# State and output feedback control for Lipschitz nonlinear systems in reciprocal state space: synthesis and real-time validation

Assem Thabet, Ghazi Bel Haj Frej, Noussaiba Gasmi, Karim Mansouri, Eric Chauveau

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Assem Thabet, Ghazi Bel Haj Frej, Noussaiba Gasmi, Karim Mansouri, Eric Chauveau. State and output feedback control for Lipschitz nonlinear systems in reciprocal state space: synthesis and real-time validation. Bulletin of Electrical Engineering and Informatics, 2022, 11 (5), pp.2518-2526. 10.11591/eei.v11i5.3521 . hal-03776216

HAL Id: hal-03776216

https://hal.science/hal-03776216

Submitted on 6 Jun 2024

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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<figure><img src="image_2.png" /></figure>
# State and output feedback control for Lipschitz nonlinear systems in reciprocal state space: synthesis and real-time validation

Assem Thabet, Ghazi Bel Haj Frej, Noussaiba Gasmi, Karim Mansouri, Eric Chauveau

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Assem Thabet, Ghazi Bel Haj Frej, Noussaiba Gasmi, Karim Mansouri, Eric Chauveau. State and output feedback control for Lipschitz nonlinear systems in reciprocal state space: synthesis and real-time validation. Bulletin of Electrical Engineering and Informatics, 2022, 11 (5), pp.2518-2526. 10.11591/eei.v11i5.3521 . hal-03776216

HAL Id: hal-03776216

https://hal.science/hal-03776216v1

Submitted on 6 Jun 2024

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>
# State and output feedback control for Lipschitz nonlinear systems in reciprocal state space: synthesis and real-time validation

Assem Thabet<fnref n="1" />, Ghazi Bel Haj Frej<fnref n="2" />, Noussaiba Gasmi<fnref n="3" />, Karim Mansouri<fnref n="4" />, Eric Chauveau<fnref n="4" />

**Article Info**

**Article history:**

Received Dec 22, 2021

Revised Apr 11, 2022

Accepted Jul 1, 2022

**Keywords:**

Arduino MEGA 2560

Cost control

Lipschitz propriety

Nonlinear systems

Reciprocal state space

State/output feedback

**ABSTRACT**

In industrials process and many studies cases, state and output derivative variables can be easily modeled in reciprocal state space (RSS) form than the standard one. Formulation of stabilization problem with a guaranteed cost control using feedback principle for Lipschitz nonlinear systems (LNS) in RSS is presented in this brief. The asymptotic stability, using the proper Lyapunov functions of the closed-loop system, is guaranteed. The control design problem is guaranteed through a resolution of linear matrix inequalities (LMI) technique under certain lemmas and minimization of non-standard cost control. Experimental validation shows the good performances of the proposed method using real-time implementation (RTI) with a digital signal processing (DSP) device (Arduino MEGA 2560).

This is an open access article under the [CC BY-SA license](https://creativecommons.org/licenses/by-sa/4.0/).

<figure><img src="image_3.png" /></figure>

**Corresponding Author:**

Assem Thabet

MACS Laboratory, National Engineering School of Gabes (ENIG), University of Gabes

Avenue Omar Ibn El Khattab, Zrig Eddakhlania 6029, Tunisia

Email: assem.thabet@issig.rnu.tn

# NOMENCLATURE

- $I_s$ is the identity matrix of dimension “s”

- $Q^T$ is the transposed matrix of Q

- The notation (*) is used for the matrix blocks induced by symmetry

- Q is a square matrix, then Q>0 (Q<0) defines that Q is positive definite (negative definite)

# 1. INTRODUCTION

The problems of monitoring and controlling the design of nonlinear systems have been exploited in many engineering areas using the state and output feedback principle [1]–[3] in standard state space (SSS) form. The synthesis problem of nonlinear control laws received remarkable attention with solid synthesis approaches: Tracking control with parametric uncertainties [4]; fuzzy and predictive control [5]; robust finite-time controller using sliding mode strategy [6]; backstepping controller using extended Kalman filter [7]; adaptive control using neural networks with input delay [8]; integral backstepping [9]; and feedback finite-time control for second-order systems [10].

In this context, the case of the Lipschitz nonlinear systems (LNS) in the SSS has been treated by many researchers considering their feasibility in real-time applications with different digital signal processors (DSP)

<footnote n="1">¹MACS Laboratory, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia</footnote>

<footnote n="2">²Department IMS Laboratory, CNRS UMR 5218, Bordeaux INP, University of Bordeaux, Talence, France</footnote>

<footnote n="3">³Laboratoire d'Informatique Télécommunications, ECAM Rennes - Louis de Broglie, Rennes, France</footnote>

<footnote n="4">⁴Laboratoire IREENA - ESEO-Tech, Université de Nantes-France, Nantes, France</footnote>
# State and output feedback control for Lipschitz nonlinear systems in reciprocal state space: synthesis and real-time validation

Assem Thabet<fnref n="1" />, Ghazi Bel Haj Frej<fnref n="2" />, Noussaiba Gasmi<fnref n="3" />, Karim Mansouri<fnref n="4" />, Eric Chauveau<fnref n="4" />

**Article Info**

**Article history:**

Received Dec 22, 2021

Revised Apr 11, 2022

Accepted Jul 1, 2022

**Keywords:**

Arduino MEGA 2560

Cost control

Lipschitz propriety

Nonlinear systems

Reciprocal state space

State/output feedback

**ABSTRACT**

In industrials process and many studies cases, state and output derivative variables can be easily modeled in reciprocal state space (RSS) form than the standard one. Formulation of stabilization problem with a guaranteed cost control using feedback principle for Lipschitz nonlinear systems (LNS) in RSS is presented in this brief. The asymptotic stability, using the proper Lyapunov functions of the closed-loop system, is guaranteed. The control design problem is guaranteed through a resolution of linear matrix inequalities (LMI) technique under certain lemmas and minimization of non-standard cost control. Experimental validation shows the good performances of the proposed method using real-time implementation (RTI) with a digital signal processing (DSP) device (Arduino MEGA 2560).

This is an open access article under the [CC BY-SA license](https://creativecommons.org/licenses/by-sa/4.0/).

<figure><img src="image_3.png" /></figure>

**Corresponding Author:**

Assem Thabet

MACS Laboratory, National Engineering School of Gabes (ENIG), University of Gabes

Avenue Omar Ibn El Khattab, Zrig Eddakhlania 6029, Tunisia

Email: assem.thabet@issig.rnu.tn

# NOMENCLATURE

- $I_s$ is the identity matrix of dimension “s”

- $Q^T$ is the transposed matrix of Q

- The notation (*) is used for the matrix blocks induced by symmetry

- Q is a square matrix, then Q>0 (Q<0) defines that Q is positive definite (negative definite)

# 1. INTRODUCTION

The problems of monitoring and controlling the design of nonlinear systems have been exploited in many engineering areas using the state and output feedback principle [1]–[3] in standard state space (SSS) form. The synthesis problem of nonlinear control laws received remarkable attention with solid synthesis approaches: Tracking control with parametric uncertainties [4]; fuzzy and predictive control [5]; robust finite-time controller using sliding mode strategy [6]; backstepping controller using extended Kalman filter [7]; adaptive control using neural networks with input delay [8]; integral backstepping [9]; and feedback finite-time control for second-order systems [10].

In this context, the case of the Lipschitz nonlinear systems (LNS) in the SSS has been treated by many researchers considering their feasibility in real-time applications with different digital signal processors (DSP)

<footnote n="1">¹MACS Laboratory, National Engineering School of Gabes (ENIG), University of Gabes, Gabes, Tunisia</footnote>

<footnote n="2">²Department IMS Laboratory, CNRS UMR 5218, Bordeaux INP, University of Bordeaux, Talence, France</footnote>

<footnote n="3">³Laboratoire d'Informatique Télécommunications, ECAM Rennes - Louis de Broglie, Rennes, France</footnote>

<footnote n="4">⁴Laboratoire IREENA - ESEO-Tech, Université de Nantes-France, Nantes, France</footnote>
cards: stabilization of one link flexible joint robot using sliding window of measurements with Arduino MEGA board [11]. Feedback stabilization using differential mean value theorem (DMVT) principle with Arduino UNO R3 board [12]. Tracking problem of the flux and the speed of induction motor using STM32 board [13]. Despite the important literature on approaches proposed in SSS (using state variables), many limitations exist. Effectively, many engineering systems and applications give measurements and data covering their basic physical phenomena or their operating principles according to derivatives state: photovoltaic systems [14]; mobile robots [15]; active suspension systems [16], [17]; and electronic impulse systems [18], [19].

This leads to the state derivatives being easily obtained in some engineering systems. Therefore, several sections have recently applied state derivative feedback controllers using the RSS reformulation. Effectively, derivative feedback principle has been addressed in many control problems for linear systems : static output feedback control [15]; robust control subject fault [16]; feedback control in linear polytypic systems [17]; pole placement for single-input single-output (SISO) linear systems [20]; sliding mode control [21]; linear optimal control [22], and linear quadratic regulator (LQR) [23]. Besides, many reliable methods and technics are used in the synthesis. For example, LQR design in the RSS framework is proposed by algebraic Riccati equations (ARE) in [24]. For descriptor systems, ARE. and linear matrix inequalities (LMI)-based solutions of LQR with state derivative feedback are developed in [25].

Similarly, for the case of nonlinear systems, recent works dealing with several control methods based on Lyapunov stability have been synthesized, such as H∞ fuzzy state feedback [14]; adaptive stabilization and tracking control [19]; state feedback stabilization of one-sided Lipschitz systems [26]; and sliding mode control for Lipschitz systems[27].

Moreover, the studies on the derivative Lipschitz nonlinearity [18], [19] are few in control design problems. As well as for the case of control by output feedback in the RSS and treating the Lipschitz nonlinear systems is not treated to the authors' knowledge. For all these reasons and motivated by the latest results of [19], [22] and to overcome these limitations, this paper proposes the design of a unified approach using the feedback principle with output/state derivatives for LNS. The main contribution of this brief is based on a combination of the guaranteed cost control problem design with the use of the Lipschitz condition in RSS. The proposed design method (using RSS feedback principle with output/state derivatives) is based on Lyapunov theory to guarantee the closed-loop stability. The control design resolution problem is ensured under certain lemmas and through LMI technique with guaranteed cost control.

This brief is organized as follows: First, formulation of problems statement with useful preliminaries are presented. Second, the synthesis method of control law design with guaranteed cost control to NLS in RSS is detailed. Finally, the experimental validations are presented which demonstrate the performance of the proposed results tusing an RTI based on a DSP device (Arduino MEGA 2560) that is used as H.I.L.

## 2. PROBLEM STATEMENT AND PRELIMINARIES

Based on the works of [15], [17], the NLS in RSS is described by:

$$ \begin{cases} x = \bar{A}\dot{x} + \bar{B}u + \bar{f}(\dot{x}) \\ \dot{y} = \bar{C}\dot{x} \end{cases} \qquad (1) $$

with:

$$ \|\bar{f}(\dot{x})\| \le \delta \|\dot{x}\| \qquad (2) $$

where $x \in \mathbb{R}^n$: state vector; $u \in \mathbb{R}^m$: input vector; $y \in \mathbb{R}^p$: output vector. $\bar{A}, \bar{B}, \bar{C}$ are constant matrices of appropriate dimensions. $\bar{f}(x) \in \mathbb{R}^{n \times m}$ is the nonlinear vector which satisfies the Lipschitz propriety (where $\delta$ is the Lipschitz constant). Throughout this paper, the following proposals and lemmas will be considered:

**Assumption 1** :[22]

Given two performance weighting matrices $\bar{Q} = \bar{Q}^T > 0$ and $\bar{R} = \bar{R}^T > 0$ of appropriate dimensions, then the non-standard quadratic cost function is as follows :

$$ \bar{J} = \int_{0}^{\infty} (\dot{x}^{T}\bar{Q}\dot{x} + u^{T}(t)\bar{R})dt = \int_{0}^{\infty} \bar{\Omega} dt \qquad (3) $$

**Assumption 2:**

Assume that $\bar{f}(0) = 0$.

**Lemma 1:** [28]
cards: stabilization of one link flexible joint robot using sliding window of measurements with Arduino MEGA board [11]. Feedback stabilization using differential mean value theorem (DMVT) principle with Arduino UNO R3 board [12]. Tracking problem of the flux and the speed of induction motor using STM32 board [13]. Despite the important literature on approaches proposed in SSS (using state variables), many limitations exist. Effectively, many engineering systems and applications give measurements and data covering their basic physical phenomena or their operating principles according to derivatives state: photovoltaic systems [14]; mobile robots [15]; active suspension systems [16], [17]; and electronic impulse systems [18], [19].

This leads to the state derivatives being easily obtained in some engineering systems. Therefore, several sections have recently applied state derivative feedback controllers using the RSS reformulation. Effectively, derivative feedback principle has been addressed in many control problems for linear systems : static output feedback control [15]; robust control subject fault [16]; feedback control in linear polytypic systems [17]; pole placement for single-input single-output (SISO) linear systems [20]; sliding mode control [21]; linear optimal control [22], and linear quadratic regulator (LQR) [23]. Besides, many reliable methods and technics are used in the synthesis. For example, LQR design in the RSS framework is proposed by algebraic Riccati equations (ARE) in [24]. For descriptor systems, ARE. and linear matrix inequalities (LMI)-based solutions of LQR with state derivative feedback are developed in [25].

Similarly, for the case of nonlinear systems, recent works dealing with several control methods based on Lyapunov stability have been synthesized, such as H∞ fuzzy state feedback [14]; adaptive stabilization and tracking control [19]; state feedback stabilization of one-sided Lipschitz systems [26]; and sliding mode control for Lipschitz systems[27].

Moreover, the studies on the derivative Lipschitz nonlinearity [18], [19] are few in control design problems. As well as for the case of control by output feedback in the RSS and treating the Lipschitz nonlinear systems is not treated to the authors' knowledge. For all these reasons and motivated by the latest results of [19], [22] and to overcome these limitations, this paper proposes the design of a unified approach using the feedback principle with output/state derivatives for LNS. The main contribution of this brief is based on a combination of the guaranteed cost control problem design with the use of the Lipschitz condition in RSS. The proposed design method (using RSS feedback principle with output/state derivatives) is based on Lyapunov theory to guarantee the closed-loop stability. The control design resolution problem is ensured under certain lemmas and through LMI technique with guaranteed cost control.

This brief is organized as follows: First, formulation of problems statement with useful preliminaries are presented. Second, the synthesis method of control law design with guaranteed cost control to NLS in RSS is detailed. Finally, the experimental validations are presented which demonstrate the performance of the proposed results tusing an RTI based on a DSP device (Arduino MEGA 2560) that is used as H.I.L.

## 2. PROBLEM STATEMENT AND PRELIMINARIES

Based on the works of [15], [17], the NLS in RSS is described by:

$$ \begin{cases} x = \bar{A}\dot{x} + \bar{B}u + \bar{f}(\dot{x}) \\ \dot{y} = \bar{C}\dot{x} \end{cases} \qquad (1) $$

with:

$$ \|\bar{f}(\dot{x})\| \le \delta \|\dot{x}\| \qquad (2) $$

where $x \in \mathbb{R}^n$: state vector; $u \in \mathbb{R}^m$: input vector; $y \in \mathbb{R}^p$: output vector. $\bar{A}, \bar{B}, \bar{C}$ are constant matrices of appropriate dimensions. $\bar{f}(x) \in \mathbb{R}^{n \times m}$ is the nonlinear vector which satisfies the Lipschitz propriety (where $\delta$ is the Lipschitz constant). Throughout this paper, the following proposals and lemmas will be considered:

**Assumption 1** :[22]

Given two performance weighting matrices $\bar{Q} = \bar{Q}^T > 0$ and $\bar{R} = \bar{R}^T > 0$ of appropriate dimensions, then the non-standard quadratic cost function is as follows :

$$ \bar{J} = \int_{0}^{\infty} (\dot{x}^{T}\bar{Q}\dot{x} + u^{T}(t)\bar{R})dt = \int_{0}^{\infty} \bar{\Omega} dt \qquad (3) $$

**Assumption 2:**

Assume that $\bar{f}(0) = 0$.

**Lemma 1:** [28]
<figure><img src="image_4.png" /></figure>

Given three real matrices, Y, Z, and F, of appropriate dimensions $F^T F \le I$, then the following inequality holds for any constant $\bar{\epsilon} > 0$:

$$2x^T Y F Z y \le \bar{\epsilon} x^T Y Y^T x + \bar{\epsilon}^{-1} y^T Z^T Z y \quad (4)$$

**Lemma 2:** [29]

Given two matrices X, Y, of appropriate dimensions, then the following inequality holds for any symmetric positive definite (SPD) matrix S of appropriate dimension:

$$X^T Y + Y^T X \le \bar{\eta} X^T S X + \bar{\eta}^{-1} Y^T S^{-1} Y \quad (5)$$

This brief aims to design a unified state and output derivatives feedback control based on optimization of non-standard cost control.

### 3. DERIVATIVE FEEDBACK CONTROLLERS DESIGN

This part presents the synthesis conditions for the design of control laws based on the principle of feedback by state and output derivatives while guaranteeing a non-standard cost control for NLS (using LMI with semi-definite programming (SDP) formulation [30]).

#### 3.1. State derivative feedback controller

The principle is to design a simple derivative-state feedback control law in the following form:

$$u(t) = \bar{K}_x \dot{x} \quad (6)$$

where $\bar{K}_x \in \mathbb{R}^{m \times n}$. Then, integrating (6) in (1), the closed-loop system becomes:

$$x = \left( \bar{A} + \bar{B} \bar{K}_x \right) \dot{x} + \bar{f}(x) \quad (7)$$

The state derivative feedback (6) controller with a guaranteed cost control laws are synthesized to minimize a non-standard quadratic cost function (3) for LNS (1) and in order to guarantee that (7) is asymptotically stable, the following state-derivative feedback control law is proposed:

**Theorem 1**

For LNS in RSS (1) with derivative-state feedback control (6), the closed-loop system (7) is asymptotically stable if there exists SPD matrices $\bar{X}$ and $\bar{M}$ of appropriate dimensions and scalar $\bar{\eta} > 0$, such that the following LMI is feasible

$$\begin{bmatrix} \bar{A}\bar{X} + \bar{A}\bar{X}^T + \bar{B}\bar{M} + \bar{M}^T\bar{B}^T + \bar{\eta}I_n & \bar{M}^T & \bar{X} \\ \dot{\bar{A}} & -\bar{R}^{-1} & 0 \\ \dot{\bar{A}} & \dot{\bar{A}} & -(\bar{Q} + \frac{\delta}{\bar{\eta}}I_n)^{-1} \end{bmatrix} < 0 \quad (8)$$

where $\bar{K}_x = \bar{M}\bar{X}^{-1}$.

**Proof:**

First, the Lyapunov function is chosen as $V = x^T P x$. Then, using the Bellman-Lyapunov inequality, the following condition must be satisfied by the designed controller with a guaranteed cost control:

$$\bar{A} = \dot{V} + \bar{\Omega} < 0 \Leftrightarrow \dot{x}^T P x + x^T P \dot{x} < -(\dot{x}^T \bar{Q} \dot{x} + u^T(t) \bar{R} u(t)) \quad (9)$$

the derivative of V with (9) gives (for simplicity $\bar{f}(\dot{x}) = \bar{f}$):

$$\dot{x}^T P \bar{A} \dot{x} + \dot{x}^T \bar{A}^T P \dot{x} + \underbrace{\dot{x}^T P \bar{f} + \bar{f}^T P \dot{x}}_{\bar{L}} < -\dot{x}^T (\bar{Q} + \bar{R}_x^T \bar{R} \bar{K}_x) \dot{x} \quad (10)$$

by applying Lemma 2 on the term $\bar{L}$ (where $X^T = \dot{x}^T P$, $Y = \bar{f}$ and $S = I$) and using the Lipschitz condition (2) with Assumption 2, (9) becomes:
<figure><img src="image_4.png" /></figure>

Given three real matrices, Y, Z, and F, of appropriate dimensions $F^T F \le I$, then the following inequality holds for any constant $\bar{\epsilon} > 0$:

$$2x^T Y F Z y \le \bar{\epsilon} x^T Y Y^T x + \bar{\epsilon}^{-1} y^T Z^T Z y \quad (4)$$

**Lemma 2:** [29]

Given two matrices X, Y, of appropriate dimensions, then the following inequality holds for any symmetric positive definite (SPD) matrix S of appropriate dimension:

$$X^T Y + Y^T X \le \bar{\eta} X^T S X + \bar{\eta}^{-1} Y^T S^{-1} Y \quad (5)$$

This brief aims to design a unified state and output derivatives feedback control based on optimization of non-standard cost control.

### 3. DERIVATIVE FEEDBACK CONTROLLERS DESIGN

This part presents the synthesis conditions for the design of control laws based on the principle of feedback by state and output derivatives while guaranteeing a non-standard cost control for NLS (using LMI with semi-definite programming (SDP) formulation [30]).

#### 3.1. State derivative feedback controller

The principle is to design a simple derivative-state feedback control law in the following form:

$$u(t) = \bar{K}_x \dot{x} \quad (6)$$

where $\bar{K}_x \in \mathbb{R}^{m \times n}$. Then, integrating (6) in (1), the closed-loop system becomes:

$$x = \underbrace{(\bar{A} + \bar{B}\bar{K}_x)}_{\bar{A}} \dot{x} + \bar{f}(\dot{x}) \quad (7)$$

The state derivative feedback (6) controller with a guaranteed cost control laws are synthesized to minimize a non-standard quadratic cost function (3) for LNS (1) and in order to guarantee that (7) is asymptotically stable, the following state-derivative feedback control law is proposed:

**Theorem 1**

For LNS in RSS (1) with derivative-state feedback control (6), the closed-loop system (7) is asymptotically stable if there exists SPD matrices $\bar{X}$ and $\bar{M}$ of appropriate dimensions and scalar $\bar{\eta} > 0$, such that the following LMI is feasible

$$\begin{bmatrix} \bar{A}\bar{X} + \bar{A}\bar{X}^T + \bar{B}\bar{M} + \bar{M}^T\bar{B}^T + \bar{\eta}I_n & \bar{M}^T & \bar{X} \\ \dot{\bar{A}} & -\bar{R}^{-1} & 0 \\ \dot{\bar{A}} & \dot{\bar{A}} & -(\bar{Q} + \frac{\delta}{\bar{\eta}}I_n)^{-1} \end{bmatrix} < 0 \quad (8)$$

where $\bar{K}_x = \bar{M}\bar{X}^{-1}$.

**Proof:**

First, the Lyapunov function is chosen as $V = x^T P x$. Then, using the Bellman-Lyapunov inequality, the following condition must be satisfied by the designed controller with a guaranteed cost control:

$$\bar{A} = \dot{V} + \bar{\Omega} < 0 \Leftrightarrow \dot{x}^T P x + x^T P \dot{x} < -(\dot{x}^T \bar{Q} \dot{x} + u^T(t) \bar{R} u(t)) \quad (9)$$

the derivative of V with (9) gives (for simplicity $\bar{f}(\dot{x}) = \bar{f}$):

$$\dot{x}^T P \bar{A} \dot{x} + \dot{x}^T \bar{A}^T P \dot{x} + \underbrace{\dot{x}^T P \bar{f} + \bar{f}^T P \dot{x}}_{\bar{L}} < -\dot{x}^T (\bar{Q} + \bar{R}_x^T \bar{R} \bar{K}_x) \dot{x} \quad (10)$$

by applying Lemma 2 on the term $\bar{L}$ (where $X^T = \dot{x}^T P$, $Y = \bar{f}$ and $S = I$) and using the Lipschitz condition (2) with Assumption 2, (9) becomes:
$$ \bar{A} = \dot{x}^T [P\bar{A} + \bar{A}^T P + \bar{Q} + \bar{K}_x^T \bar{R} \bar{K}_x + \bar{\eta} PP + \frac{\delta}{\bar{\eta}} I_n] \dot{x} = \dot{x}^T \bar{\Theta}_x \dot{x} < 0 \quad (11) $$

Second, the use Schur complements on $\bar{\Theta}_x$ leads to:

$$ \bar{\Theta}_x = \begin{bmatrix} P\bar{A} + \bar{A}^T P + \bar{Q} + \bar{\eta} PP + \frac{\delta}{\bar{\eta}} I_n & \bar{K}_x^T \\ \bar{\delta} & -\bar{R}^{-1} \end{bmatrix} \quad (12) $$

introduce a new variable $\bar{X} = \bar{X}^T = P^{-1}$ and pre-and post-multiply (12) by $\begin{bmatrix} \bar{X} & 0 \\ 0 & I \end{bmatrix}$. Then, (12) is congruent to:

$$ \bar{\Theta}_x = \begin{bmatrix} \bar{A}\bar{X} + \bar{A}\bar{A}^T + \bar{X}(\bar{Q} + \frac{\bar{\delta}}{\bar{\eta}}I_n)\bar{X} + \bar{\eta}I_n & \bar{X}\bar{K}_x^T \\ \bar{\delta} & -\bar{R}^{-1} \end{bmatrix} \quad (12a) $$

then, applying the Schur complement [25] and considering the expression of $\bar{A}$ leads to:

$$ \bar{\Theta}_x = \begin{bmatrix} \bar{A}\bar{X} + \bar{B}\bar{K}_x\bar{X} + \bar{X}\bar{A}^T + (\bar{B}\bar{K}_x\bar{X})^T + \bar{\eta}I_n & \bar{X}\bar{K}_x^T & \bar{X} \\ \bar{\delta} & -\bar{R}^{-1} & 0 \\ \bar{\delta} & \bar{\delta} & -(\bar{Q} + \frac{\bar{\delta}}{\bar{\eta}}I_n)^{-1} \end{bmatrix} \quad (12b) $$

now, let us consider the variable change $\bar{M} = \bar{K}_x\bar{X}$. The expression of (12b) can be rewritten as:

$$ \bar{\Theta}_x = \begin{bmatrix} \bar{A}\bar{X} + \bar{B}\bar{M} + \bar{X}\bar{A}^T + (\bar{B}\bar{M})^T + \bar{\eta}I_n & \bar{M}^T & \bar{X} \\ \bar{\delta} & -\bar{R}^{-1} & 0 \\ \bar{\delta} & \bar{\delta} & -(\bar{Q} + \frac{\bar{\delta}}{\bar{\eta}}I_n)^{-1} \end{bmatrix} \quad (12c) $$

finally, it is easy to find the inequality (8).

## 3.2. Output derivative feedback controller

The principle is to design a simple derivative-output feedback control law in the following form:

$$ u(t) = \bar{K}_y \dot{y} = \bar{K}_y \bar{C} \dot{x} \quad (13) $$

where $\bar{K}_y \in \mathbb{R}^{m \times p}$. Then, integrating (13) in (1), the closed-loop system becomes:

$$ x = (\bar{A} + \bar{B}\bar{K}_y\bar{C})\dot{x} + \bar{f}(\dot{x}) \quad (14) $$

The problem formulation now is to design an output derivative feedback (13) which ensures the asymptotic stability of (14) as well as the minimization of the criterion (3). Then, the following output-derivative feedback control law is proposed:

### Theorem 2

For the nonlinear RSS system (1) with derivative-output feedback control (13), the closed-loop system (14) is asymptotically stable if there exists SPD Matrices $\bar{X}$, $\bar{U}$ and $\bar{T}$ of appropriate dimensions and positives scalars $\bar{\eta}$; $\bar{\alpha}$, such that the following LMIs are feasible:

$$ \begin{bmatrix} \bar{A}\bar{X} + \bar{X}\bar{A}^T + \bar{B}\bar{U}\bar{C} + \bar{C}^T\bar{U}^T\bar{B}^T + \bar{\eta}I_n & \bar{C}^T\bar{U}^T & \bar{X} \\ \bar{\delta} & -\bar{R}^{-1} & 0 \\ \bar{\delta} & \bar{\delta} & -(\bar{Q} + \frac{\bar{\delta}}{\bar{\eta}}I_n)^{-1} \end{bmatrix} < 0 \quad (15) $$

$$ \begin{bmatrix} \bar{\alpha}I_n & (\bar{C}\bar{X} - \bar{T}\bar{C})^T \\ \bar{\delta} & \bar{\alpha}I_n \end{bmatrix} > 0 \quad (16) $$

where $\bar{K}_y = \bar{U}\bar{T}^{-1}$.

### Proof:

First, the same Lyapunov function given in the previous section is chosen. Then using the same analogy and to synthesis a controller with guaranteed cost control law, the following inequality must be satisfied:
$$ \bar{A} = \dot{x}^T [P\bar{A} + \bar{A}^T P + \bar{Q} + \bar{K}_x^T \bar{R}\bar{K}_x + \bar{\eta} PP + \frac{\delta}{\bar{\eta}} I_n] \dot{x} = \dot{x}^T \bar{\Theta}_x \dot{x} < 0 \quad (11) $$

Second, the use Schur complements on $\bar{\Theta}_x$ leads to:

$$ \bar{\Theta}_x = \begin{bmatrix} P\bar{A} + \bar{A}^T P + \bar{Q} + \bar{\eta} PP + \frac{\delta}{\bar{\eta}} I_n & \bar{K}_x^T \\ \bar{\delta} & -\bar{R}^{-1} \end{bmatrix} \quad (12) $$

introduce a new variable $\bar{X} = \bar{X}^T = P^{-1}$ and pre-and post-multiply (12) by $\begin{bmatrix} \bar{X} & 0 \\ 0 & I \end{bmatrix}$. Then, (12) is congruent to:

$$ \bar{\Theta}_x = \begin{bmatrix} \bar{A}\bar{X} + \bar{A}\bar{A}^T + \bar{X}(\bar{Q} + \frac{\delta}{\bar{\eta}}I_n)\bar{X} + \bar{\eta}I_n & \bar{X}\bar{K}_x^T \\ \bar{\delta} & -\bar{R}^{-1} \end{bmatrix} \quad (12a) $$

then, applying the Schur complement [25] and considering the expression of $\bar{A}$ leads to:

$$ \bar{\Theta}_x = \begin{bmatrix} \bar{A}\bar{X} + \bar{B}\bar{K}_x\bar{X} + \bar{X}\bar{A}^T + (\bar{B}\bar{K}_x\bar{X})^T + \bar{\eta}I_n & \bar{X}\bar{K}_x^T & \bar{X} \\ \bar{\delta} & -\bar{R}^{-1} & 0 \\ \bar{\delta} & \bar{\delta} & -(\bar{Q} + \frac{\delta}{\bar{\eta}}I_n)^{-1} \end{bmatrix} \quad (12b) $$

now, let us consider the variable change $\bar{M} = \bar{K}_x\bar{X}$. The expression of (12b) can be rewritten as:

$$ \bar{\Theta}_x = \begin{bmatrix} \bar{A}\bar{X} + \bar{B}\bar{M} + \bar{X}\bar{A}^T + (\bar{B}\bar{M})^T + \bar{\eta}I_n & \bar{M}^T & \bar{X} \\ \bar{\delta} & -\bar{R}^{-1} & 0 \\ \bar{\delta} & \bar{\delta} & -(\bar{Q} + \frac{\delta}{\bar{\eta}}I_n)^{-1} \end{bmatrix} \quad (12c) $$

finally, it is easy to find the inequality (8).

## 3.2. Output derivative feedback controller

The principle is to design a simple derivative-output feedback control law in the following form:

$$ u(t) = \bar{K}_y \dot{y} = \bar{K}_y \bar{C} \dot{x} \quad (13) $$

where $\bar{K}_y \in \mathbb{R}^{m \times p}$. Then, integrating (13) in (1), the closed-loop system becomes:

$$ x = (\bar{A} + \bar{B}\bar{K}_y\bar{C})\dot{x} + \bar{f}(\dot{x}) \quad (14) $$

The problem formulation now is to design an output derivative feedback (13) which ensures the asymptotic stability of (14) as well as the minimization of the criterion (3). Then, the following output-derivative feedback control law is proposed:

### Theorem 2

For the nonlinear RSS system (1) with derivative-output feedback control (13), the closed-loop system (14) is asymptotically stable if there exists SPD Matrices $\bar{X}$, $\bar{U}$ and $\bar{T}$ of appropriate dimensions and positives scalars $\bar{\eta}$; $\bar{\alpha}$, such that the following LMIs are feasible:

$$ \begin{bmatrix} \bar{A}\bar{X} + \bar{X}\bar{A}^T + \bar{B}\bar{U}\bar{C} + \bar{C}^T\bar{U}^T\bar{B}^T + \bar{\eta}I_n & \bar{C}^T\bar{U}^T & \bar{X} \\ \bar{\delta} & -\bar{R}^{-1} & 0 \\ \bar{\delta} & \bar{\delta} & -(\bar{Q} + \frac{\delta}{\bar{\eta}}I_n)^{-1} \end{bmatrix} < 0 \quad (15) $$

$$ \begin{bmatrix} \bar{\alpha}I_n & (\bar{C}\bar{X} - \bar{T}\bar{C})^T \\ \bar{\delta} & \bar{\alpha}I_n \end{bmatrix} > 0 \quad (16) $$

where $\bar{K}_y = \bar{U}\bar{T}^{-1}$.

### Proof:

First, the same Lyapunov function given in the previous section is chosen. Then using the same analogy and to synthesis a controller with guaranteed cost control law, the following inequality must be satisfied:
<figure><img src="image_5.png" /></figure>

$$ \begin{gathered} \bar{\Lambda} = \dot{V} + \bar{\Omega} = \dot{x}^T \bar{\Theta}_y \dot{x} < 0 \\ \Leftrightarrow \\ \dot{x}^T P x + x^T P \dot{x} < -\dot{x}^T (\bar{Q} + \bar{C}^T \bar{K}_y^T \bar{R} \bar{K}_y \bar{C}) \dot{x} \end{gathered} \tag{17} $$

according to (13)-(14) and applying the Schur complement [25] on (17), $\bar{\Theta}_y$ it becomes:

$$ \bar{\Theta}_y = \left[ \begin{array}{c} P(\bar{A} + \bar{B}\bar{K}_y\bar{C}) + (\bar{A} + \bar{B}\bar{K}_y\bar{C})P + \bar{Q} + \bar{\eta}PP + \frac{\bar{\delta}}{\bar{\eta}}I_n \bar{C}^T\bar{K}_y^T \\ \qquad \qquad -\bar{R}^{-1} \end{array} \right] < 0 \quad (18) $$

Next, according to the same steps presented in the previous section: i) use a slack variable $\bar{X} = \bar{X}^T = P^{-1}$ and pre and post multiply (18) by $\begin{bmatrix} \bar{X} & 0 \\ 0 & I \end{bmatrix}$; ii) apply the Schur complement [25]. Moreover, by applying them to (18), $\bar{\Theta}_y$ it becomes:

$$ \left[ \begin{array}{cc} \bar{A}\bar{X} + \bar{X}\bar{A}^T + \bar{B}\bar{K}_y\bar{C}\bar{X} + (\bar{B}\bar{K}_y\bar{C}\bar{X})^T + \bar{\eta}I_n & \bar{X} \\ * & -\bar{R}^{-1} \\ * & * \end{array} \right] \left[ \begin{array}{c} \bar{X} \\ 0 \\ -(\bar{Q} + \frac{\bar{\delta}}{\bar{\eta}}I_n)^{-1} \end{array} \right] < 0 \tag{19} $$

since the two variables $\bar{K}_y\bar{X}$ are multiplied directly, (19) is a BMI. For the linearization, we consider the following new variables:

$$ \begin{cases} \bar{C}\bar{X} \triangleq \bar{T}\bar{C} \\ \bar{U} \triangleq \bar{K}_y\bar{T} \end{cases} \tag{20} $$

it is easy to find the inequality (15). Then, the equality constraint is approximated with the proposed choice of a new variable [26] which must satisfy :

$$ (\bar{C}\bar{X} - \bar{T}\bar{C})(\bar{C}\bar{X} - \bar{T}\bar{C})^T < \bar{\alpha}^2 I_n \tag{21} $$

finally, this leads to inequality (16).

### Remark

A condition has been proposed by [27] in order to ensure the optimization of the criterion $\bar{J}$ in the case where the system has non-zero initial conditions as follows:

Introduce a decision variable $\bar{Y} \in \mathbb{R}^{n \times p}$ $\bar{Y} - I_n \bar{X}^{-1} I_n > 0$: Then, minimize trace $(\bar{Y})$ the subject to minimize the cost function $\bar{J}$.

## 4. ANALYSIS AND INTERPRETATION OF THE EXPERIMENTAL RESULTS

The following section is interested in the practical validation throughout a RTI by using the DSP device Arduino MEGA 2560 in HIL mode as well as the demonstration of the good performances offered by the proposed approaches with two examples.

In these examples, Arduino I/O mode is chosen where all technical implementation details are given in [12]. In a similar way to the implementation conditions given in [27], the same noise signals are considered whose amplitudes and frequencies are variable from 7 to 80 Hz and $(\pm 0.35V)$ respectively. In this section, we present the variation of the different variables in the closed-loop mode to show the stability offered by the proposed approach.

Note: All magnitudes are in Volts [V].

### 4.1. State feedback controller

The dynamic equation for the considered system is given by: $\bar{A} = \begin{bmatrix} -2.5 & 2 \\ 1.5 & -2 \end{bmatrix}$; $\bar{B} = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix}$; $\bar{f}(\dot{x}) = \begin{bmatrix} 0.25\sin(\dot{x}_1(t)) \\ 0 \end{bmatrix}$. The different parameters are as follows: $\bar{\delta} = 0.25$; $\bar{Q} = 100 * I_2$ and $\bar{R} = 0.01$.

Subsequently, by applying the **Theorem 1**, the resolution of the LMI (8) with YALMIP® gives the following solution of a guaranteed cost state feedback controller:
<figure><img src="image_5.png" /></figure>

$$ \begin{gathered} \bar{\Lambda} = \dot{V} + \bar{\Omega} = \dot{x}^T \bar{\Theta}_y \dot{x} < 0 \\ \Leftrightarrow \\ \dot{x}^T P x + x^T P \dot{x} < -\dot{x}^T (\bar{Q} + \bar{C}^T \bar{K}_y^T \bar{R} \bar{K}_y \bar{C}) \dot{x} \end{gathered} \tag{17} $$

according to (13)-(14) and applying the Schur complement [25] on (17), $\bar{\Theta}_y$ it becomes:

$$ \bar{\Theta}_y = \left[ \begin{array}{c} P(\bar{A} + \bar{B}\bar{K}_y\bar{C}) + (\bar{A} + \bar{B}\bar{K}_y\bar{C})P + \bar{Q} + \bar{\eta}PP + \frac{\bar{\delta}}{\bar{\eta}}I_n \bar{C}^T\bar{K}_y^T \\ \qquad \qquad -\bar{R}^{-1} \end{array} \right] < 0 \quad (18) $$

Next, according to the same steps presented in the previous section: i) use a slack variable $\bar{X} = \bar{X}^T = P^{-1}$ and pre and post multiply (18) by $\begin{bmatrix} \bar{X} & 0 \\ 0 & I \end{bmatrix}$; ii) apply the Schur complement [25]. Moreover, by applying them to (18), $\bar{\Theta}_y$ it becomes:

$$ \left[ \begin{array}{cc} \bar{A}\bar{X} + \bar{X}\bar{A}^T + \bar{B}\bar{K}_y\bar{C}\bar{X} + (\bar{B}\bar{K}_y\bar{C}\bar{X})^T + \bar{\eta}I_n & \bar{X}\bar{C}^T\bar{K}_y^T \\ * & -\bar{R}^{-1} \\ * & * \\ \end{array} \right] < 0 \quad (19) $$

since the two variables $\bar{K}_y\bar{X}$, are multiplied directly, (19) is a BMI. For the linearization, we consider the following new variables:

$$ \begin{cases} \bar{C}\bar{X} \triangleq \bar{T}\bar{C} \\ \bar{U} \triangleq \bar{K}_y\bar{T} \end{cases} \tag{20} $$

it is easy to find the inequality (15). Then, the equality constraint is approximated with the proposed choice of a new variable [26] which must satisfy :

$$ (\bar{C}\bar{X} - \bar{T}\bar{C})(\bar{C}\bar{X} - \bar{T}\bar{C})^T < \bar{\alpha}^2 I_n \tag{21} $$

finally, this leads to inequality (16).

### Remark

A condition has been proposed by [27] in order to ensure the optimization of the criterion $\bar{J}$ in the case where the system has non-zero initial conditions as follows:

Introduce a decision variable $\bar{Y} \in \mathbb{R}^{n \times p}$ $\bar{Y} - I_n \bar{X}^{-1} I_n > 0$: Then, minimize trace $(\bar{Y})$ the subject to minimize the cost function $\bar{J}$.

## 4. ANALYSIS AND INTERPRETATION OF THE EXPERIMENTAL RESULTS

The following section is interested in the practical validation throughout a RTI by using the DSP device Arduino MEGA 2560 in HIL mode as well as the demonstration of the good performances offered by the proposed approaches with two examples.

In these examples, Arduino I/O mode is chosen where all technical implementation details are given in [12]. In a similar way to the implementation conditions given in [27], the same noise signals are considered whose amplitudes and frequencies are variable from 7 to 80 Hz and $(\pm 0.35V)$ respectively. In this section, we present the variation of the different variables in the closed-loop mode to show the stability offered by the proposed approach.

Note: All magnitudes are in Volts [V].

### 4.1. State feedback controller

The dynamic equation for the considered system is given by: $\bar{A} = \begin{bmatrix} -2.5 & 2 \\ 1.5 & -2 \end{bmatrix}$; $\bar{B} = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix}$; $\bar{f}(\dot{x}) = \begin{bmatrix} 0.25\sin(\dot{x}_1(t)) \\ 0 \end{bmatrix}$. The different parameters are as follows: $\bar{\delta} = 0.25$; $\bar{Q} = 100 * I_2$ and $\bar{R} = 0.01$.

Subsequently, by applying the **Theorem 1**, the resolution of the LMI (8) with YALMIP® gives the following solution of a guaranteed cost state feedback controller:
$$ \bar{X} = \begin{bmatrix} 0.4738 & 0.1391 \\ 0.1391 & 0.1733 \end{bmatrix}; \bar{K}_x = \begin{bmatrix} -33.169 & -31.0135 \end{bmatrix} $$

Now, Figures 1 and 2 plot the evolution of control and state of the closed-loop systems (with $x_0 = \begin{bmatrix} 1 \\ 1.5 \end{bmatrix}$). Based on the findings of Figure 1 and Figure 2, it is clear that the states are asymptotically stable with the state derivative feedback law under the minimization of cost control.

<figure><img src="image_6.png" /><figcaption>Figure 1. Variation of u(t) with state derivative feedback control</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 2. Evolution of x(t) with state derivative feedback control</figcaption></figure>

## 4.2. Output feedback controller

The considered system is described by:

$$ \bar{A} = \begin{bmatrix} -0.1 & 0 & -0.0202 & -0.0375 \\ 0 & -0.1 & -0.0022 & -0.0375 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}; \bar{B} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0.0111 & 0 \\ -0.0007 & 0.0007 \end{bmatrix}; \bar{f}(\dot{x}) = \begin{bmatrix} 0 \\ 0.33\sin(\dot{x}_2(t)) \\ 0 \\ 0 \end{bmatrix} $$

and

$$ \bar{C} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}. $$

The different parameters are as follows: $\bar{\delta} = 0.33$; $\bar{Q} = 10^3 * I_4$ and $\bar{R} = 0.01 * I_2$. In the same way, by applying the **Theorem 2.**, the resolution of the LMIs (15)-(16) with YALMIP® gives the following solution of a guaranteed cost output feedback controller:

$$ \bar{X} = \begin{bmatrix} 1.404 & 0.014 & 3.586 & 0.239 \\ 0.014 & 1.385 & 0.005 & 3.48 \\ 3.586 & 0.005 & 4.009 & 0.042 \\ 0.239 & 3.48 & 0.042 & 3.967 \end{bmatrix}; \bar{K}_y = \begin{bmatrix} 1.452 & -0.06 \\ 0.07 & 0.793 \end{bmatrix} $$
$$ \bar{X} = \begin{bmatrix} 0.4738 & 0.1391 \\ 0.1391 & 0.1733 \end{bmatrix}; \bar{K}_x = \begin{bmatrix} -33.169 & -31.0135 \end{bmatrix} $$

Now, Figures 1 and 2 plot the evolution of control and state of the closed-loop systems (with $x_0 = \begin{bmatrix} 1 \\ 1.5 \end{bmatrix}$). Based on the findings of Figure 1 and Figure 2, it is clear that the states are asymptotically stable with the state derivative feedback law under the minimization of cost control.

<figure><img src="image_6.png" /><figcaption>Figure 1. Variation of u(t) with state derivative feedback control</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 2. Evolution of x(t) with state derivative feedback control</figcaption></figure>

## 4.2. Output feedback controller

The considered system is described by:

$$ \bar{A} = \begin{bmatrix} -0.1 & 0 & -0.0202 & -0.0375 \\ 0 & -0.1 & -0.0022 & -0.0375 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}; \bar{B} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0.0111 & 0 \\ -0.0007 & 0.0007 \end{bmatrix}; \bar{f}(\dot{x}) = \begin{bmatrix} 0 \\ 0.33\sin(\dot{x}_2(t)) \\ 0 \\ 0 \end{bmatrix} $$

and

$$ \bar{C} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}. $$

The different parameters are as follows: $\bar{\delta} = 0.33$; $\bar{Q} = 10^3 * I_4$ and $\bar{R} = 0.01 * I_2$. In the same way, by applying the **Theorem 2.**, the resolution of the LMIs (15)-(16) with YALMIP® gives the following solution of a guaranteed cost output feedback controller:

$$ \bar{X} = \begin{bmatrix} 1.404 & 0.014 & 3.586 & 0.239 \\ 0.014 & 1.385 & 0.005 & 3.48 \\ 3.586 & 0.005 & 4.009 & 0.042 \\ 0.239 & 3.48 & 0.042 & 3.967 \end{bmatrix}; \bar{K}_y = \begin{bmatrix} 1.452 & -0.06 \\ 0.07 & 0.793 \end{bmatrix} $$
<figure><img src="image_8.png" /></figure>

Next, Figure 3 and Figure 4 show the evolution of control and state of the closed-loop systems with an output derivative feedback control (with $x_0 = \begin{bmatrix} 0.1 \\ 0.1 \\ 0.25 \\ 0.25 \end{bmatrix}$).

<figure><img src="image_9.png" /><figcaption>Figure 3. Evolution of $u(t)$ with output derivative feedback control</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 4. Evolution of $x(t)$ with output derivative feedback control</figcaption></figure>

It is clear from Figure 3 and Figure 4 that the designed controller with output derivative feedback reduce the effect of destabilization caused by the amplitudes and perturbations with a guarantee of closed loop stability of the system. This results in minimization of nonstandard cost control where the evolution of the performance index $\bar{J}_0$ [22] is described by:

$$ \bar{J}_0 = E\{\int_0^\infty (\dot{x}^T \bar{Q} \dot{x} + u^T(t) \bar{R}) dt\} + E\{x_0^T \bar{X} x_0\} $$

The value of $\bar{J}_0$ decreases from 3,58791.10⁵ to 3,49662.10⁵ (as given in **Remark 1**). As can be observed from all Figures 1-4, the proposed unified controllers with state or output feedback in RSS form are less sensitive to externals perturbations (noises).

## 5. CONCLUSION

This paper presents a state and output feedback design methodology for a class of LNS in RSS. The proposed control law ensures the minimization of non-standard cost control unsing the Lyapunov theory to guarentee the asymptotic stability of the closed-loop system. The proposed controllers are deduced from a convex resolution problem with the use of LMIs and some lemmas. RTI with ARDUINO MEGA 2560 board used as a real-time emulator in HIL mode further demonstrates the performances of stabilization offered by the proposed approach. Expanding proposed robust synthesis methods with uncertain parameters and the generalization of the proposed method for one-sided Lipschitz systems might be a direction for further studies.

## ACKNOWLEDGEMENTS

The authors would like to thank “l'Institut Supérieur des Systèmes Industriels de Gabes- Tunisie” and “L'Ecole Supérieure d'Electronique d'Ouest-France” for the realization of the practical tests.
<figure><img src="image_8.png" /></figure>

Next, Figure 3 and Figure 4 show the evolution of control and state of the closed-loop systems with an output derivative feedback control (with $x_0 = \begin{bmatrix} 0.1 \\ 0.1 \\ 0.25 \\ 0.25 \end{bmatrix}$).

<figure><img src="image_9.png" /><figcaption>Figure 3. Evolution of $u(t)$ with output derivative feedback control</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 4. Evolution of $x(t)$ with output derivative feedback control</figcaption></figure>

It is clear from Figure 3 and Figure 4 that the designed controller with output derivative feedback reduce the effect of destabilization caused by the amplitudes and perturbations with a guarantee of closed loop stability of the system. This results in minimization of nonstandard cost control where the evolution of the performance index $\bar{J}_0$ [22] is described by:

$$ \bar{J}_0 = E\{\int_0^\infty (\dot{x}^T \bar{Q} \dot{x} + u^T(t) \bar{R}) dt\} + E\{x_0^T \bar{X} x_0\} $$

The value of $\bar{J}_0$ decreases from 3,58791.10⁵ to 3,49662.10⁵ (as given in **Remark 1**). As can be observed from all Figures 1-4, the proposed unified controllers with state or output feedback in RSS form are less sensitive to externals perturbations (noises).

## 5. CONCLUSION

This paper presents a state and output feedback design methodology for a class of LNS in RSS. The proposed control law ensures the minimization of non-standard cost control unsing the Lyapunov theory to guarentee the asymptotic stability of the closed-loop system. The proposed controllers are deduced from a convex resolution problem with the use of LMIs and some lemmas. RTI with ARDUINO MEGA 2560 board used as a real-time emulator in HIL mode further demonstrates the performances of stabilization offered by the proposed approach. Expanding proposed robust synthesis methods with uncertain parameters and the generalization of the proposed method for one-sided Lipschitz systems might be a direction for further studies.

## ACKNOWLEDGEMENTS

The authors would like to thank “l'Institut Superieur des Systemes Industriels de Gabes- Tunisie” and “L'Ecole Superieur d'Electronique d'Ouest-France” for the realization of the practical tests.
<figure><img src="image_11.png" /></figure>

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[21] Y.-W. Tseng, "Sliding Mode Control with State Derivative Feedback in Novel Reciprocal State Space Form," in *Studies in Computational Intelligence*, vol. 635, 2016, pp. 159–184.

[22] T. H. S. Abdelaziz, "Optimal control using derivative feedback for linear systems," *Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering*, vol. 224, no. 2, pp. 185–202, Mar. 2010, doi: 10.1243/09596518JSCE886.

[23] T. H. S. Abdelaziz and M. Valášek, "State derivative feedback by LQR for linear time-invariant systems," *IFAC Proceedings Volumes (IFAC-PapersOnline)*, vol. 16, no. 1, pp. 435–440, 2005, doi: 10.3182/20050703-6-cz-1902.00934.

[24] S. K. Kwak, G. Washington, and R. K. Yedavalli, "Acceleration-based vibration control of distributed parameter systems using the 'reciprocal state-space framework,'" *Journal of Sound and Vibration*, vol. 251, no. 3, pp. 543–557, Mar. 2002, doi: 10.1006/jsvi.2001.3842.

[25] Y.-W. Tseng and J.-G. Hsieh, "Optimal Control for a Family of Systems in Novel State Derivative Space Form with Experiment in a Double Inverted Pendulum System," *Abstract and Applied Analysis*, vol. 2013, pp. 1–8, 2013, doi: 10.1155/2013/715026.

[26] A. Thabet, G. B. H. Frej, N. Gasmi, and M. Boutayeb, "Feedback stabilization for one sided Lipschitz nonlinear systems in reciprocal state space: Synthesis and experimental validation," *Journal of Electrical Engineering*, vol. 70, no. 5, pp. 412–417, Sep. 2019, doi: 10.2478/jee-2019-0074.

[27] A. Thabet, N. Gasmi, G. B. H. Frej, and M. Boutayeb, "Sliding Mode Control for Lipschitz Nonlinear Systems in Reciprocal State Space: Synthesis and Experimental Validation," *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 68, no. 3, pp. 948–952, Mar. 2021, doi: 10.1109/TCSII.2020.3018016.

[28] I. R. Petersen, "A stabilization algorithm for a class of uncertain linear systems," *Systems & Control Letters*, vol. 8, no. 4, pp. 351–
<figure><img src="image_11.png" /></figure>

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[2] Y. Su and C. Zheng, "A Simple Unified Control for Output Feedback Finite-Time Stabilization of Uncertain Planar Systems Without Controllable/O sabervale Linearization," *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 68, no. 1, pp. 326–330, Jan. 2021, doi: 10.1109/TCSII.2020.2996208.

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[4] H. Min, N. Duan, X. Yu, and S. Fei, "Tracking-Based Control for Constrained Nonlinear Systems Under Parametric Uncertainties," *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 68, no. 3, pp. 973–977, Mar. 2021, doi: 10.1109/TCSII.2020.3010100.

[5] Z. Massaq, A. Abounada, and M. Ramzi, "Fuzzy and predictive control of a photovoltaic pumping system based on three-level boost converter," *Bulletin of Electrical Engineering and Informatics*, vol. 10, no. 3, pp. 1183–1192, Jun. 2021, doi: 10.11591/eei.v10i3.2605.

[6] S. He, J. Song, and F. Liu, "Robust Finite-Time Bounded Controller Design of Time-Delay Conic Nonlinear Systems Using Sliding Mode Control Strategy," *IEEE Transactions on Systems, Man, and Cybernetics: Systems*, vol. 48, no. 11, pp. 1863–1873, Nov. 2018, doi: 10.1109/TSMC.2017.2695483.

[7] A. Kirad, S. Groini, and Y. Soufi, "Improved sensorless backstepping controller using extended Kalman filter of a permanent magnet synchronous machine," *Bulletin of Electrical Engineering and Informatics*, vol. 11, no. 2, pp. 658–671, Apr. 2022, doi: 10.11591/eei.v11i2.3560.

[8] D.-P. Li, Y.-J. Liu, S. Tong, C. L. P. Chen, and D.-J. Li, "Neural Networks-Based Adaptive Control for Nonlinear State Constrained Systems With Input Delay," *IEEE Transactions on Cybernetics*, vol. 49, no. 4, pp. 1249–1258, Apr. 2019, doi: 10.1109/TCYB.2018.2799683.

[9] M. El Malah, A. Ba-Razzouk, E. Abdelmounim, and M. Madark, "Nonlinear control for an optimized grid connection system of renewable energy resources," *Bulletin of Electrical Engineering and Informatics*, vol. 10, no. 5, pp. 2355–2366, Oct. 2021, doi: 10.11591/eei.v10i5.3151.

[10] H. Liu, T. Zhang, and X. Tian, "Continuous output-feedback finite-time control for a class of second-order nonlinear systems with disturbances," *International Journal of Robust and Nonlinear Control*, vol. 26, no. 2, pp. 218–234, Jan. 2016, doi: 10.1002/rmc.3305.

[11] N. Gasmi, M. Boutayeb, A. Thabet, and M. Aoun, "Sliding Window Based Nonlinear H∞ Filtering: Design and Experimental Results," *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 66, no. 2, pp. 302–306, Feb. 2019, doi: 10.1109/TCSII.2018.2859484.

[12] A. Thabet, G. B. H. Frej and M. Boutayeb, "Observer-Based Feedback Stabilization for Lipschitz Nonlinear Systems With Extension to H∞ Performance Analysis: Design and Experimental Results," in *IEEE Transactions on Control Systems Technology*, vol. 26, no. 1, pp. 321-328, Jan. 2018, doi: 10.1109/TCST.2017.2669143.

[13] V. T. Ha, T. L. Nguyen, and V. T. Ha, "Hardware-in-the-loop based comparative analysis of speed controllers for a two-mass system using an induction motor drive with ideal stator current performance," *Bulletin of Electrical Engineering and Informatics*, vol. 10, no. 2, pp. 569–579, Apr. 2021, doi: 10.11591/eei.v10i2.2370.

[14] N. Kaewpraeak and W. Assawinchaichote, "H∞ Fuzzy State-Feedback Control Plus State-Derivative-Feedback Control Synthesis for Photovoltaic Systems," *Asian Journal of Control*, vol. 18, no. 4, pp. 1441–1452, Jul. 2016, doi: 10.1002/asjc.1233.

[15] E. K. Boukas, "Static output feedback control for linear descriptor systems: LMI approach," in *IEEE International Conference Mechatronics and Automation*, 2005, vol. 3, pp. 1230–1234, doi: 10.1109/ICMA.2005.1626729.

[16] E. R. P. da Silva, E. Assuncao, M. C. M. Teixeira, and R. Cardim, "Robust controller implementation via state-derivative feedback in an active suspension system subjected to fault," in *2013 Conference on Control and Fault-Tolerant Systems (SysTol)*, Oct. 2013, pp. 752–757, doi: 10.1109/SysTol.2013.6693904.

[17] E. R. P. da Silva, E. Assunção, M. C. M. Teixeira, F. A. Faria, and L. F. S. Buzachero, "Parameter-dependent Lyapunov functions for state-derivative feedback control in polytopic linear systems," *International Journal of Control*, vol. 84, no. 8, pp. 1377–1386, Aug. 2011, doi: 10.1080/00207179.2011.598948.

[18] Y.-W. Tseng, "Control Design for System with Lipschitz Nonlinearity of State Derivative Variables in Reciprocal State Space Form," Jul. 2015, doi: 10.17758/ER715217.

[19] A. Thabet, "Adaptive-state feedback control for Lipschitz nonlinear systems in reciprocal-state space: Design and experimental results," *Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering*, vol. 233, no. 2, pp. 144–152, Feb. 2019, doi: 10.1177/0959651818786374.

[20] T. H. S. Abdelaziz and M. Valášek, "Pole-placement for SISO linear systems by state-derivative feedback," *IEE Proceedings - Control Theory and Applications*, vol. 151, no. 4, pp. 377–385, Jul. 2004, doi: 10.1049/ip-cta:20040660.

[21] Y.-W. Tseng, "Sliding Mode Control with State Derivative Feedback in Novel Reciprocal State Space Form," in *Studies in Computational Intelligence*, vol. 635, 2016, pp. 159–184.

[22] T. H. S. Abdelaziz, "Optimal control using derivative feedback for linear systems," *Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering*, vol. 224, no. 2, pp. 185–202, Mar. 2010, doi: 10.1243/09596518JSCE886.

[23] T. H. S. Abdelaziz and M. Valášek, "State derivative feedback by LQR for linear time-invariant systems," *IFAC Proceedings Volumes (IFAC-PapersOnline)*, vol. 16, no. 1, pp. 435–440, 2005, doi: 10.3182/20050703-6-cz-1902.00934.

[24] S. K. Kwak, G. Washington, and R. K. Yedavalli, "Acceleration-based vibration control of distributed parameter systems using the 'reciprocal state-space framework,'" *Journal of Sound and Vibration*, vol. 251, no. 3, pp. 543–557, Mar. 2002, doi: 10.1006/jsvi.2001.3842.

[25] Y.-W. Tseng and J.-G. Hsieh, "Optimal Control for a Family of Systems in Novel State Derivative Space Form with Experiment in a Double Inverted Pendulum System," *Abstract and Applied Analysis*, vol. 2013, pp. 1–8, 2013, doi: 10.1155/2013/715026.

[26] A. Thabet, G. B. H. Frej, N. Gasmi, and M. Boutayeb, "Feedback stabilization for one sided Lipschitz nonlinear systems in reciprocal state space: Synthesis and experimental validation," *Journal of Electrical Engineering*, vol. 70, no. 5, pp. 412–417, Sep. 2019, doi: 10.2478/jee-2019-0074.

[27] A. Thabet, N. Gasmi, G. B. H. Frej, and M. Boutayeb, "Sliding Mode Control for Lipschitz Nonlinear Systems in Reciprocal State Space: Synthesis and Experimental Validation," *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 68, no. 3, pp. 948–952, Mar. 2021, doi: 10.1109/TCSII.2020.3018016.

[28] I. R. Petersen, "A stabilization algorithm for a class of uncertain linear systems," *Systems & Control Letters*, vol. 8, no. 4, pp. 351–
<figure><img src="image_12.png" /></figure>

## BIOGRAPHIES OF AUTHORS

<figure><img src="image_13.png" /></figure>

**Assem Thabet** received his Electrical Engineer degree from the National Engineering School of Gabes, Tunisia, in 2006, and his Master's and Ph.D. degrees in automatic control from the University of Gabes, Tunisia, in 2008 and 2012, respectively. Since 2012, he is an Associate Professor at the University of Gabes - Tunisia and a member of the MACS Laboratory (Modeling, Analysis, and Control of Systems) of The National Engineering School of Gabes. His research interests are identification, state estimation, and control of dynamical systems. He can be contacted at email: sem.thabet@yahoo.fr and assem.thabet@issig.rnu.tn.

<figure><img src="image_14.png" /></figure>

**Ghazi Bel Haj Frej** received his degree in electrical and Automatic engineering and Ph.D. degrees in automatic control from the National Engineering School of Gabes, Tunisia, in 2013 and 2017. Since 2012 is with the Centre de Recherche en Automatique de Nancy, University of Lorraine, France, and the MACS Laboratory (Modeling, Analysis, and Control of Systems), National Engineering School of Gabes, University of Gabes, Tunisia. His research interests are identification, state estimation, and robust control of dynamical systems. He can be contacted at email: ghazi.bel_haj frej@bordeaux-inp.fr.

<figure><img src="image_15.png" /></figure>

**Noussaiba Gasmi** received her degree in electrical and Automatic engineering and Ph. D. degrees in automatic control from the National Engineering School of Gabes, Tunisia, in 2014 and 2018. Since 2014, she is with the Centre de Recherche en Automatique de Nancy, University of Lorraine, France, and with the MACS Laboratory (Modeling, Analysis, and Control of Systems), National Engineering School of Gabes, University of Gabes, Tunisia. Her research interests are identification, state estimation, and robust control of dynamical systems. She can be contacted at email: noussaiba.gasmi@ecam-rennes.fr.

<figure><img src="image_16.png" /></figure>

**Karim Mansouri** received his degree in electrical and Automatic engineering and Ph.D. degrees in electrical engineering from the National Engineering School of Gabes, Tunisia, in 2008 and 2018 respectively. Since 2019, he is member of the research team AGE-ESEO-Tech, ESEO-Angers-France. His research interests are power systems, control, renewable energy, and control of dynamic systems. He can be contacted at email: karim.mansouri@eseo.fr.

<figure><img src="image_17.png" /></figure>

**Eric Chauveau** received his degree in electrical engineering in 1996 from ESAIGELEC; DEA degree in 1996 and Ph.D. degrees in electrical and electronic in 2001 from Nante University-Frane. Since 2007, he is with Laboratoire IREENA - ESEO-Tech, Université de Nantes-France. His research interests are power systems, control, renewable energy, and control of dynamic systems. He can be contacted at email: eric.chauveau@eseo.fr.
<figure><img src="image_12.png" /></figure>

## BIOGRAPHIES OF AUTHORS

<figure><img src="image_13.png" /></figure>

**Assem Thabet** received his Electrical Engineer degree from the National Engineering School of Gabes, Tunisia, in 2006, and his Master's and Ph.D. degrees in automatic control from the University of Gabes, Tunisia, in 2008 and 2012, respectively. Since 2012, he is an Associate Professor at the University of Gabes - Tunisia and a member of the MACS Laboratory (Modeling, Analysis, and Control of Systems) of The National Engineering School of Gabes. His research interests are identification, state estimation, and control of dynamical systems. He can be contacted at email: sem.thabet@yahoo.fr and assem.thabet@issig.rnu.tn.

<figure><img src="image_14.png" /></figure>

**Ghazi Bel Haj Frej** received his degree in electrical and Automatic engineering and Ph.D. degrees in automatic control from the National Engineering School of Gabes, Tunisia, in 2013 and 2017. Since 2012 is with the Centre de Recherche en Automatique de Nancy, University of Lorraine, France, and the MACS Laboratory (Modeling, Analysis, and Control of Systems), National Engineering School of Gabes, University of Gabes, Tunisia. His research interests are identification, state estimation, and robust control of dynamical systems. He can be contacted at email: ghazi.bel_haj frej@bordeaux-inp.fr.

<figure><img src="image_15.png" /></figure>

**Noussaiba Gasmi** received her degree in electrical and Automatic engineering and Ph. D. degrees in automatic control from the National Engineering School of Gabes, Tunisia, in 2014 and 2018. Since 2014, she is with the Centre de Recherche en Automatique de Nancy, University of Lorraine, France, and with the MACS Laboratory (Modeling, Analysis, and Control of Systems), National Engineering School of Gabes, University of Gabes, Tunisia. Her research interests are identification, state estimation, and robust control of dynamical systems. She can be contacted at email: noussaiba.gasmi@ecam-rennes.fr.

<figure><img src="image_16.png" /></figure>

**Karim Mansouri** received his degree in electrical and Automatic engineering and Ph.D. degrees in electrical engineering from the National Engineering School of Gabes, Tunisia, in 2008 and 2018 respectively. Since 2019, he is member of the research team AGE-ESEO-Tech, ESEO-Angers-France. His research interests are power systems, control, renewable energy, and control of dynamic systems. He can be contacted at email: karim.mansouri@eseo.fr.

<figure><img src="image_17.png" /></figure>

**Eric Chauveau** received his degree in electrical engineering in 1996 from ESAIGELEC; DEA degree in 1996 and Ph.D. degrees in electrical and electronic in 2001 from Nante University-Frane. Since 2007, he is with Laboratoire IREENA - ESEO-Tech, Université de Nantes-France. His research interests are power systems, control, renewable energy, and control of dynamic systems. He can be contacted at email: eric.chauveau@eseo.fr.