8)
with GSORE (6) is calculated. Then, it is shown that $A_{\rho}^{T}\Varrho
A_{\rho}-\Varrho<0$.

  * •

Step 2: It is shown that
$\displaystyle\lim_{\omega\to\infty}\omega^{2}(H(j\omega)+H(-j\omega)^{T})>0.$

  * •

Step 3: For systems with poles at the origin it is shown that
$\displaystyle\lim_{\omega\to 0}H(j\omega)+H(-j\omega)^{T}>0$.

  * •

Step 4: It is shown that $H(j\omega)+H(-j\omega)^{T}>0$, for all
$\omega\in\mathbb{R}^{+}$.

Step 1: In the case of GSORE let
$\beta=-\begin{bmatrix}\beta_{1}&\beta_{2}\end{bmatrix}$ and
$\Varrho=\begin{bmatrix}\Varrho_{1}&\Varrho_{2}\\\
\Varrho_{2}&\Varrho_{3}\end{bmatrix}>0$ be such that

$\beta_{i}\in\mathbb{R},\ \Varrho_{3}>0,\ \Varrho_{1}>0,\
\Varrho_{1}\Varrho_{3}>\Varrho_{2}^{2}.$ (39)

In addition, since $A_{\rho}=\begin{bmatrix}\gamma_{1}&0\\\
0&\gamma_{2}\end{bmatrix}$, we have the condition

$A_{\rho}^{T}\Varrho
A_{\rho}-\Varrho=\begin{bmatrix}(\gamma_{1}^{2}-1)\Varrho_{1}&(\gamma_{1}\gamma_{2}-1)\Varrho_{2}\\\
(\gamma_{1}\gamma_{2}-1)\Varrho_{2}&(\gamma_{2}^{2}-1)\Varrho_{3}\end{bmatrix}<0.$
(40)

Since $-1<\gamma_{i}<1$, using (39) and (40), yields

$\dfrac{\Varrho_{1}\Varrho_{3}}{\Varrho_{2}^{2}}>\Gamma(\gamma_{1},\gamma_{2})=\dfrac{(\gamma_{1}\gamma_{2}-1)^{2}}{(\gamma_{1}^{2}-1)(\gamma_{2}^{2}-1)}\geq
1.$ (41)

With the considered matrix $\Varrho$ and vector $\beta$, $H(s)$ in (10) with
$C_{0}$ as in (15) is equal to (see also Fig. 5)

$H(s)=\begin{bmatrix}H_{11}(s)&H_{12}(s)\\\ H_{21}(s)&H_{22}(s)\end{bmatrix}.$
(42)

$-$$r_{1}$$y_{1}$$e$$-$$\dfrac{1}{s}$$2\omega_{r\alpha}$$\dfrac{1}{s}$$\gamma_{1}$$\omega^{2}_{r\alpha}$$+$$C_{L_{2}}$$G$$+$$C_{L_{1}}$$\mathcal{C}_{s}$$\Varrho_{1}$$\beta_{1}$$+$$r_{2}$$+$$\gamma_{2}$$\Varrho_{2}$

$-$$r_{1}$$y_{2}$$e$$-$$\dfrac{1}{s}$$2\omega_{r\alpha}$$\dfrac{1}{s}$$\gamma_{1}$$\omega^{2}_{r\alpha}$$+$$C_{L_{2}}$$G$$+$$C_{L_{1}}$$\mathcal{C}_{s}$$\Varrho_{2}$$\beta_{2}$$+$$r_{2}$$+$$\gamma_{2}$$\Varrho_{3}$

Figure 5: The block diagram of the $H_{\beta}$ condition for the closed-loop
architecture Fig. 1 with GSORE

in which $H_{ij}(s)$ with $i,j=1,2$ is transfer function from $r_{j}$ to
$y_{i}$. Thus, $H(j\omega)+H(-j\omega)^{T}$ is equal to

$\begin{array}[]{*{35}{c}}\begin{bmatrix}2\Re(H_{11}(j\omega))&\Re(H_{12}(j\omega)+H_{21}(j\omega))\\\
\Re(H_{12}(j\omega)+H_{21}(j\omega))&2\Re(H_{22}(j\omega))\end{bmatrix}>0\Rightarrow\end{array}$
(43)
$\dfrac{1}{|\kappa(\omega)|^{2}}\left[\begin{matrix}2f_{1}(\Varrho_{1},\Varrho_{2},\beta_{1},\omega)\\\
f_{1}(\Varrho_{2},\Varrho_{1},\beta_{2},\omega)+f_{2}(\Varrho_{2},\Varrho_{1},\beta_{2},\omega)\end{matrix}\right.\\\
\left.\begin{matrix}f_{1}(\Varrho_{2},\Varrho_{1},\beta_{2},\omega)+f_{2}(\Varrho_{2},\Varrho_{1},\beta_{2},\omega)\\\
2f_{2}(\Varrho_{3},\Varrho_{2},\beta_{2},\omega)\end{matrix}\right]>0.$ (44)

Step 2: Since the transfer functions $\dfrac{y_{i}}{r_{j}}$, with $i,j=1,2$,
are strictly proper, $\displaystyle\lim_{s\to\infty}H(s)=0$. Therefore, it is
necessary to have
$\displaystyle\lim_{\omega\to\infty}\omega^{2}(H(j\omega)+H(-j\omega)^{T})>0.$
Note that in the case of SORE, $n-m\geq 3$. By (44), if $n-m>3$,
$\displaystyle\lim_{\omega\to\infty}\omega^{2}(H(j\omega)+H(-j\omega)^{T})$ is
equal to

$\begin{bmatrix}4\Varrho_{1}\xi\omega_{r}-2\Varrho_{2}&\omega_{r}^{2}\Varrho_{1}+2\Varrho_{2}\xi\omega_{r}-\Varrho_{3}\\\
\omega_{r}^{2}\Varrho_{1}+2\Varrho_{2}\xi\omega_{r}-\Varrho_{3}&2\omega_{r}^{2}\Varrho_{2}\end{bmatrix}.$
(45)

Therefore, the condition
$\displaystyle\lim_{\omega\to\infty}\omega^{2}(H(j\omega)+H(-j\omega)^{T})>0$
is equivalent to

$2\Varrho_{1}\xi\omega_{r}>\Varrho_{2},\quad\Varrho_{2}>0,$ (46)

and

$\begin{array}[]{*{35}{c}}4(2\Varrho_{1}\xi\omega_{r}-\Varrho_{2})(\omega_{r}^{2}\Varrho_{2})>(\omega_{r}^{2}\Varrho_{1}+2\Varrho_{2}\xi\omega_{r}-\Varrho_{3})^{2}\\\
\Downarrow\\\
\Varrho_{3}^{2}-2\Varrho_{3}(\omega_{r}^{2}\Varrho_{1}+2\xi\omega_{r}\Varrho_{2})+(\omega_{r}^{2}\Varrho_{1}-2\xi\omega_{r}\Varrho_{2})^{2}+4\omega_{r}^{2}\Varrho_{2}^{2}<0\\\
\Downarrow\\\
\left(\Varrho_{3}>(\omega_{r}^{2}\Varrho_{1}+2\xi\omega_{r}\Varrho_{2})-2\omega_{r}\sqrt{2\xi\omega_{r}\Varrho_{1}\Varrho_{2}-\Varrho_{2}^{2}}\right)\
\land\\\
\left(\Varrho_{3}<(\omega_{r}^{2}\Varrho_{1}+2\xi\omega_{r}\Varrho_{2})+2\omega_{r}\sqrt{2\xi\omega_{r}\Varrho_{1}\Varrho_{2}-\Varrho_{2}^{2}}\right).\end{array}$

(47)

When $n-m=3$, condition (45) is re-written as

$\begin{bmatrix}4\Varrho_{1}\xi\omega_{r}-2\Varrho_{2}&\omega_{r}^{2}\Varrho_{1}+2\Varrho_{2}\xi\omega_{r}-\Varrho_{3}-K_{n}\beta_{1}\\\
\omega_{r}^{2}\Varrho_{1}+2\Varrho_{2}\xi\omega_{r}-\Varrho_{3}-K_{n}\beta_{1}&2\omega_{r}^{2}\Varrho_{2}-2K_{n}\beta_{2}\end{bmatrix}>0,$

(48)

which is equivalent to

$2\Varrho_{1}\xi\omega_{r}>\Varrho_{2},\quad\omega_{r}^{2}\Varrho_{2}>K_{n}\beta_{2},$
(49)

and

$\begin{array}[]{*{35}{c}}4(2\Varrho_{1}\xi\omega_{r}-\Varrho_{2})(\omega_{r}^{2}\Varrho_{2}-K_{n}\beta_{2})>(\omega_{r}^{2}\Varrho_{1}+2\Varrho_{2}\xi\omega_{r}-\Varrho_{3}-K_{n}\beta_{1})^{2}\\\
\Downarrow\\\
\left(\Varrho_{3}>(\omega_{r}^{2}\Varrho_{1}-K_{n}\beta_{1}+2\xi\omega_{r}\Varrho_{2})-2\sqrt{2\xi\omega_{r}^{3}\Varrho_{1}\Varrho_{2}+K_{n}\Varrho_{2}\beta_{2}-\omega_{r}^{2}\Varrho_{2}^{2}}\right)\
\land\\\
\left(\Varrho_{3}<(\omega_{r}^{2}\Varrho_{1}-K_{n}\beta_{1}+2\xi\omega_{r}\Varrho_{2})+2\sqrt{2\xi\omega_{r}^{3}\Varrho_{1}\Varrho_{2}+K_{n}\Varrho_{2}\beta_{2}-\omega_{r}^{2}\Varrho_{2}^{2}}\right)\
\land\\\
\left(2\xi\omega_{r}^{3}\Varrho_{1}\Varrho_{2}+K_{n}\Varrho_{2}\beta_{2}>\omega_{r}^{2}\Varrho_{2}^{2}\right).\end{array}$

(50)

Step 3: When $L(s)$ has at least one pole at the origin, by (44)
$displaystyle\lim_{\omega\to 0}H(j\omega)+H(-j\omega)^{T}$ is equal to

$\begin{bmatrix}2K_{s_{0}}\beta_{1}&K_{s_{0}}\beta_{2}+2K_{s_{0}}\beta_{1}\xi\omega_{r}-\Varrho_{1}\\\
K_{s_{0}}\beta_{2}+2K_{s_{0}}\beta_{1}\xi\omega_{r}-\Varrho_{1}&4K_{s_{0}}\beta_{2}\xi\omega_{r}-2\Varrho_{2}\end{bmatrix}>0,$
(51)

which is equivalent to

$K_{s_{0}}\beta_{1}>0,\quad 2K_{s_{0}}\beta_{2}\xi\omega_{r}>\Varrho_{2},$
(52)

and

$\begin{array}[]{*{35}{c}}4(K_{s_{0}}\beta_{1})(2K_{s_{0}}\beta_{2}\xi\omega_{r}-\Varrho_{2})>(K_{s_{0}}\beta_{2}+2K_{s_{0}}\beta_{1}\xi\omega_{r}-\Varrho_{1})^{2}\\\
\Downarrow\\\
\left(\Varrho_{1}>K_{s_{0}}(2\beta_{1}\xi\omega_{r}+\beta_{2})-2\sqrt{2K_{s_{0}}^{2}\xi\omega_{r}\beta_{1}\beta_{2}-K_{s_{0}}\beta_{1}\Varrho_{2}}\right)\
\land\\\
\left(\Varrho_{1}<K_{s_{0}}(2\beta_{1}\xi\omega_{r}+\beta_{2})+2\sqrt{2K_{s_{0}}^{2}\xi\omega_{r}\beta_{1}\beta_{2}-K_{s_{0}}\beta_{1}\Varrho_{2}}\right).\end{array}$

(53)

Step 4: In the case in which $L(s)$ has poles at the origin, denote
$Q_{1}=\dfrac{\Varrho_{1}}{\beta_{1}}$,
$Q_{2}=\dfrac{\Varrho_{2}}{\beta_{1}}$, $Q_{3}=\dfrac{\Varrho_{3}}{\beta_{2}}$
and $Q_{4}=\dfrac{\Varrho_{2}}{\beta_{2}}$. Furthermore, since
$K_{s_{0}}\beta_{1}$, $K_{s_{0}}\beta_{2}$, and $|\kappa(\omega)|^{2}$ are
positive, condition (44) is equal to

$\left[\begin{matrix}2K_{s_{0}}f_{1}(Q_{1},Q_{2},1,\omega)\\\
f_{1}(Q_{4},Q_{3},1,\omega)+f_{2}(Q_{4},\dfrac{Q_{1}Q_{4}}{Q_{2}},\dfrac{Q_{4}}{Q_{2}},\omega)\end{matrix}\right.\\\
\left.\begin{matrix}f_{1}(Q_{2},\dfrac{Q_{2}Q_{3}}{Q_{4}},\dfrac{Q_{2}}{Q_{4}},\omega)+f_{2}(Q_{2},Q_{1},1,\omega)\\\
\dfrac{2}{K_{s_{0}}}f_{2}(Q_{3},Q_{4},1,\omega)\end{matrix}\right]>0.$ (54)

Therefore, for all $\omega\in(0,\infty)$, there exist $Q_{1},\ Q_{2},\ Q_{3}$,
and $Q_{4}$ such that

$K_{s_{0}}f_{1}(Q_{1},Q_{2},1,\omega)>0,\
K_{s_{0}}f_{2}(Q_{3},Q_{4},1,\omega)>0,$ (55)

and since $f_{1}(Q_{1},Q_{2},1,\omega)f_{2}(Q_{3},Q_{4},1,\omega)>0$,

$\displaystyle 4$ $\displaystyle{>}$
$\displaystyle\dfrac{f_{1}(Q_{2},\dfrac{Q_{2}Q_{3}}{Q_{4}},\dfrac{Q_{2}}{Q_{4}},\omega)+f_{2}(Q_{2},Q_{1},1,\omega)}{f_{1}(Q_{1},Q_{2},1,\omega)}$
(56)
$\displaystyle{\times}\dfrac{f_{1}(Q_{4},Q_{3},1,\omega)+f_{2}(Q_{4},\dfrac{Q_{1}Q_{4}}{Q_{2}},\dfrac{Q_{4}}{Q_{2}},\omega)}{f_{2}(Q_{3},Q_{4},1,\omega)}.$

Thus, since the condition (56) must hold for all $\omega\in(0,\infty)$,
$\underset{Q_{i}}{\min}\ G_{1}(Q_{1},Q_{2},Q_{3},Q_{4})<4$, with $i=1,2,3,4$.
Moreover, re-writing equations (39), (41) (47), and (53) using the variables
$Q_{1},\ Q_{2},\ Q_{3}$, and $Q_{4}$, the constraints
$\mathcal{S}_{3}-\mathcal{S}_{8}$ of Definition 4.2 are obtained.
When $L(s)$ does not have any pole at the origin, let
$Q^{\prime}_{1}=\dfrac{\beta_{1}}{\Varrho_{1}}$,
$Q^{\prime}_{2}=\dfrac{\Varrho_{2}}{\Varrho_{1}}$,
$Q^{\prime}_{3}=\dfrac{\beta_{2}}{\Varrho_{3}}$ and
$Q^{\prime}_{4}=\dfrac{\Varrho_{2}}{\Varrho_{3}}$. With this change of
variables, since $\Varrho_{3}$, $\Varrho_{1}$ and $|\kappa(\omega)|^{2}$ are
positive, condition (44) is equivalent to

$\left[\begin{matrix}2f_{1}(1,Q^{\prime}_{2},Q^{\prime}_{1},\omega)\\\
f_{1}(Q^{\prime}_{4},1,Q^{\prime}_{3},\omega)+f_{2}(Q^{\prime}_{4},\dfrac{Q^{\prime}_{4}}{Q^{\prime}_{2}},\dfrac{Q^{\prime}_{1}Q^{\prime}_{4}}{Q^{\prime}_{2}},\omega)\end{matrix}\right.\\\
\left.\begin{matrix}f_{1}(Q^{\prime}_{2},\dfrac{Q^{\prime}_{2}}{Q^{\prime}_{4}},\dfrac{Q^{\prime}_{2}Q^{\prime}_{3}}{Q^{\prime}_{4}},\omega)+f_{2}(Q^{\prime}_{2},1,Q^{\prime}_{1},\omega)\\\
2f_{2}(1,Q^{\prime}_{4},Q^{\prime}_{3},\omega)\end{matrix}\right]>0.$ (58)

This implies that $f_{1}(1,Q^{\prime}_{2},Q^{\prime}_{1},\omega)>0,\
f_{2}(1,Q^{\prime}_{4},Q^{\prime}_{3},\omega)>0,$ and since
$f_{1}(1,Q^{\prime}_{2},Q^{\prime}_{1},\omega)f_{2}(1,Q^{\prime}_{4},Q^{\prime}_{3},\omega)>0$,

$\displaystyle 4$ $\displaystyle{>}$
$\displaystyle\dfrac{f_{1}(Q^{\prime}_{2},\dfrac{Q^{\prime}_{2}}{Q^{\prime}_{4}},\dfrac{Q^{\prime}_{2}Q^{\prime}_{3}}{Q^{\prime}_{4}},\omega)+f_{2}(Q^{\prime}_{2},1,Q^{\prime}_{1},\omega)}{f_{1}(1,Q^{\prime}_{2},Q^{\prime}_{1},\omega)}$
(59)
$\displaystyle{\times}\dfrac{f_{1}(Q^{\prime}_{4},1,Q^{\prime}_{3},\omega)+f_{2}(Q^{\prime}_{4},\dfrac{Q^{\prime}_{4}}{Q^{\prime}_{2}},\dfrac{Q^{\prime}_{1}Q^{\prime}_{4}}{Q^{\prime}_{2}},\omega)}{f_{2}(1,Q^{\prime}_{4},Q^{\prime}_{3},\omega)}.$

Therefore, since condition (59) must hold for all $\omega\in[0,\infty)$,
$\underset{Q^{\prime}_{i}}{\min}\
G_{2}(Q^{\prime}_{1},Q^{\prime}_{2},Q^{\prime}_{3},Q^{\prime}_{4})<4$, with
$i=1,2,3,4$. Re-writing equations (41) and (47) with the variables
$Q^{\prime}_{1}$, $Q^{\prime}_{2}$, $Q^{\prime}_{3}$, and $Q^{\prime}_{4}$,
the constraints $\mathcal{S}_{3}-\mathcal{S}_{5}$ of Definition 4.3 are
achieved. Similarly, using these variables in equations (41) and (50), the
constraints $\mathcal{S}_{3}-\mathcal{S}_{6}$ of Definition 4.4 are obtained.
By Steps 1-4, $A_{\rho}^{T}\Varrho A_{\rho}-\Varrho<0$, $H(s)$ is SPR [32],
$(\bar{A},C_{0})$ is observable and $(\bar{A},B_{0})$ is controllable, and the
base linear system is stable. Thus, the $H_{\beta}$ condition is satisfied for
the reset control system (8) with GSORE (6). Hence, the zero equilibrium of
the system is globally uniformly asymptotically stable when $w=0$ and
according to Lemma 1, it has the UBIBS property for any initial condition
$x_{0}$ and any input $w$ which is a Bohl function.

######  Remark 4.5.

The minimum value of the function $\Gamma(\gamma_{1},\gamma_{2})$ occurs when
$\gamma_{1}=\gamma_{2}$. In other words, if
$\gamma_{1}=\gamma_{2}\Rightarrow\Gamma(\gamma_{1},\gamma_{2})=\underset{\gamma_{1},\
\gamma_{2}}{\min\ }\Gamma(\gamma_{1},\gamma_{2})=1.$ Thus, if Theorem 3 holds
for a pair of ($\gamma_{1}$, $\gamma_{2}$), it also holds for $A_{\rho}=\gamma
I,\ -1<\gamma<1$.

Note that unlike linear controllers, the GSORE (5) with a different state-
space realization yields different performance, and Theorem 3 can not be used
for such realizations. For example, the GSORE (5) can also be realized in
observable canonical form, that is with

$A_{r}=\begin{bmatrix}0&-\omega_{r}^{2}\\\ 1&-2\xi\omega_{r}\end{bmatrix},\
B_{r}=\begin{bmatrix}1\\\ 0\end{bmatrix},\
C_{r}=\begin{bmatrix}0&1\end{bmatrix},\ D_{r}=0,$ (61)

or it can be realized with two GFORE yielding the realization (provided
$\xi\geq 1$)

$\begin{array}[]{*{35}{c}}A_{r}=\begin{bmatrix}-\omega_{r_{1}}&0\\\
1&-\omega_{r_{2}}\end{bmatrix},\ B_{r}=\begin{bmatrix}1\\\ 0\end{bmatrix},\
C_{r}=\begin{bmatrix}0&1\end{bmatrix},\ D_{r}=0,\\\
\omega_{r_{1}}+\omega_{r_{2}}=2\xi\omega_{r},\
\omega_{r_{1}}\omega_{r_{2}}=\omega_{r}^{2},\end{array}$ (62)

which results in different closed-loop performance.

######  Corollary 4.6.

Suppose hypotheses of Theorem 3 hold for the reset control system (8) with the
GSORE (5) in the controllable canonical form (6) for a pair ($\gamma_{1}$,
$\gamma_{2}$). Then the reset control system (8) with GSORE (5) with
realization (61) or (62) and $A_{\rho}=\gamma I,\ -1<\gamma<1$ has the
following property. For each initial condition $x_{0}$ such that
$x_{0}=\begin{bmatrix}0&0&\zeta_{0}^{T}\end{bmatrix}^{T}$ and each bounded
input $w$ which is a Bohl function, there exists $\epsilon>0$ such that
$||x(t,x_{0},w(t))||<\epsilon$ for $t\geq 0$.

See Appendix D.

###  4.2 Reset control systems with (SOSRE)

In this section stability analysis for the reset control system (8) with the
SOSRE [29] is presented. In [29] GSORE (6) with
$A_{\rho}=\begin{bmatrix}\gamma&0\\\ 0&1\end{bmatrix}$, which is termed SOSRE,
is used to improve the performance of the reset control system (8). In the
case of SOSRE one state of GSORE is reset and the other state is utilized to
reduce the high order harmonics of the reset element.

######  Corollary 4.7.

Consider the reset control system (8) with SOSRE. Define the NSV vector as

$\vv{\mathcal{N}}_{SOS}(\omega)=[\mathcal{N}_{SOS_{\chi}}\quad\mathcal{N}_{SOS_{\Upsilon}}]^{T}=$
$\begin{bmatrix}\Re(L(j\omega)\kappa(j\omega)\mathcal{C}_{s}(j\omega))&-\Im(\omega\kappa(j\omega)C_{R}(j\omega))\end{bmatrix}^{T}.$

Suppose that the reset instants have the well-posedness property and
$-1<\gamma<1$. Then, with this definition of NSV the zero equilibrium of the
reset control system (8) with SOSRE is globally uniformly asymptotically
stable when $w=0$, and the system has the UBIBS property for any input $w$
which is a Bohl function if all of the following conditions are satisfied.

  * •

The base linear system is stable and the open-loop transfer function does not
have any pole-zero cancellation.

  * •

The reset control system (8) is either of Type I and/or of Type II.

Let $\beta^{\prime}=-\beta$. The transfer function (10) with $C_{0}$ as in
(15) can be rewritten as (see also Fig. 5, transfer function from $r_{1}$ to
$y_{1}$ with $\Varrho_{2}=0$)

$H(s)=\dfrac{\beta^{\prime}L(s)\mathcal{C}_{s}(s)+\Varrho sC_{R}(s)}{1+L(s)}.$
(63)

Step 1 and Step 4 of the proof of Theorem 1 are repeated with small
modifications. When the open-loop system has poles at the origin

$\displaystyle\lim_{\omega\to 0}\Re(H(j\omega))=K_{s_{0}}\beta^{\prime}>0.$
(64)

In the case of SOSRE one has $n-m\geq 3$. Consequently,

$\displaystyle\lim_{\omega\to\infty}\omega^{2}\Re(H(j\omega))=2\Varrho\xi\omega_{r}>0,$
(65)

and the proof is complete. Note that it is impossible to satisfy Assumption 2
for this configuration. Thus, the reset instants must have the well-posedness
property.

##  5 Illustrative Examples

In this section two examples showing how the proposed methods can be used to
study stability properties of reset control systems are presented. In
particular, stability properties of a precision positioning system [38] (knows
as a spider stage) controlled by a reset controller are considered. In this
system (see Fig. 6), three actuators are angularly spaced to actuate three
masses (labeled as B1, B2, and B3) which are constrained by parallel flexures
and connected to the central mass D through leaf flexures. Only one of the
actuators (A1) is considered and used for controlling the position of the mass
B1 attached to the same actuator, which results in a SISO system. For using
these stability methods the FRF measurement of the plant (Fig. 7) is needed.

Figure 6: The whole setup including computer, CompactRio, power supply, sensor
power, amplifier, isolator, sensor and, stage Figure 7: FRF measurement of
considered SISO position of the Spyder stage

In [38] a non-linear phase compensator, which is termed “Constant in gain Lead
in phase” (CgLp) (for more detail see [35, 16, 38]), has been used to improve
the performance of this precision positioning stage. CgLp compensators,
consisting of a first/second order lead filter and a GFORE/GSORE, have been
utilized along with a PID controller to enhance the precision of the system.
In the following, stability properties of two CgLp+PID controllers, one of
which has GSORE and the other has SOSRE, are assessed with the proposed
methods. The general structure of the controller is

$\displaystyle C(s)$ $\displaystyle{=}$ $\displaystyle
K_{p}\underbrace{\overbrace{\left(\cancelto{A_{\rho}}{\dfrac{1}{s^{2}+2\xi\omega_{r}s+\omega_{r}^{2}}}\right)}^{\text{GSORE}}\overbrace{\left(\dfrac{s^{2}+2\xi_{d}\omega_{d}s+\omega_{d}^{2}}{s^{2}+20\omega_{c}+100\omega_{c}^{2}}\right)}^{\text{Lead}}}_{\text{CgLp}}$
(66)
$\displaystyle{\times}\underbrace{\overbrace{\left(1+\frac{\omega_{c}}{10s}\right)}^{\text{PI}}\overbrace{\left(\dfrac{\frac{3s}{\omega_{c}}+1}{\frac{s}{3\omega_{c}}+1}\right)}^{\text{Lead}}}_{\text{PID}},$

in which $\omega_{c}$ is the cross-over frequency and $K_{p}$, $\gamma$,
$\omega_{d}$, $\omega_{r}$, $\xi$, and $\xi_{d}$ are tuning parameters. The
PID part is tuned on the basis of [39, 18] and the CgLp part is tuned on the
basis of [38, 29, 31], and $K_{p}$ is set so that $\omega_{c}=200\pi$,
considering the Describing Function (DF) method [38]. In addition, no shaping
filter is used for modifying the performance of the reset controller (i.e.
$\mathcal{C}_{s}(s)=1$). Note that the tuning of the CgLp compensator is not
within the scope of this paper, and we only discuss how to assess stability
properties of reset control systems with these compensators.

######  Remark 5.1.

Suppose that the $H_{\beta}$ condition is/is not satisfied for the reset
control system (8) with $\mathcal{C}_{s}(s)$, $C_{L_{1}}(s)$, $C_{L_{2}}(s)$,
$C_{R}(s)$, $G(s)$, and $A_{\rho}$. Then the $H_{\beta}$ condition is/is not
satisfied for the reset control system (8) with $\mathcal{C}_{s}(s)$,
$C^{\prime}_{L_{1}}(s)$, $C^{\prime}_{L_{2}}(s)$, $C_{R}(s)$, $G^{\prime}(s)$,
and $A_{\rho}$ if
$C^{\prime}_{L_{1}}(s)C^{\prime}_{L_{2}}(s)G^{\prime}(s)=C_{L_{1}}(s)C_{L_{2}}(s)G(s)$
and $G^{\prime}(s)$ is strictly proper. In other words, the “position” of the
reset element does not change in the $H_{\beta}$ condition. However, the
“position” of the reset element has effects on the performance of the reset
control systems [14]. In the two following examples, the sequence of control
filters is such that the tracking error is the input of the reset element and
other linear parts following in series.

###  5.1 A Reset control system with GSORE

In the case of GSORE, the control parameters are $\gamma_{1}=\gamma_{2}=0.5$,
$\omega_{r}=800\pi$, $\omega_{d}=720\pi$, $K_{p}=8.5273e^{7}$, and
$\xi=\xi_{d}=1$. Since the controller has a pole at the origin, we use
Definition 4.2 to assess stability properties of this reset control system.
Using Proposition 2 yields $340<\dfrac{Q_{2}}{Q_{1}}<5057$ and
$1132<\dfrac{Q_{3}}{Q_{4}}$ for $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$,
respectively. Thus, we have to solve the optimization problem
$M=\underset{Q_{1},Q_{2},Q_{3},Q_{4}}{\min}\ G_{1}(Q_{1},Q_{2},Q_{3},Q_{4})$
such that the following constraints hold

$\begin{aligned} \mathcal{S}_{1}:\ &\forall\omega\in(0,\infty):\
f_{1}(Q_{1},Q_{2},1,\omega)>0\\\ \mathcal{S}_{2}:\
&\forall\omega\in(0,\infty):\ f_{2}(Q_{3},Q_{4},1,\omega)>0\\\
\mathcal{S}_{3}:\
&\dfrac{1600\pi}{Q_{1}}+\dfrac{Q_{2}}{Q_{1}Q_{4}}+\dfrac{2}{Q_{1}}\sqrt{\dfrac{1600\pi
Q_{2}}{Q_{4}}-Q_{2}}>1\\\ \mathcal{S}_{4}:\
&\dfrac{1600\pi}{Q_{1}}+\dfrac{Q_{2}}{Q_{1}Q_{4}}-\dfrac{2}{Q_{1}}\sqrt{\dfrac{1600\pi
Q_{2}}{Q_{4}}-Q_{2}}<1\\\ \mathcal{S}_{5}:\
&\dfrac{640000\pi^{2}Q_{1}}{Q_{2}}+1600\pi\left(1+2\sqrt{\dfrac{1600\pi
Q_{1}}{Q_{2}}-1}\right)>\dfrac{Q_{3}}{Q_{4}}\\\ \mathcal{S}_{6}:\
&\dfrac{640000\pi^{2}Q_{1}}{Q_{2}}+1600\pi\left(1-2\sqrt{\dfrac{1600\pi
Q_{1}}{Q_{2}}-1}\right)<\dfrac{Q_{3}}{Q_{4}}\\\ \mathcal{S}_{7}:\ &Q_{i}>0,\
1600\pi>Q_{4},\ 1600\pi<\dfrac{Q_{2}}{Q_{1}}<5057,\
1132<\dfrac{Q_{3}}{Q_{4}},\\\ \mathcal{S}_{8}:\
&\dfrac{Q_{1}Q_{3}}{Q_{2}Q_{4}}>1,\end{aligned}$

(67)

This optimization problem is solved using Genetic Algorithm and Proposition 2.
The optimal solution is $Q_{1}=13172$, $Q_{2}=12001144$, $Q_{3}=8113151$, and
$Q_{4}=1055$, yielding $M=3.5$ (note that it is not necessary to find the
global minimum in these methods). Furthermore, $(\bar{A},C_{0})$ is observable
and $(\bar{A},B_{0})$ is controllable. Hence, the reset control system is of
Type III and using Theorem 3 this GSORE has the UBIBS property for
$A_{\rho}=\gamma I,\ -1<\gamma<1$. Furthermore, since
$\dfrac{Q_{1}Q_{3}}{Q_{2}Q_{4}}>\Gamma(-0.5,0.5)$ and
$\dfrac{Q_{1}Q_{3}}{Q_{2}Q_{4}}>\Gamma(0.5,-0.5)$, Theorem 3 holds for the
considered closed-loop system with $A_{\rho}=\begin{bmatrix}0.5&0\\\
0&-0.5\end{bmatrix}$ or $A_{\rho}=\begin{bmatrix}-0.5&0\\\
0&0.5\end{bmatrix}$. In Fig. 8 the step responses of the closed-loop Spider
stage (Fig. 6) with the designed controller for different values of
$\gamma_{i}$ are displayed. As it can be observed, the values of $\gamma_{i}$
have effect on the performance of the system. In the sense of transient
response, the reset controller with $\gamma_{1}=\gamma_{2}=0.5$ has better
performance among other configurations (for more detail see [38, 29]).

Figure 8: Step response of the closed-loop system with the designed GSORE for
different values of $\gamma_{i}$

###  5.2 Reset control system with SOSRE

In the case in which the controller is a SOSRE the control parameters are
$-1<\gamma<1$, $\omega_{r}=150\pi$, $\omega_{d}=96\pi$, $K_{p}=1.135e^{6}$,
and $\xi=\xi_{d}=1$. Since the controller has a pole at the origin, we use
Definition 3 with the NSV defined in Corollary 4.7 to assess stability
properties. The phase of the NSV for this example is shown in Fig. 9. Since
the phase of the NSV for this example is between $(-\dfrac{\pi}{2},\pi)$ and
the difference between its maximum and its minimum is less than $\pi$, by
Remark 2 the reset control system is of Type I. Moreover, the time
regularization technique (to prevent successive reset instants, i.e. if the
reset happened at one sample time before, the system does not reset) is used
to guarantee the well-posedness property. Consequently, by Corollary 4.7 the
designed SOSRE yields a closed-loop system which has the UBIBS property. The
step responses of the closed-loop Spider stage (Fig. 6) with the designed
controller for different values of $\gamma$ are shown in Fig. 10. In the sense
of transient response, reset control system with $\gamma=0.5$ has better
performance among other controllers. For deeper insights on the performance of
closed-loop reset control systems with SOSRE see [31, 29].

$\boldmath{\phase{\vv{\mathcal{N}}(\omega)}(^{\circ})}$

Figure 9: $\phase{\vec{\mathcal{N}}(\omega)}$ for the reset control systems
with SOSORE Figure 10: Step response of the closed-loop system with the
designed SOSRE for different values of $\gamma$

##  6 Conclusion

In this paper a novel frequency-domain approach based on the $H_{\beta}$
condition for assessing stability properties of reset control systems has been
proposed. This method can be used to determine stability properties of control
systems with first and second order reset elements using FRF measurements of
their base linear open-loop system. Consequently, the methods do not need an
accurate parametric model of the system and the solution of LMIs. In addition,
these methods are applicable to the case in which partial reset techniques are
used. The effectiveness of the proposed methods have been illustrated with a
practical example. This work has been partially supported by NWO through OTP
TTW project $\\#$16335, by the EACEA, by the European Union’s Horizon 2020
Research and Innovation Programme under grant agreement No 739551 (KIOS CoE),
and by the Italian Ministry for Research in the framework of the 2017 Program
for Research Projects of National Interest (PRIN), Grant no. 2017YKXYXJ.

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##  Appendix A Proof of Lemma 1

It has been shown in [12] that when $A_{\rho}=0$, $\mathcal{C}_{s}(s)=1$,
Assumption 2 holds, and the $H_{\beta}$ condition is satisfied, the reset
control system has the UBIBS property. In what follows, we provide a slight
modification of the proof in [12] to deal with the case $A_{\rho}\neq 0$. The
base linear dynamic of the reset control system is given by

$\begin{cases}\dot{x}_{l}(t)=\bar{A}x_{l}(t)+\bar{B}w(t),\\\
y_{l}(t)=\bar{C}x_{l}(t),\end{cases}$ (68)

where $x_{l}(t)=[x_{r_{l}}(t)^{T}\
\zeta_{l}(t)^{T}]^{T}\in\mathbb{R}^{n_{p}+n_{r}}$. Denoting
$z(t):x(t)-x_{l}(t)=[z_{p}(t)^{T}\ \ z_{r}(t)^{T}]^{T}$, yields

$\begin{cases}\dot{z}(t)=\bar{A}z(t),&e(t)\neq 0,\\\
z(t^{+})=\bar{A}_{\rho}z(t)+(\bar{A}_{\rho}-I)x_{l}(t),&e(t)=0.\\\
\end{cases}$ (69)

According to [12], it is sufficient to show that $z(t)$ is bounded. Since the
$H_{\beta}$ condition is satisfied, there exists a matrix $P=P^{T}>0$ such
that

$P=\begin{bmatrix}P_{1}&(\beta\bar{C}_{e})^{T}\\\
\beta\bar{C}_{e}&\Varrho\end{bmatrix},\ P_{1}=P_{1}^{T}>0.$ (70)

Consider now the quadratic Lyapunov function $V(t)=z(t)^{T}Pz(t)$. Using the
same procedure as in [12] yields

$V(t)\leq e^{-\epsilon(t-t_{i})}V(t_{i}),\quad t\in(t_{i},t_{i+1}],\
\epsilon>0,$ (71)

and

$\displaystyle V(t_{i}^{+})$ $\displaystyle{=}$ $\displaystyle
V(t_{i})+x_{r}^{T}(t_{i})(A_{\rho}^{T}\Varrho A_{\rho}-\Varrho)x_{r}(t_{i})$
$\displaystyle{+}\>2(A_{\rho}^{T}-I)x_{r}^{T}(t_{i})\beta\bar{C}_{e}z_{p}(t_{i})-2x^{T}_{r}(t_{i})A_{\rho}^{T}\Varrho
x_{r_{l}}(t_{i}),$

in which $t_{i}$ are the reset instants. Now, let the maximum eigenvalue of
$A_{\rho}^{T}\Varrho A_{\rho}-\Varrho$ be $\lambda_{\max}$ and note that
$\lambda_{\max}<0$ since $A_{\rho}^{T}\Varrho A_{\rho}-\Varrho<0$. As a result

$\displaystyle V(t_{i}^{+})$ $\displaystyle{\leq}$ $\displaystyle
V(t_{i})-|\lambda_{\max}|x_{r}^{T}(t_{i})x_{r}(t_{i})+2(A_{\rho}^{T}-I)x_{r}^{T}(t_{i})\beta\bar{C}_{e}z_{p}(t_{i})$
$\displaystyle{-}\>2x^{T}_{r}(t_{i})A_{\rho}^{T}\Varrho
x_{r_{l}}(t_{i})\Rightarrow$ $\displaystyle V(t_{i}^{+})$
$\displaystyle{\leq}$ $\displaystyle V(t_{i})+2\left\lVert
x_{r}(t_{i})\right\rVert(||A_{\rho}^{T}-I||\left\lVert\beta\bar{C}_{e}z_{p}(t_{i})\right\rVert$
(73) $\displaystyle{+}\>||A_{\rho}\Varrho x_{r_{l}}(t_{i})||).$

At the reset instants $|\bar{C}_{e}z_{p}(t_{i})|\leq|D_{e}r(t)|$ which implies
that $|\bar{C}_{e}z_{p}(t_{i})|$ is bounded. Moreover, since the base linear
system is stable, $x_{r_{l}}(t_{i})$ is bounded. Assume now that
$x_{r}(t_{i})$ is unbounded. By (71) and (A), we conclude that
$\displaystyle\lim_{i\to\infty}V(t_{i})=0$. This is a contradiction because
$z(t)=0\Rightarrow x(t)=x_{l}(t)$ which implies that the system is a stable
linear system with bounded state. Therefore, $x_{r}(t_{i})$ is bounded. Now,
we prove that $\dot{x}_{r}(t_{i})$ is bounded. If reset happens when the input
of the reset element is zero (i.e. $\mathcal{C}_{s}(s)=1$) and Assumption 2
holds, then

$\displaystyle\dfrac{dx_{r}(t)}{dt}\Big{|}_{t=t_{i}^{-}}$ $\displaystyle{=}$

$A_{r}\left(e^{A_{r}(t_{i}-t_{i-1})}x_{r}(t_{i-1})+\displaystyle\int_{t_{i-1}}^{t_{i}}e^{A_{r}(t_{i}-\tau)}B_{r}e(\tau)d\tau\right)$

(74) $\displaystyle{=}$ $\displaystyle
A_{r}x_{r}(t_{i})\Rightarrow\left\lvert\dot{x}_{r}(t_{i}^{-})\right\rvert=\left\lvert
A_{r}x_{r}(t_{i})\right\rvert.$

Thus, since $|x_{r}(t_{i})|$ is bounded,
$\left\lvert\dot{x}_{r}(t_{i}^{-})\right\rvert$ is bounded. As a result, since
$|x_{r}(t_{i-1}^{+})|\leq|A_{\rho}||x_{r}(t_{i-1})|$, $|x_{r}(t_{i})|$ and
$\left\lvert\dot{x}_{r}(t_{i}^{-})\right\rvert$ are bounded,

$\exists\ K_{1}>0,\ \alpha>0\ \text{such that }|x_{r}(t_{i})|\leq
K_{1}\left(1-e^{\alpha(t_{i}-t_{i-1})}\right),\ \forall\ t_{i}.$ (75)

Now assume that there exist $t_{i}$ and $t_{i-1}$ such that for any
$\epsilon>0$, $t_{i}-t_{i-1}<\epsilon$. Thus, by (75) and for sufficient small
$\epsilon$, $x_{r}(t_{i})\to 0$. This is a contradiction because
$(I-\bar{A}_{\rho})x_{r}(t_{i})\to 0$ which means that $t_{i}$ is not a reset
instant. Thus, there exists $\lambda>0$ such that, for all $k\in\mathbb{N}$,
$\lambda\leq t_{k+1}-t_{k}$. Therefore, the reset instants have the the well-
posedness property (see Definition 1).
In the case in which $\mathcal{C}_{s}=1$ or Assumption 2 does not hold, (74)
can not be concluded. However, if the well-posedness property of the reset
instants holds, then there exists $\lambda>0$ such that, for all
$k\in\mathbb{N}$, $\lambda\leq t_{k+1}-t_{k}$. In addition, since
$|x_{r}(t_{i-1}^{+})|\leq|A_{\rho}||x_{r}(t_{i-1})|$ and $|x_{r}(t_{i})|$ are
bounded, we conclude (75). Since the system has the well-posedness property,
the reset control system (8) has an unique well-defined solution for any
initial condition $x_{0}$ and any input $w$ which is a Bohl function [7]. The
rest of the proof is the same as the proof in [12].

##  Appendix B Proof of Corollary 3.2

Let $\beta^{\prime}=-\beta$ and $\Varrho^{\prime}=\dfrac{\Varrho}{C_{r}}$. By
the proof of the $H_{\beta}$ condition in [12] the transfer function (10) for
the configuration shown in Fig. 4) can be rewritten as (see also Fig. 11)

$H(s)=\dfrac{\beta^{\prime}\dfrac{L^{\prime}(s)}{\mathcal{C}_{s}(s)}+\Varrho^{\prime}C_{R}(s)}{1+L^{\prime}(s)}.$
(76)

Let
$C_{L_{1}}(s)C_{L_{2}}(s)C_{R}(s)G(s)=\dfrac{k_{m}s^{m}+k_{m-1}s^{m-1}+...+k_{0}}{s^{n}+k^{\prime}_{n-1}s^{n-1}+...+k^{\prime}_{0}}$.
Using the NSV defined in (4.1), one could repeat Steps 1 to 4 of the proof of
Theorem 1. Note that $K_{s_{0}}\beta^{\prime}$ in (20)-(24) and (28) has to be
replaced by $\dfrac{\beta^{\prime}}{K_{s_{0}}}$ and $K_{n}$ has also to be
replaced by $k_{n}$ in (25).

$-$$y$$\mathbf{C_{\mathbf{R}}}$$\mathbf{C_{\mathbf{L}_{2}}}$$\mathbf{G}$$\mathbf{C_{\mathbf{L}_{1}}}$$r$$\beta^{\prime}$$\Varrho^{\prime}$$+$$\mathcal{C}_{s}$

Figure 11: The block diagram of $H_{\beta}$ condition for the modified
architecture Fig. 4 with GFORE or PCI

##  Appendix C Proof of Proposition 2

Consider $Q_{1}\mathcal{F}_{1}(\omega)+Q_{2}\mathcal{F}_{2}(\omega)$ as the
scalar product of the two vectors $\vv{\mathcal{F}}(\omega)$ and
$\vv{\mathcal{Q}}$. Thus, for all $\omega\in\mathbb{R}^{+}$, the condition
(33) can be re-written as

$\sqrt{Q_{1}^{2}+Q_{2}^{2}}\sqrt{\mathcal{F}_{1}^{2}(\omega)+\mathcal{F}_{2}^{2}(\omega)}\cos(\vartheta)>\mathcal{F}_{3}(\omega).$
(77)

As a result, when $\mathcal{F}_{3}(\omega)\geq 0$, $\cos(\vartheta)$ must be
positive and

$\sqrt{Q_{1}^{2}+Q_{2}^{2}}>\underset{\omega\in\omega_{p}}{\max\
}\dfrac{\mathcal{F}_{3}(\omega)}{\cos(\vartheta)\sqrt{\mathcal{F}^{2}_{1}(\omega)+\mathcal{F}^{2}_{2}(\omega)}}=\eta_{1}(\dfrac{Q_{2}}{Q_{1}}).$
(78)

Positivity of $\cos(\vartheta)$ implies $\dfrac{Q_{2}}{Q_{1}}\in g_{p}$. When
$\mathcal{F}_{3}(\omega)<0$, there are two solutions for condition (77).
$\cos(\vartheta)\geq 0$ which requiers $\dfrac{Q_{2}}{Q_{1}}\in g_{N}$, or

$\sqrt{Q_{1}^{2}+Q_{2}^{2}}<\underset{\omega\in\omega_{N}}{\min\
}\dfrac{\mathcal{F}_{3}(\omega)}{\cos(\vartheta)\sqrt{\mathcal{F}^{2}_{1}(\omega)+\mathcal{F}^{2}_{2}(\omega)}}=\eta_{2}(\dfrac{Q_{2}}{Q_{1}}).$
(79)

Therefore, by (78) and (79)
$\eta_{2}(\dfrac{Q_{2}}{Q_{1}})>\eta_{1}(\dfrac{Q_{2}}{Q_{1}})$ and the proof
is complete.

##  Appendix D Proof of Corollary 4.6

First, a preliminary result is stated and proved.

######  Lemma D.1.

Consider the reset controllers $C_{R_{1}}$ and $C_{R_{2}}$ shown in Fig. 12.
Suppose $C_{R_{1}}$ and $C_{R_{2}}$ have the same base linear system, are
strictly proper, and have different state-space realizations. Then if
$A_{\rho}=\gamma I$ and their initial conditions are zero,
$y_{1}(t)=y_{2}(t)$, for $t\geq 0$.

$\mathbf{C_{\mathbf{R}_{1}}}$$u(t)$$\mathbf{C_{\mathbf{R}_{2}}}$$y_{1}(t)$$y_{2}(t)$

Figure 12: Two reset controllers $C_{R_{1}}$ and $C_{R_{2}}$ with different
realizations

Let $A_{r_{1}}$, $B_{r_{1}}$, and $C_{r_{1}}$ be the state-space realization
of $C_{R_{1}}(s)$ and $A_{r_{2}}$, $B_{r_{2}}$, and $C_{r_{2}}$ be the state-
space realization of $C_{R_{2}}(s)$. Since the base transfer function of
$C_{R_{1}}(s)$ and $C_{R_{2}}(s)$ are the same,

$C_{r_{1}}\displaystyle\int_{t_{0}}^{t}e^{A_{r_{1}}(t-\tau)}B_{r_{1}}u(\tau)d\tau=C_{r_{2}}\displaystyle\int_{t_{0}}^{t}e^{A_{r_{2}}(t-\tau)}B_{r_{2}}u(\tau)d\tau.$
(80)

Let $t_{k},\ k\in\mathbb{N}$, be the reset instants of $C_{R_{1}}$ and
$C_{R_{2}}$. Since the initial condition is zero,

$\displaystyle y_{1}(t)$ $\displaystyle{=}$

$C_{r_{1}}\Big{(}\displaystyle\int_{t_{k}}^{t}e^{A_{r_{1}}(t-\tau)}B_{r_{1}}u(\tau)d\tau+\gamma\displaystyle\int_{t_{k-1}}^{t_{k}}e^{A_{r_{1}}(t-\tau)}B_{r_{1}}u(\tau)d\tau$

(81)
$\displaystyle{+}\>\cdots+\gamma^{k}\displaystyle\int_{t_{0}}^{t_{1}}e^{A_{r_{1}}(t-\tau)}B_{r_{1}}u(\tau)d\tau\Big{)},$
$\displaystyle y_{2}(t)$ $\displaystyle{=}$

$C_{r_{2}}\Big{(}\displaystyle\int_{t_{k}}^{t}e^{A_{r_{2}}(t-\tau)}B_{r_{2}}u(\tau)d\tau+\gamma\displaystyle\int_{t_{k-1}}^{t_{k}}e^{A_{r_{2}}(t-\tau)}B_{r_{2}}u(\tau)d\tau$

(82)
$\displaystyle{+}\>\cdots+\gamma^{k}\displaystyle\int_{t_{0}}^{t_{1}}e^{A_{r_{2}}(t-\tau)}B_{r_{2}}u(\tau)d\tau\Big{)}.$

As a result, $y_{1}(t)=y_{2}(t)$, for all $t\geq 0$ by (80). The corollary is
now proved. Using Lemma D.1, since the initial condition of the reset part are
zero and $A_{\rho}=\gamma I$, the states $\zeta(t)$ are the same for different
state-space realizations of $C_{R}(s)$ of the reset control system (8). Thus,
since Theorem 3 holds for the reset control system (8) with GSORE (5) with the
controllable realization (6), $\zeta(t)$, which contains $u_{1}(t)$ and
$u_{r}(t)$, is also bounded in the reset control system (8) with GSORE (5)
with realization configurations (61) and (62). Therefore, it is just needed to
show that the reset state $x_{r}(t)$ is bounded. Consider the reset controller
(1). Now assume that $x_{r}(t)$ is unbounded, since $u_{1}(t)$ is bounded and
by state-space realization (61) and (62), $u_{r}(t)$ becomes unbounded which
is a contradiction. Therefore, $x_{r}(t)$ must be bounded and the proof is
complete.