# LMI-Based Robust Stabilization of a Class of Input-Constrained Uncertain Nonlinear Systems with Application to a Helicopter Model

Authors: Hassène Gritli
Journal: Complexity
 (2020)

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## Abstract

This paper is concerned with the robust stabilization of a class of continuous-time nonlinear systems, with an application to the pitch dynamics of a simple helicopter model, via an affine state-feedback control law using the linear matrix inequality (LMI) approach. The nonlinear dynamics is subject to norm-bounded parametric uncertainties and disturbances. In addition, the problem of actuator nonlinearity is addressed by considering the saturation effect of the control law. We demonstrate first that the synthesis problem of the saturated controller is expressed in terms of bilinear matrix inequalities (BMIs). Thanks to the Schur complement lemma and the matrix inversion lemma, we convert these BMIs into LMIs allowing the simultaneous computation of the two gains of the affine controller. Furthermore, we address in this work the estimation problem of the domain of attraction using the invariant set concept. This is solved by computing the largest attractive invariant ellipsoid. Compared with previous works, the research procedure of such ellipsoidal set is achieved in a single step with a reduced number of LMI constraints and then with less conservative conditions. A portfolio of numerical results is presented. The effectiveness and robustness of the proposed saturated controller in the stabilization of the adopted helicopter pitch model toward parametric uncertainties and disturbances are illustrated through simulation results.


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## Body

## 1. Introduction

### 1.1. Background and Literature Review

Mechatronics, such as aircraft, spacecrafts, launch vehicles, unmanned autonomous vehicles, missiles, walking robots, robot manipulators, electronic vehicles, and unmanned aerial vehicles, has played a very important role in modern industry-related applications. Nowadays, there is an ever-increasing demand of advanced control strategies for mechatronic systems with enhanced performances.It is known, on the one hand, that almost all existing physical and mechatronic systems unavoidably include uncertainties and disturbances due to inaccurate modeling, measurement errors, exterior conditions, or parameter variations. The presence of uncertainties may cause instability and bad performances on a controlled system. Thus, considerable efforts have been assigned to the robust stability and stabilization of linear and nonlinear systems with parametric uncertainties. For a recent literature, we refer the readers to [1–8]. Two types of parametric uncertainties are very often considered in systems: norm-bounded uncertainty and polytopic uncertainty. In recent years, the linear matrix inequality (LMI) technique [9] has been widely used to solve the robust control for uncertain linear and nonlinear systems with polytopic uncertain parameters and norm-bounded uncertain parameters. However, most control synthesis problems cannot be written in a LMI form. However, they are written in terms of a more general form known as a bilinear matrix inequality (BMI), which is usually not exploitable numerically to solve. For some BMI problems with simple bilinearities, the YALMIP toolbox [10] was used to solve such problems. There are several approaches and different relaxed synthesis conditions proposed for conducting the BMIs into LMIs, such as in [5–7, 9, 11–16], just to mention a few.On the other hand, the actuator saturation nonlinearity exists commonly in practical control systems. Indeed, the signal amplitude that an actuator can deliver is usually limited by physical or safety constraints. Thus, the neglect of the actuator constraints in the controller design may cause instability of the closed-loop system [17–19]. The design of controllers for continuous-time and discrete-time systems by taking into consideration the actuator saturation has been extensively studied in the past few decades [17–36]. Some works have investigated the problem of uncertainties and actuator saturation in mechatronic systems such as the vehicle lateral dynamics [37], the vehicle active suspension systems [38–41], the rigid spacecraft [42, 43], the inertia wheel inverted pendulum [12], the supercavitating vehicle [44], the flexible robotic manipulator [45], and the marine surface vessel [46], among others.The concept of attractive invariant sets has been generally proposed for linear systems involving actuator saturation [5, 20, 28, 29, 36, 44, 47–51]. Moreover, through such concept, numerous studies have focused on the characterization of the maximum attraction set for linear systems subject to saturated linear feedback. Thus, some research works focused on writing the saturation function of the control input as a linear convex combination of some constrained variables [18, 20, 27, 36, 47, 48, 50–53]. This approach leads to sufficient conditions in terms of a very large number of LMIs that are difficult to solve numerically [27–29]. The common goal of almost all of these previous research works is to find the largest attractive invariant ellipsoidal set in order to provide an estimation of the domain of attraction for initial conditions.In addition, it is well recognized that most practical control systems are inherently nonlinear. The number of available results by taking the actuator saturation nonlinearity into account in the design and analysis of nonlinear control systems is still limited. Because of the difficulty of the problem itself, most research works have been focused on particular classes of nonlinear systems to design saturated controllers [5, 25, 26]. Furthermore, the simultaneous presence of parametric uncertainties and actuator saturation nonlinearity in physical systems has led many authors to combine the techniques of the robust control theory and those of constrained control [5, 54].


### 1.2. Objective of the Paper

In this paper, we are interested in the design of a robust controller to stabilize a class of nonlinear systems, for which the zero state is not the equilibrium point. As a motivation application, we consider the pitch dynamics of a simple helicopter model [55–57]. In fact, such helicopter pitch model has been considered as a testbed used in order to develop new advanced control strategies. Authors in [56] approximated the pitch dynamics of the helicopter with a set of piecewise affine stochastic systems. The proposed stochastic affine feedback controller is designed for the approximated model and implemented on the main nonlinear system. Authors in [57] designed an observer-based minimum variance control for the same objective by taking into consideration norm-bounded uncertainties. Such uncertainties arise in the form of the difference between the actual nonlinear dynamics and its piecewise-affine approximation. Furthermore, authors in [55] approximated also the simplified pitch model of the helicopter via a piecewise affine system. In these works [55–57], the saturation effect in the actuators was not taken into consideration in the control design.In the present work, we propose an affine state-feedback control law for the adopted class of nonlinear systems, with a particular interest to the pitch dynamics of the helicopter model, where all the states are assumed to be available for direct measurement. Our main objective is to control, via such affine feedback control law, the adopted nonlinear systems to the zero state. Moreover, we consider the problem of norm-bounded parametric uncertainties and also the problem of external disturbances. Furthermore, we take into consideration the problem of actuator saturation in the design of the robust affine state-feedback control law. Thus, the main role of such control law is to asymptotically bring the trajectory of the nonlinear system to the zero state even if it is subject to the parametric uncertainties, the disturbances, and the actuator saturation. In addition, in this work, the problem of searching for appropriate feedback gain matrices of the affine state-feedback control law is realized also by solving the problem of estimating the largest attractive invariant set. Our design methodology of the saturated stabilizing affine state-feedback control law is based on the framework of LMIs. We show at first that the affine state-feedback controller is designed by solving an optimization problem subject to three BMI constraints. Then, by using some judicious congruence transformations and some technical lemmas, we convert these BMIs into three LMI constraints. In the end of this work, we demonstrate through simulations the effectiveness of the proposed affine state-feedback control law in the robust stabilization of the pitch dynamics of the helicopter model while the largest attractive invariant set is guaranteed.


### 1.3. Contributions and Innovations

The main contributions and innovations of the work in this paper can be summarized as follows:(1)
The problem of robust stabilization of a class of nonlinear systems, with an application to the pitch dynamics of a simple helicopter model, under norm-bounded parametric uncertainties, external disturbance, and input saturation nonlinearity, is considered using the concept of the invariant set. The main objective is to control the disturbed uncertain nonlinear system to the zero state, which is not the equilibrium point in the open loop.(2)
A saturated affine state-feedback controller is designed based on the LMI approach. To the best of the authors’ knowledge, such problem was not developed in the literature. Only the linear state/output feedback controllers have been considered.(3)
A transformation of the BMI stability condition, related to the input saturation problem, into an LMI condition is achieved using the matrix inversion lemma and the Schur complement lemma.(4)
An estimation of the ellipsoidal region of attraction for the nonlinear system under study is also realized. Compared with previous works [20, 47–50, 52, 58], our research procedure of the largest invariant attractive ellipsoid is achieved in only one step with a reduced number of LMI constraints.


### 1.4. Structure of the Paper

The rest of this paper is organized as follows. In Section2, we present first some technical lemmas that will be used in this work, the adopted class of the nonlinear systems, and the problem formulation. The simple helicopter model, as a motivation application, and its pitch dynamics are also described in this section. The design of the robust affine state-feedback controller under saturation, based on the LMI approach, is discussed in Section 3. Transformation of the BMIs into LMIs is also realized in this section. The problem of computation/estimation of the domain of attraction (the largest attractive ellipsoid) is addressed in Section 4. Simulation results and some comparisons are presented in Section 5. Finally, concluding remarks and future works are drawn in Section 6.


### 1.5. Notations

Throughout this paper,AT represents the transpose of A, the symbol ⋆ in matrix inequality denotes the symmetric term of the matrix, for example, XY⋆Z=XYYTZ and X+⋆=X+XT, X>0 <0 means X is a symmetric positive (negative) definite matrix, and diagA,B,…,Z represents a diagonal matrix. Moreover, O and ℐ denote the zero matrix and the identity matrix, respectively, with appropriate dimensions.


## 1.1. Background and Literature Review

Mechatronics, such as aircraft, spacecrafts, launch vehicles, unmanned autonomous vehicles, missiles, walking robots, robot manipulators, electronic vehicles, and unmanned aerial vehicles, has played a very important role in modern industry-related applications. Nowadays, there is an ever-increasing demand of advanced control strategies for mechatronic systems with enhanced performances.It is known, on the one hand, that almost all existing physical and mechatronic systems unavoidably include uncertainties and disturbances due to inaccurate modeling, measurement errors, exterior conditions, or parameter variations. The presence of uncertainties may cause instability and bad performances on a controlled system. Thus, considerable efforts have been assigned to the robust stability and stabilization of linear and nonlinear systems with parametric uncertainties. For a recent literature, we refer the readers to [1–8]. Two types of parametric uncertainties are very often considered in systems: norm-bounded uncertainty and polytopic uncertainty. In recent years, the linear matrix inequality (LMI) technique [9] has been widely used to solve the robust control for uncertain linear and nonlinear systems with polytopic uncertain parameters and norm-bounded uncertain parameters. However, most control synthesis problems cannot be written in a LMI form. However, they are written in terms of a more general form known as a bilinear matrix inequality (BMI), which is usually not exploitable numerically to solve. For some BMI problems with simple bilinearities, the YALMIP toolbox [10] was used to solve such problems. There are several approaches and different relaxed synthesis conditions proposed for conducting the BMIs into LMIs, such as in [5–7, 9, 11–16], just to mention a few.On the other hand, the actuator saturation nonlinearity exists commonly in practical control systems. Indeed, the signal amplitude that an actuator can deliver is usually limited by physical or safety constraints. Thus, the neglect of the actuator constraints in the controller design may cause instability of the closed-loop system [17–19]. The design of controllers for continuous-time and discrete-time systems by taking into consideration the actuator saturation has been extensively studied in the past few decades [17–36]. Some works have investigated the problem of uncertainties and actuator saturation in mechatronic systems such as the vehicle lateral dynamics [37], the vehicle active suspension systems [38–41], the rigid spacecraft [42, 43], the inertia wheel inverted pendulum [12], the supercavitating vehicle [44], the flexible robotic manipulator [45], and the marine surface vessel [46], among others.The concept of attractive invariant sets has been generally proposed for linear systems involving actuator saturation [5, 20, 28, 29, 36, 44, 47–51]. Moreover, through such concept, numerous studies have focused on the characterization of the maximum attraction set for linear systems subject to saturated linear feedback. Thus, some research works focused on writing the saturation function of the control input as a linear convex combination of some constrained variables [18, 20, 27, 36, 47, 48, 50–53]. This approach leads to sufficient conditions in terms of a very large number of LMIs that are difficult to solve numerically [27–29]. The common goal of almost all of these previous research works is to find the largest attractive invariant ellipsoidal set in order to provide an estimation of the domain of attraction for initial conditions.In addition, it is well recognized that most practical control systems are inherently nonlinear. The number of available results by taking the actuator saturation nonlinearity into account in the design and analysis of nonlinear control systems is still limited. Because of the difficulty of the problem itself, most research works have been focused on particular classes of nonlinear systems to design saturated controllers [5, 25, 26]. Furthermore, the simultaneous presence of parametric uncertainties and actuator saturation nonlinearity in physical systems has led many authors to combine the techniques of the robust control theory and those of constrained control [5, 54].


## 1.2. Objective of the Paper

In this paper, we are interested in the design of a robust controller to stabilize a class of nonlinear systems, for which the zero state is not the equilibrium point. As a motivation application, we consider the pitch dynamics of a simple helicopter model [55–57]. In fact, such helicopter pitch model has been considered as a testbed used in order to develop new advanced control strategies. Authors in [56] approximated the pitch dynamics of the helicopter with a set of piecewise affine stochastic systems. The proposed stochastic affine feedback controller is designed for the approximated model and implemented on the main nonlinear system. Authors in [57] designed an observer-based minimum variance control for the same objective by taking into consideration norm-bounded uncertainties. Such uncertainties arise in the form of the difference between the actual nonlinear dynamics and its piecewise-affine approximation. Furthermore, authors in [55] approximated also the simplified pitch model of the helicopter via a piecewise affine system. In these works [55–57], the saturation effect in the actuators was not taken into consideration in the control design.In the present work, we propose an affine state-feedback control law for the adopted class of nonlinear systems, with a particular interest to the pitch dynamics of the helicopter model, where all the states are assumed to be available for direct measurement. Our main objective is to control, via such affine feedback control law, the adopted nonlinear systems to the zero state. Moreover, we consider the problem of norm-bounded parametric uncertainties and also the problem of external disturbances. Furthermore, we take into consideration the problem of actuator saturation in the design of the robust affine state-feedback control law. Thus, the main role of such control law is to asymptotically bring the trajectory of the nonlinear system to the zero state even if it is subject to the parametric uncertainties, the disturbances, and the actuator saturation. In addition, in this work, the problem of searching for appropriate feedback gain matrices of the affine state-feedback control law is realized also by solving the problem of estimating the largest attractive invariant set. Our design methodology of the saturated stabilizing affine state-feedback control law is based on the framework of LMIs. We show at first that the affine state-feedback controller is designed by solving an optimization problem subject to three BMI constraints. Then, by using some judicious congruence transformations and some technical lemmas, we convert these BMIs into three LMI constraints. In the end of this work, we demonstrate through simulations the effectiveness of the proposed affine state-feedback control law in the robust stabilization of the pitch dynamics of the helicopter model while the largest attractive invariant set is guaranteed.


## 1.3. Contributions and Innovations

The main contributions and innovations of the work in this paper can be summarized as follows:(1)
The problem of robust stabilization of a class of nonlinear systems, with an application to the pitch dynamics of a simple helicopter model, under norm-bounded parametric uncertainties, external disturbance, and input saturation nonlinearity, is considered using the concept of the invariant set. The main objective is to control the disturbed uncertain nonlinear system to the zero state, which is not the equilibrium point in the open loop.(2)
A saturated affine state-feedback controller is designed based on the LMI approach. To the best of the authors’ knowledge, such problem was not developed in the literature. Only the linear state/output feedback controllers have been considered.(3)
A transformation of the BMI stability condition, related to the input saturation problem, into an LMI condition is achieved using the matrix inversion lemma and the Schur complement lemma.(4)
An estimation of the ellipsoidal region of attraction for the nonlinear system under study is also realized. Compared with previous works [20, 47–50, 52, 58], our research procedure of the largest invariant attractive ellipsoid is achieved in only one step with a reduced number of LMI constraints.


## 1.4. Structure of the Paper

The rest of this paper is organized as follows. In Section2, we present first some technical lemmas that will be used in this work, the adopted class of the nonlinear systems, and the problem formulation. The simple helicopter model, as a motivation application, and its pitch dynamics are also described in this section. The design of the robust affine state-feedback controller under saturation, based on the LMI approach, is discussed in Section 3. Transformation of the BMIs into LMIs is also realized in this section. The problem of computation/estimation of the domain of attraction (the largest attractive ellipsoid) is addressed in Section 4. Simulation results and some comparisons are presented in Section 5. Finally, concluding remarks and future works are drawn in Section 6.


## 1.5. Notations

Throughout this paper,AT represents the transpose of A, the symbol ⋆ in matrix inequality denotes the symmetric term of the matrix, for example, XY⋆Z=XYYTZ and X+⋆=X+XT, X>0 <0 means X is a symmetric positive (negative) definite matrix, and diagA,B,…,Z represents a diagonal matrix. Moreover, O and ℐ denote the zero matrix and the identity matrix, respectively, with appropriate dimensions.


## 2. Preliminaries and Problem Statement

In this section, we define the class of continuous-time nonlinear systems that will be investigated in this paper. The pitch dynamics of a simple helicopter model is given as an illustrative example. First of all, we present some technical lemmas that will be used subsequently.


### 2.1. Some Technical Lemmas

Lemma 1 (see [41]).
The Lyapunov candidate functionVt is bounded given that the initial condition V0 is bounded, Vt≥0 is continuous, and if the following equation is true:(1)V˙t≤−μVt+η,where μ>0 and η>0.Lemma 2 (the Young relation [12]).
Given constant matricesX and Y with appropriate dimensions, the following inequality holds:(2)XTY+YTX≤εXTX+ε−1YTY,for all positive scalar ε.Lemma 3 (the matrix inversion lemma [9, 12]).
Given invertible matricesA and B such that A∈ℝn×n and B∈ℝm×m. Moreover, given matrices C and D with appropriate dimensions: C∈ℝn×m and D∈ℝm×n. Then,(3)A+CBD−1=A−1−A−1CB−1+DA−1C−1DA−1.Lemma 4 (the Schur complement lemma [9, 12]).
Given matricesQ=QT, R=RT, and S with appropriate dimensions, the following propositions are equivalent:(4)QS⋆R>0,R>0,Q−SR−1ST>0..Lemma 5 (the S-procedure lemma [9, 12]).
LetF0, … , Fp ∈ℝn×n be symmetric matrices. We consider the following conditions on F0, …, Fp:(5)ζTF0ζ>0,∀ζ≠0,s.t.ζTFiζ≥0,∀i=1,…,p.
If there exist scalar variablesτ1≥0, … , τp≥0, such that(6)F0−∑i=1pτiFi>0,then (5) holds.


### 2.2. Class of Nonlinear Systems

A general class of continuous-time nonlinear systems with a control inputu and under disturbance signal w is defined by the following form:(7)x˙=Fx,u,w,where x∈ℝna, u∈ℝnu, and w∈ℝnw.In this work, we will consider a particular class of these nonlinear systems, for which the mathematical model is given by the following differential equation:(8)x˙=Ax+Bu+∑i=1nfCifix+Ew,where A∈ℝnx×nx, B∈ℝnx×nu, Ci∈ℝnx×1, E∈ℝnx×nw, and the nonlinear functions fix are scalars, i.e., fix∈ℝ for all i=1,…,nf. Moreover, we will consider that, in the nonlinear system (7), we have F0,0,0≠0. This means that the state x=0 is not the equilibrium point. Hence, in the dynamics (8), we have(9)∑i=1nfCifi0≠0.In addition, we will consider, without loss of generality, that(10)fix≤1,∀i=1,…,nf.Remark 1.
In this work, we consider a class of nonlinear systems (8), in which the nonlinear terms fix, for all i=1,…,nf, satisfy the constraints in (10). Our LMI-based approach for the design of a stabilizing control law u, which will be designed in the sequel, can be applied for a general class of fully nonlinear systems such as (hyper) chaotic systems as in [59, 60], robotic/mechatronic systems as in [43, 46], and underactuated mechanical systems [61, 62], just to mention a few. Thus, the nonlinearity term, saying hx=∑i=1nfCifix, can be assumed to satisfy the following quadratic inequality [12]:(11)hTxhx≤λ2xTHTHx,where λ is a positive scalar and H is a constant matrix having appropriate dimension.
Some other research papers have considered ellipsoidal condition or one-sided Lipschitz condition on the nonlinearityhx. For this subject, we refer the readers to, for example, [3, 8, 60, 63–65] and references therein.


### 2.3. Motivation Application: The Simplified Pitch Dynamics of the Helicopter Model

As an illustrative motivation application, we will consider in this work a simplified helicopter pitch dynamics model. The schematic model of the helicopter is shown in Figure1. We refer our readers to [55–57] for further details on this simplified model. The pitch dynamics of the simplified helicopter model has two degrees of freedom with only one actuator. Such model was employed to develop new control strategies for mechatronic systems [55–57].Figure 1
Simplified pitch model of the helicopter, from [55–57].The nonlinear model of the pitch dynamics of the simplified helicopter is described by the following differential equations [55–57]:(12)x˙1=x2,x˙2=1Iyy−mhelilcgxcosx1−mhelilcgzgsinx1−FvMx2+u+sIyyw,where x1 and x2 represent the pitch angle (θ in Figure 1) and pitch rate, respectively; Iyy is the second moment around the y-axis; mheli is the mass of the helicopter; lcgx and lcgz are displacements from the center of mass (GC in Figure 1) relative to the rotation joint B shown in Figure 1; FvM is the pitch damping; s is the variance of the moment disturbance; u is the control torque exerted by the main blade of the helicopter around the y-axis; and w is the external disturbance, which is modeled as an additive white noise representing the turbulent moments on the helicopter [55–57].For simplicity, let us introduce the following change of variables:a=−FvM/Iyy, b=1/Iyy, c=−mhelilcgx/Iyy, d=−mhelilcgzg/Iyy, and e=s/Iyy. Then, the nonlinear differential equations in (12) can be rewritten under the particular class (8), where x=x1x2, A=010a, B=0b, C1=0c, C2=0d, E=0e, f1x=f1x1=cosx1, and f2x=f2x1=sinx1.Notice that the two nonlinear termsf1x and f2x satisfy condition (10) and also constraint (9). Hence, the zero state, x=0, is not the equilibrium (singular) point of the pitch dynamics of the helicopter model and then of the undisturbed uncontrolled nonlinear system (8), i.e., for w=0 and u=0.Remark 2.
It is worth to note that the model of the simple pitch dynamics (8) of the helicopter in Figure 1 was adopted from [55–57]. It is a simple model where the propeller dynamics (including motor, propeller interaction, air stream interaction, gyroscopic moments, and propeller flexing), the perturbation modelling (including wind gust modelling and payload modelling), and the cyclic/collective mechanism modelling are not considered in this study. Only the turbulent moments are taken into account as an external disturbance input, w. We emphasize that all these previous unmodelled dynamics can be lumped into a single term, saying that δx can be majored according to condition (11).
For more general and basic pitch models for helicopters that reproduce basic dynamic properties of these vehicles, we can refer to [66] for typical helicopter models or [67] for more modern applications.


### 2.4. Problem Statement

In the sequel, and for simplicity in developing and designing LMI stability conditions, we will consider that, in (8), nf=2, and then the class of nonlinear systems to consider is as follows:(13)x˙=Ax+Bu+Cfx+Dgx+Ew,where fx=f1x and gx=f2x are the two scalar nonlinearities satisfying conditions (9) and (10).As noted before, the zero statex=0 is not the equilibrium point of the undisturbed uncontrolled nonlinear system (8) and then the simplified nonlinear system (13). Thus, we obtain Cf0+Dg0≠0.Our objective in this work is to control dynamics (13) to the zero state by designing a feedback controller, even if the dynamics is under parametric uncertainties or subject to the disturbance vector w. Thus, we assume that the disturbance signal w is bounded such that(14)wTw≤ρ,for some positive scalar ρ.Furthermore, we assume that all parameters in the nonlinear system (13) are uncertain and are with a bounded norm. The values of these parameters presented in the nonlinear dynamics (13) are considered to be the nominal ones. Thus, the nonlinear system (13) will be considered as the nominal system, for which A,B is controllable.Hence, under parametric uncertainties, the nominal system (13) will be rewritten as follows:(15)x˙=A+ΔAx+B+ΔBu+C+ΔCfx+D+ΔDgx+E+ΔEw,where ΔA, ΔB, ΔC, ΔD, and ΔE are matrices containing parametric uncertainties and satisfying the following expressions:(16a)ΔA=∑i1=1q1δai1A1i1A2i1,(16b)ΔB=∑i2=1q2δbi2B1i2B2i2,(16c)ΔC=∑i3=1q3δci3C1i3C2i3,(16d)ΔD=∑i4=1q4δdi4D1i4D2i4,(16e)ΔE=∑i5=1q5δei5E1i5E2i5,where the matrices A1i1, A2i1, B1i2, B2i2, C1i3, C2i3, D1i4, D2i4, E1i5, and E2i5 are with appropriate dimensions. Moreover, all the uncertainties δai1, δbi2, δci3, δdi4, and δei5, for all i1=1,…,q1, i2=1,…,q2, i3=1,…,q3, i4=1,…,q4, and i5=1,…,q5, are with a bounded norm, i.e.,(17)δij≤δ¯ij,for all i∈a,b,c,d,e and for all j∈i1,i2,i3,i4,i5.In addition, we will consider that the uncertain nonlinear dynamics (15) is subject to an input saturation of the controller u:(18)−umaxi≤ui≤umaxi,∀i=1,…,nu,where umaxi, for all i=1,…,nu, are prescribed positive scalars and ui is the ith element of the control vector u.The problem we are addressing in this work is to find, for the uncertain disturbed nonlinear system (15), an affine state-feedback control law(19)u=Kx+m,where K∈ℝnu×nx and m∈ℝnu are the feedback gains to design, for which the control law u is constrained according to condition (18). The proposed control law (19) is actually a linear state-feedback law augmented with the constant term, m, to keep the system state around an operating point, that is, the zero-equilibrium point x=0.Let us define the following set:(20)ℒK,m,umax=x∈ℝnx:Kx+m≤umax,as the region in the state space where the feedback control law m in (19) is linear (affine) in terms of the state vector x.In addition, letP∈ℝnx×nx be a positive-definite symmetric matrix and denote the Lyapunov level set as(21)εP,γ=x∈ℝnx:xTPx≤γ,where γ is a prescribed positive scalar.The level setεP,γ defined by (21) is an invariant set [5, 52]. Moreover, it is associated with the Lyapunov function Vx=xTPx for the closed-loop system. Consequently, εP,γ is an invariant set of the nonlinear system (15) with the control law (19) in the sense of the following definition.Definition 1.
A setS⊂ℝnx is said to be invariant with respect to motion of system (14), if for all initial conditions x0∈S, the motion of the system xt remains in S for all t>0.Accordingly, the ellipsoidεP,γ is said to be contractively invariant if V˙x<0 for all x∈εP,γ╲0. Clearly, if εP,γ is contractively invariant, then it is inside the domain of attraction [5, 52].According to Lemma1 and by posing η=μγ, we will have(22)V˙x≤−μVx−γ.Thus, on the boundary of the setεP,γ, we have xTPx=γ. Hence, V˙x<0. It follows that εP,γ is a strictly ellipsoidal invariant set [5, 47].Condition (22) will be used in the sequel in order to design the robust stabilization conditions of the closed-loop nonlinear system (15) with the norm-bounded parametric uncertainties (17) under the bounded external disturbance (14) and subject to the saturation (18) of the affine state-feedback control law (19) under condition (35).As we look for stability conditions for the uncertain disturbed nonlinear system (15) under the saturated feedback control law u inside the invariant ellipsoid (21), we should have the following constraint:(23)εP,γ⊂ℒK,m,umax.Therefore, if for a certain Lyapunov matrixP=PT>0 satisfying conditions (22) and (23), then taking any initial condition in the invariant ellipsoid εP,γ, we will have u=Kx+m belongs to the set ℒK,m,umax.In the sequel, we will develop conditions under which the ellipsoidεP,γ is contractively invariant and the disturbed uncertain nonlinear system (15) is robustly stable by means of the saturated affine state-feedback control law (19). Moreover, we will show how to compute the gains K and m and the Lyapunov matrix P, which determine the largest ellipsoid εP,γ and hence obtain an estimate of the domain of attraction.It is worth noting that, for the case of the pitch dynamics of the helicopter model, the uncertainty matrices areΔA=000δa, ΔB=0δb, ΔC=0δc, ΔD=0δd, and ΔE=0δe.Moreover, it is easy to show that these uncertainty matricesΔA, ΔB, ΔC, ΔD, and ΔE can be rewritten like so:(24)ΔA=δaFFT,ΔB=δbF,ΔC=δcF,ΔD=δdF,ΔE=δeF,where F=01T.Thus, in (16a)–(16e), we have q1=1, q2=1, q3=1, q4=1, and q5=1 and then A11=F, A21=FT, B11=F, B21=1, C11=F, C21=1, D11=F, D21=1, E11=F, and E21=1.Furthermore, the uncertaintiesδa, δb, δc, δd, and δe should satisfy condition (17) and then δi≤δ¯i, for all i∈a,b,c,d,e.In the sequel of this work, without loss of generality, we will take in (16a)–(16e), q1=1, q2=1, q3=1, q4=1, and q5=1. Moreover, we will consider a single control input u and also a single disturbance signal w, i.e., nu=1 and nw=1. We will also consider the expression of the uncertainty matrices in (24).


## 2.1. Some Technical Lemmas

Lemma 1 (see [41]).
The Lyapunov candidate functionVt is bounded given that the initial condition V0 is bounded, Vt≥0 is continuous, and if the following equation is true:(1)V˙t≤−μVt+η,where μ>0 and η>0.Lemma 2 (the Young relation [12]).
Given constant matricesX and Y with appropriate dimensions, the following inequality holds:(2)XTY+YTX≤εXTX+ε−1YTY,for all positive scalar ε.Lemma 3 (the matrix inversion lemma [9, 12]).
Given invertible matricesA and B such that A∈ℝn×n and B∈ℝm×m. Moreover, given matrices C and D with appropriate dimensions: C∈ℝn×m and D∈ℝm×n. Then,(3)A+CBD−1=A−1−A−1CB−1+DA−1C−1DA−1.Lemma 4 (the Schur complement lemma [9, 12]).
Given matricesQ=QT, R=RT, and S with appropriate dimensions, the following propositions are equivalent:(4)QS⋆R>0,R>0,Q−SR−1ST>0..Lemma 5 (the S-procedure lemma [9, 12]).
LetF0, … , Fp ∈ℝn×n be symmetric matrices. We consider the following conditions on F0, …, Fp:(5)ζTF0ζ>0,∀ζ≠0,s.t.ζTFiζ≥0,∀i=1,…,p.
If there exist scalar variablesτ1≥0, … , τp≥0, such that(6)F0−∑i=1pτiFi>0,then (5) holds.


## 2.2. Class of Nonlinear Systems

A general class of continuous-time nonlinear systems with a control inputu and under disturbance signal w is defined by the following form:(7)x˙=Fx,u,w,where x∈ℝna, u∈ℝnu, and w∈ℝnw.In this work, we will consider a particular class of these nonlinear systems, for which the mathematical model is given by the following differential equation:(8)x˙=Ax+Bu+∑i=1nfCifix+Ew,where A∈ℝnx×nx, B∈ℝnx×nu, Ci∈ℝnx×1, E∈ℝnx×nw, and the nonlinear functions fix are scalars, i.e., fix∈ℝ for all i=1,…,nf. Moreover, we will consider that, in the nonlinear system (7), we have F0,0,0≠0. This means that the state x=0 is not the equilibrium point. Hence, in the dynamics (8), we have(9)∑i=1nfCifi0≠0.In addition, we will consider, without loss of generality, that(10)fix≤1,∀i=1,…,nf.Remark 1.
In this work, we consider a class of nonlinear systems (8), in which the nonlinear terms fix, for all i=1,…,nf, satisfy the constraints in (10). Our LMI-based approach for the design of a stabilizing control law u, which will be designed in the sequel, can be applied for a general class of fully nonlinear systems such as (hyper) chaotic systems as in [59, 60], robotic/mechatronic systems as in [43, 46], and underactuated mechanical systems [61, 62], just to mention a few. Thus, the nonlinearity term, saying hx=∑i=1nfCifix, can be assumed to satisfy the following quadratic inequality [12]:(11)hTxhx≤λ2xTHTHx,where λ is a positive scalar and H is a constant matrix having appropriate dimension.
Some other research papers have considered ellipsoidal condition or one-sided Lipschitz condition on the nonlinearityhx. For this subject, we refer the readers to, for example, [3, 8, 60, 63–65] and references therein.


## 2.3. Motivation Application: The Simplified Pitch Dynamics of the Helicopter Model

As an illustrative motivation application, we will consider in this work a simplified helicopter pitch dynamics model. The schematic model of the helicopter is shown in Figure1. We refer our readers to [55–57] for further details on this simplified model. The pitch dynamics of the simplified helicopter model has two degrees of freedom with only one actuator. Such model was employed to develop new control strategies for mechatronic systems [55–57].Figure 1
Simplified pitch model of the helicopter, from [55–57].The nonlinear model of the pitch dynamics of the simplified helicopter is described by the following differential equations [55–57]:(12)x˙1=x2,x˙2=1Iyy−mhelilcgxcosx1−mhelilcgzgsinx1−FvMx2+u+sIyyw,where x1 and x2 represent the pitch angle (θ in Figure 1) and pitch rate, respectively; Iyy is the second moment around the y-axis; mheli is the mass of the helicopter; lcgx and lcgz are displacements from the center of mass (GC in Figure 1) relative to the rotation joint B shown in Figure 1; FvM is the pitch damping; s is the variance of the moment disturbance; u is the control torque exerted by the main blade of the helicopter around the y-axis; and w is the external disturbance, which is modeled as an additive white noise representing the turbulent moments on the helicopter [55–57].For simplicity, let us introduce the following change of variables:a=−FvM/Iyy, b=1/Iyy, c=−mhelilcgx/Iyy, d=−mhelilcgzg/Iyy, and e=s/Iyy. Then, the nonlinear differential equations in (12) can be rewritten under the particular class (8), where x=x1x2, A=010a, B=0b, C1=0c, C2=0d, E=0e, f1x=f1x1=cosx1, and f2x=f2x1=sinx1.Notice that the two nonlinear termsf1x and f2x satisfy condition (10) and also constraint (9). Hence, the zero state, x=0, is not the equilibrium (singular) point of the pitch dynamics of the helicopter model and then of the undisturbed uncontrolled nonlinear system (8), i.e., for w=0 and u=0.Remark 2.
It is worth to note that the model of the simple pitch dynamics (8) of the helicopter in Figure 1 was adopted from [55–57]. It is a simple model where the propeller dynamics (including motor, propeller interaction, air stream interaction, gyroscopic moments, and propeller flexing), the perturbation modelling (including wind gust modelling and payload modelling), and the cyclic/collective mechanism modelling are not considered in this study. Only the turbulent moments are taken into account as an external disturbance input, w. We emphasize that all these previous unmodelled dynamics can be lumped into a single term, saying that δx can be majored according to condition (11).
For more general and basic pitch models for helicopters that reproduce basic dynamic properties of these vehicles, we can refer to [66] for typical helicopter models or [67] for more modern applications.


## 2.4. Problem Statement

In the sequel, and for simplicity in developing and designing LMI stability conditions, we will consider that, in (8), nf=2, and then the class of nonlinear systems to consider is as follows:(13)x˙=Ax+Bu+Cfx+Dgx+Ew,where fx=f1x and gx=f2x are the two scalar nonlinearities satisfying conditions (9) and (10).As noted before, the zero statex=0 is not the equilibrium point of the undisturbed uncontrolled nonlinear system (8) and then the simplified nonlinear system (13). Thus, we obtain Cf0+Dg0≠0.Our objective in this work is to control dynamics (13) to the zero state by designing a feedback controller, even if the dynamics is under parametric uncertainties or subject to the disturbance vector w. Thus, we assume that the disturbance signal w is bounded such that(14)wTw≤ρ,for some positive scalar ρ.Furthermore, we assume that all parameters in the nonlinear system (13) are uncertain and are with a bounded norm. The values of these parameters presented in the nonlinear dynamics (13) are considered to be the nominal ones. Thus, the nonlinear system (13) will be considered as the nominal system, for which A,B is controllable.Hence, under parametric uncertainties, the nominal system (13) will be rewritten as follows:(15)x˙=A+ΔAx+B+ΔBu+C+ΔCfx+D+ΔDgx+E+ΔEw,where ΔA, ΔB, ΔC, ΔD, and ΔE are matrices containing parametric uncertainties and satisfying the following expressions:(16a)ΔA=∑i1=1q1δai1A1i1A2i1,(16b)ΔB=∑i2=1q2δbi2B1i2B2i2,(16c)ΔC=∑i3=1q3δci3C1i3C2i3,(16d)ΔD=∑i4=1q4δdi4D1i4D2i4,(16e)ΔE=∑i5=1q5δei5E1i5E2i5,where the matrices A1i1, A2i1, B1i2, B2i2, C1i3, C2i3, D1i4, D2i4, E1i5, and E2i5 are with appropriate dimensions. Moreover, all the uncertainties δai1, δbi2, δci3, δdi4, and δei5, for all i1=1,…,q1, i2=1,…,q2, i3=1,…,q3, i4=1,…,q4, and i5=1,…,q5, are with a bounded norm, i.e.,(17)δij≤δ¯ij,for all i∈a,b,c,d,e and for all j∈i1,i2,i3,i4,i5.In addition, we will consider that the uncertain nonlinear dynamics (15) is subject to an input saturation of the controller u:(18)−umaxi≤ui≤umaxi,∀i=1,…,nu,where umaxi, for all i=1,…,nu, are prescribed positive scalars and ui is the ith element of the control vector u.The problem we are addressing in this work is to find, for the uncertain disturbed nonlinear system (15), an affine state-feedback control law(19)u=Kx+m,where K∈ℝnu×nx and m∈ℝnu are the feedback gains to design, for which the control law u is constrained according to condition (18). The proposed control law (19) is actually a linear state-feedback law augmented with the constant term, m, to keep the system state around an operating point, that is, the zero-equilibrium point x=0.Let us define the following set:(20)ℒK,m,umax=x∈ℝnx:Kx+m≤umax,as the region in the state space where the feedback control law m in (19) is linear (affine) in terms of the state vector x.In addition, letP∈ℝnx×nx be a positive-definite symmetric matrix and denote the Lyapunov level set as(21)εP,γ=x∈ℝnx:xTPx≤γ,where γ is a prescribed positive scalar.The level setεP,γ defined by (21) is an invariant set [5, 52]. Moreover, it is associated with the Lyapunov function Vx=xTPx for the closed-loop system. Consequently, εP,γ is an invariant set of the nonlinear system (15) with the control law (19) in the sense of the following definition.Definition 1.
A setS⊂ℝnx is said to be invariant with respect to motion of system (14), if for all initial conditions x0∈S, the motion of the system xt remains in S for all t>0.Accordingly, the ellipsoidεP,γ is said to be contractively invariant if V˙x<0 for all x∈εP,γ╲0. Clearly, if εP,γ is contractively invariant, then it is inside the domain of attraction [5, 52].According to Lemma1 and by posing η=μγ, we will have(22)V˙x≤−μVx−γ.Thus, on the boundary of the setεP,γ, we have xTPx=γ. Hence, V˙x<0. It follows that εP,γ is a strictly ellipsoidal invariant set [5, 47].Condition (22) will be used in the sequel in order to design the robust stabilization conditions of the closed-loop nonlinear system (15) with the norm-bounded parametric uncertainties (17) under the bounded external disturbance (14) and subject to the saturation (18) of the affine state-feedback control law (19) under condition (35).As we look for stability conditions for the uncertain disturbed nonlinear system (15) under the saturated feedback control law u inside the invariant ellipsoid (21), we should have the following constraint:(23)εP,γ⊂ℒK,m,umax.Therefore, if for a certain Lyapunov matrixP=PT>0 satisfying conditions (22) and (23), then taking any initial condition in the invariant ellipsoid εP,γ, we will have u=Kx+m belongs to the set ℒK,m,umax.In the sequel, we will develop conditions under which the ellipsoidεP,γ is contractively invariant and the disturbed uncertain nonlinear system (15) is robustly stable by means of the saturated affine state-feedback control law (19). Moreover, we will show how to compute the gains K and m and the Lyapunov matrix P, which determine the largest ellipsoid εP,γ and hence obtain an estimate of the domain of attraction.It is worth noting that, for the case of the pitch dynamics of the helicopter model, the uncertainty matrices areΔA=000δa, ΔB=0δb, ΔC=0δc, ΔD=0δd, and ΔE=0δe.Moreover, it is easy to show that these uncertainty matricesΔA, ΔB, ΔC, ΔD, and ΔE can be rewritten like so:(24)ΔA=δaFFT,ΔB=δbF,ΔC=δcF,ΔD=δdF,ΔE=δeF,where F=01T.Thus, in (16a)–(16e), we have q1=1, q2=1, q3=1, q4=1, and q5=1 and then A11=F, A21=FT, B11=F, B21=1, C11=F, C21=1, D11=F, D21=1, E11=F, and E21=1.Furthermore, the uncertaintiesδa, δb, δc, δd, and δe should satisfy condition (17) and then δi≤δ¯i, for all i∈a,b,c,d,e.In the sequel of this work, without loss of generality, we will take in (16a)–(16e), q1=1, q2=1, q3=1, q4=1, and q5=1. Moreover, we will consider a single control input u and also a single disturbance signal w, i.e., nu=1 and nw=1. We will also consider the expression of the uncertainty matrices in (24).


## 3. Design of the Robust Saturated Affine State-Feedback Controller

In this section, we develop conditions satisfying robust stabilization of the uncertain disturbed nonlinear dynamics (15) under the affine state-feedback control law (19) subject to the saturation constraint (18). Thus, these stability conditions must verify constraints (22) and (23). We show first that the stability conditions are expressed in terms of BMIs. Next, by means of previously provided technical lemmas, we transform these BMIs into LMIs. Finally, we present an optimization problem providing the allowable maximum values of the parametric uncertainties δa, δb, δc, δd, and δe.


### 3.1. BMI-Based Stability Conditions

These stability conditions of the disturbed uncertain nonlinear system (15) under the saturated affine state-feedback control law defined by expression (19), constraint (18), and condition (35) are presented in the following theorem.Theorem 1.
The nonlinear system (15) with norm-bounded parametric uncertainties (17), under the bounded external disturbance (14), is robustly stabilizable by implementing the affine state-feedback control law (19) subject to the saturation constraint (18), if for some fixed positive parameters γ, εm, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e, there exist a symmetric matrix P>0, a feedback matrix K, a feedback constant m, and some positive scalars ε1, ε2, ε3, ε4, ε5, ε6, ε7, ε8, ε9, μ, and η such that the following set of matrix inequalities is feasible:(25)PA+PBK+⋆+μP+A1PBm⋆−μγ+ξ<0,(26)KTK−ηPKTm⋆ηγ+mTm−umax2<0,(27)Bm+C^TBm+C^<εm,where(28)A1=ε1PCCTP+ε2PDDTP+ε3PEETP+ε4+ε5+ε6+ε7+ε8+ε9PFFTP+ε4−1δ¯a2FFT+ε5−1δ¯b2KTK,ξ=ε1−1+ε2−1+ε3−1+ε6−1δ¯b2mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1δ¯e2ρ2.Proof.
By using system (15) and control law (19), the derivative of the Lyapunov function, Vx=xTPx, is given by(29)V˙x=2xTPA¯x+2xTPBm+2xTPCfx+2xTPDgx+2xTPEw+2xTPΔAx+2xTPΔBKx+2xTPΔBm+2xTPΔCfx+2xTPΔDgx+2xTPΔEw,with A¯=A+BK.
We now consider condition (22) on the Lyapunov function Vx and also conditions (10), (14), and (17). Then, using expressions of the uncertainties matrices in (24) and relying on the Young relation (Lemma 2), we obtain the following condition:(30)V˙x+μVx−γ≤μxTPx−μγ+2xTPA¯x+2xTPBm+ε1xTPCCTPx+ε1−1+ε2−1+ε3−1+ε2xTPDDTPx+ε3xTPEETPx+ε4+ε5+ε6+ε7+ε8+ε9xTPFFTPx+ε4−1δ¯a2xTFFTx+ε5−1δ¯b2xTKTKx+ε6−1δ¯b2mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1δ¯e2ρ2<0.
This last inequality constraint can be rewritten as follows:(31)x1TA2+μPPBm⋆−μγ+ξx1<0,where A2=PA¯+⋆+ε1PCCTP+ε2PDDTP+ε3PEETP+ε4+ε5+ε6+ε7+ε8+ε9PFFTP+ε4−1δ¯a2FFT+ε5−1δ¯b2KTK and ξ=ε1−1+ε2−1+ε3−1+ε6−1δ¯b2mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1δ¯e2ρ2.
Hence, we obtain the matrix inequality (25).
Furthermore, from expressions (20) and (21), rewrite x∈εP,γ as(32)x1TPO⋆−γx1≤0,and x∈ℒK,m,umax as(33)x1TKTKKTm⋆mTm−umax2x1≤0.
It is worth to note that the conditionεP,γ⊂ℒK,m,umax (constraint (23)) is nothing than the implication (32) ⟹ (33). Therefore, by using the S-procedure lemma, this implication condition is equivalent to the existence of a positive scalar η such that the matrix inequality (26) holds.
As noted in the beginning of the previous section, the zero statex=0 is not the equilibrium point. Then, under the affine state-feedback control law (19) and for a certain and undisturbed dynamics (that is, for the nominal dynamics (13)), the zero-equilibrium point should verify the following condition:(34)Bm+Cf0+Dg0=0.
PosingC^=Cf0+Dg0, then relation (34) is recast as follows:(35)Bm+C^=0.
It is worth to note that, in condition (35), the term Bm+C^ is a column vector. Then, such relation in (35) can be written in a (scalar) matrix form like so:(36)Bm+C^TBm+C^=0.
For a small enough positive scalarεm, equality (36) can be transformed into condition (27). This completes the proof of Theorem 1.
It is worth mentioning that the three matrix inequalities (25)–(27) in Theorem 1 are BMIs, which are hardly tractable numerically. Thus, our objective is to convert these BMIs (25)–(27) into LMIs, which is the objective of the next section.Remark 3.
Recall that, in the previous development of the BMI stability conditions, we have considered a single control inputu and then nu=1. Hence, we used one row matrix gain K and one scalar gain m. This choice has led to the design of only one condition (26) satisfying the saturation condition of the controller u. For a general case and then for nu>1, K∈ℝnu×nx, m∈ℝnu, and umax∈ℝnu, the stability condition (26) becomes(37)KTeiTeiK−ηiPKTeiTeim⋆ηiγ+mTeiTeim−umaxTeiTeiumax<0,∀i=1,…,nu,where ei=0,…,0,1︷ith,0,…,0︸nucomponents∈ℝ1×nu and ηi>0 for all i=1,…,nu.


### 3.2. LMI-Based Stability Conditions

In this part, we aim at linearizing the three BMIs (25)–(27) in Theorem 1. Thus, we first linearize the BMI (25). After that, we linearize the BMI (26) and the BMI (27).


#### 3.2.1. Linearization of the BMI (25)

Let us put firstS=P−1. We premultiply and postmultiply the BMI (25) by the matrix diagS,1. Then, we obtain(38)AS+BR+⋆+μS+A3Bm⋆−μγ+ξ<0,where R=KS and A3=ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT+ε4−1δ¯a2SFFTS+ε5−1δ¯b2RTR.Based on the Schur complement lemma, matrix inequality (38) is equivalent to(39)A4+μSBmA5O⋆−μγOA7⋆⋆−A6O⋆⋆⋆−A8<0,where A4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT, A5=δ¯aSFδ¯bRT, A6=diagε4,ε5, A7=111δ¯bmTδ¯cδ¯dδ¯eρ, and A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.


#### 3.2.2. Linearization of the BMI (26)

We premultiply and postmultiply the BMI (26) by the matrix diagS,1. As a result, we obtain(40)RTR−ηSRTm⋆−umax2+mTm+ηγ<0.Let us introduce a new change of variables by posingG=RTO, L=m1, M=1001/ηγ, and α=−umax2. Then, we have RTR=GGT=GMGT, RTm=GL, and −umax2+mTm+ηγ=α+LTM−1L.Then, by applying the Schur lemma to inequality (40), we obtain(41)−ηS+GMGT−GLψ−1LTGT<0,(42)ψ=α+LTM−1L<0.Relying on the matrix inversion lemma (see Lemma3), its follows that(43)ψ−1=α−1−α−2LTH−1L,with(44)H=M+α−1LLT.Then, by substituting expression (43) in inequality (41), this last inequality (41) can be rewritten like so:(45)−ηS+GMGT−α−1GLLTGT+α−2GLLTH−1LLTGT<0.Through expression (44), it is straightforward to show that(46)α−1GLLT=GH−GM.Therefore, based on this relation, we can note the following:(47)α−1GLLTGT=GHGT−GMGT,(48)α−2GLLTH−1LLTGT=GHGT−2GMGT+GMH−1MGT.Thus, asGM=G, substituting relations (47) and (48) in inequality (45) yields(49)ηS−GH−1GT>0.Accordingly, inequality (41) is transformed into inequality (49).Our concern now is the auxiliary condition (42). As M>0, then through the Schur lemma, inequality (42) is equivalent to(50)αLT⋆−M<0.By applying again the Schur lemma on inequality (50), we obtain(51)−M−α−1LLT<0.Hence, it follows thatH>0. Thus, by multiplying inequality (49) by η−1 and relying on the Schur lemma, we obtain the following matrix inequality:(52)SG⋆ηH>0.Sinceα=−umax2, then we can write the following:(53)ηH=ηM−ηumax2LLT,with ηM=η001/γ.Therefore, by taking into account relation (53), the Schur complement lemma states that inequality (52) is equivalent to(54)SGO⋆ηML⋆⋆umax2η>0.Hence, the BMI (26) is converted into the LMI (54), where the two parameters γ and η should be fixed a priori.


#### 3.2.3. Linearization of the BMI (27)

We now consider the BMI (27). It can be rewritten like so:(55)mTBTBm+2CTBm+C^TC^<εm.By applying the Schur lemma, this matrix inequality (55) is equivalent to the following LMI:(56)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0.


#### 3.2.4. LMI Conditions

According to the previous linearization results, we state the following new theorem.Theorem 2.
Assume that, for some positive parametersγ, μ, η, εm, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e fixed a priori, there exist a symmetric matrix S, a matrix R, a scalar m, and some scalars ε1, ε2, ε3, ε4, ε5, ε6, ε7, ε8, and ε9 so that the following set of LMIs is feasible:(57)A4+μSBmA5O⋆−μγOA7⋆⋆−A6O⋆⋆⋆−A8<0,(58)SRTOO⋆ηOm⋆⋆1γ1⋆⋆⋆umax2η>0,(59)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0,where(60)A4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT,A5=δ¯aSFδ¯bRT,A6=diagε4,ε5,A7=111δ¯bmTδ¯cδ¯dδ¯eρ,A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.Then, the nonlinear system (15) with norm-bounded parametric uncertainties (17) and bounded disturbance (14) is robustly stabilizable by means of the saturated affine state-feedback control law (19) subject to constraint (18). Moreover, the matrix gain K of the control law (19) is(61)K=RS−1.Proof.
The proof of this theorem was already achieved in the previous sections.
As noted in Theorem2, the parameters γ, μ, η, εm, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e should be fixed a priori to obtain a solution of LMIs (57)–(59). Thus, the computation complexity for solving these three LMIs is reduced. However, in this way, the design conservatism will increase and obtaining a solution becomes more difficult. Indeed, we should test several times the feasibility of these LMIs (57)–(59) with different values of all these fixed parameters. Therefore, in order to reduce the conservatism of the designed LMIs, we should consider these parameters as decision variables, which should be optimized numerically. To this end, we next present improved LMI conditions. We note that the parameter εm can be selected as a decision variable to be minimized. Nevertheless, we will consider it as a predefined constant scalar, which will be fixed to be very small.Remark 4.
As noted in Remark3, for a general case where nu>1, we obtain the BMI (37) instead of the BMI (26). The linearization of the BMI stability condition (37) leads to the following LMI:(62)S⋆OOeiRηiOeim⋆⋆1γ1⋆⋆⋆1ηiumaxTeiTeiumax>0,∀i=1,2,…,nu.


### 3.3. Enhanced LMI Stability Conditions

Our immediate concern is to convert the constant parametersγ, μ, η, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e into decision variables. First of all, it is evident that the two parameters δ¯c and δ¯d presented in the LMI (57) are independent of other unknown variables. Then, they can be considered as decision variables. However, the two main difficulties lie on the one hand in the parameters γ, μ, and η and on the other hand in the three parameters δ¯a, δ¯b, δ¯e, and ρ. Transformation of δ¯a, δ¯e, and ρ is simple [12]. Furthermore, transformation of the parameters γ and δ¯b requires a certain judicious mathematical manipulation. Note that the parameter γ appears in both LMI (57) and LMI (58). However, transformation of the two remaining parameters μ and η is quite difficult and perhaps impossible. To simplify computation, we can pose μ=η. Hence, we state the following enhanced version of Theorem 2.Theorem 3.
Assume that, for some positive parametersλ and ϵm fixed a priori such that ϵm≪1, there exist a symmetric matrix S, a matrix R, a gain m, and some scalars γ, α>0, β>0, φ>0, ϵ1, ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8, ϵ9, δ¯c, and δ¯d so that the following optimization problem with LMI constraints is feasible:(63)minimizeε4+ε5+ε6+ε9−α+β+φ+δ¯c+δ¯d+ρsubject toA^4+λSBmA^5O⋆−λγOA^7⋆⋆−A6O⋆⋆⋆−A8<0,(64)SRTO⋆λm⋆⋆umax2λ−γ>0,(65)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0,where(66)A^4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+α+β+φ+ε7+ε8FFT,A^5=SFRT,A6=diagε4,ε5,A^7=111mTδ¯cδ¯dρ,A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.Then, the uncertain disturbed nonlinear system (14) is robustly stabilizable via the saturated affine state-feedback control law (18), with K=RS−1, and(67)δ¯a=αε4,δ¯b=βε5+ε6,δ¯e=φε9.Proof.
We first consider expression (29). Thus, by applying the Young relation, we can obtain another related condition instead of inequality (30):(68)V˙x+μVx−γ≤μxTPx−μγ+2xTPA¯x+2xTPBm+ε1xTPCCTPx+ε1−1+ε2−1+ε3−1+ε2xTPDDTPx+ε3xTPEETPx+ε4δ¯a2+ε5+ε6δ¯b2+ε7+ε8+ε9δ¯e2xTPFFTPx+ε4−1xTFFTx+ε5−1xTKTKx+ε6−1mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1ρ2<0.
Let us introduce a new change of variables by posingα=ε4δ¯a2, β=ε5+ε6δ¯b2, and φ=ε9δ¯e2. Then, according to the previous section, we obtain the LMI (64), in which α, β, and φ are three decision variables. Hence, once the values of α, β, φ, ε4, ε5, ε6, and ε9 are obtained, we calculate δ¯a, δ¯b, and δ¯e according to the following relations: δ¯a=α/ε4, δ¯b=β/ε5+ε6, and δ¯e=φ/ε9.
The remaining problem is the parameterγ, which appears in both LMI (57) and LMI (58). We premultiply and postmultiply LMI (58) by the matrix ℐOOO⋆ℐOO⋆⋆Oℐ⋆⋆⋆O and then we obtain(69)SRTOO⋆ηmO⋆⋆umax2η1⋆⋆⋆1γ>0.
By applying the Schur lemma, inequality (69) is equivalent to the LMI (64). Now, the parameter γ is linear in the two LMIs (63) and (64).
In addition, the parameterμ appears in the LMI (57), whereas the parameter η appears in the LMI (58). As noted before, it is difficult to transform μ and η into two decision variables. Then, to simplify the computation procedure, we can pose μ=η=λ.
The objective is to look for maximum values of the uncertainty boundsδ¯a, δ¯b, δ¯c, δ¯d, and δ¯e and the disturbance bound ρ. The three parameters δ¯c, δ¯d, and ρ appear directly in the LMI (63) (matrix A^7). Then, these parameters should be maximized. However, the variables δ¯a, δ¯b, and δ¯e are incorporated, respectively, in the parameters α, β, and φ: α=ε4δ¯a2, β=ε5+ε6δ¯b2, and φ=ε9δ¯e2. Then, to maximize δ¯a, δ¯b, and δ¯e, we should maximize α, β, and φ. Moreover, as α (resp. β, φ) depends on ε4 (resp. ε5 and ε6, ε9), we should minimize ε4, ε5, ε6, and ε9 in order to ensure maximized values of δ¯a, δ¯b, and δ¯e. We emphasize that, in order to obtain a minimization objective function that can be handled with the LMI toolbox, we should transform the maximization problem of the variables α, β, φ, δ¯c, δ¯d, and ρ into a minimization problem of the quantity −α+β+φ+δ¯c+δ¯d+ρ. This completes the proof of Theorem 3.
It is worth to mention that Theorem 3 offers a wider choice of invariant ellipsoidsεP,γ (as γ varies) for optimization and will lead to less conservative estimation of the domain of attraction.
In addition, we stress that, in Theorem3, we imposed the two free parameters μ and η in Theorem 2 to be equal, i.e., μ=η=λ, in order to obtain an LMI with a single parameter λ. Nevertheless, this choice, which is the only solution to simplify computation, injects some degree of conservatism in the LMI conditions (63)–(65). Furthermore, the parameter λ is the only one that should be fixed a priori to obtain a solution of the optimization problem in Theorem 3.


## 3.1. BMI-Based Stability Conditions

These stability conditions of the disturbed uncertain nonlinear system (15) under the saturated affine state-feedback control law defined by expression (19), constraint (18), and condition (35) are presented in the following theorem.Theorem 1.
The nonlinear system (15) with norm-bounded parametric uncertainties (17), under the bounded external disturbance (14), is robustly stabilizable by implementing the affine state-feedback control law (19) subject to the saturation constraint (18), if for some fixed positive parameters γ, εm, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e, there exist a symmetric matrix P>0, a feedback matrix K, a feedback constant m, and some positive scalars ε1, ε2, ε3, ε4, ε5, ε6, ε7, ε8, ε9, μ, and η such that the following set of matrix inequalities is feasible:(25)PA+PBK+⋆+μP+A1PBm⋆−μγ+ξ<0,(26)KTK−ηPKTm⋆ηγ+mTm−umax2<0,(27)Bm+C^TBm+C^<εm,where(28)A1=ε1PCCTP+ε2PDDTP+ε3PEETP+ε4+ε5+ε6+ε7+ε8+ε9PFFTP+ε4−1δ¯a2FFT+ε5−1δ¯b2KTK,ξ=ε1−1+ε2−1+ε3−1+ε6−1δ¯b2mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1δ¯e2ρ2.Proof.
By using system (15) and control law (19), the derivative of the Lyapunov function, Vx=xTPx, is given by(29)V˙x=2xTPA¯x+2xTPBm+2xTPCfx+2xTPDgx+2xTPEw+2xTPΔAx+2xTPΔBKx+2xTPΔBm+2xTPΔCfx+2xTPΔDgx+2xTPΔEw,with A¯=A+BK.
We now consider condition (22) on the Lyapunov function Vx and also conditions (10), (14), and (17). Then, using expressions of the uncertainties matrices in (24) and relying on the Young relation (Lemma 2), we obtain the following condition:(30)V˙x+μVx−γ≤μxTPx−μγ+2xTPA¯x+2xTPBm+ε1xTPCCTPx+ε1−1+ε2−1+ε3−1+ε2xTPDDTPx+ε3xTPEETPx+ε4+ε5+ε6+ε7+ε8+ε9xTPFFTPx+ε4−1δ¯a2xTFFTx+ε5−1δ¯b2xTKTKx+ε6−1δ¯b2mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1δ¯e2ρ2<0.
This last inequality constraint can be rewritten as follows:(31)x1TA2+μPPBm⋆−μγ+ξx1<0,where A2=PA¯+⋆+ε1PCCTP+ε2PDDTP+ε3PEETP+ε4+ε5+ε6+ε7+ε8+ε9PFFTP+ε4−1δ¯a2FFT+ε5−1δ¯b2KTK and ξ=ε1−1+ε2−1+ε3−1+ε6−1δ¯b2mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1δ¯e2ρ2.
Hence, we obtain the matrix inequality (25).
Furthermore, from expressions (20) and (21), rewrite x∈εP,γ as(32)x1TPO⋆−γx1≤0,and x∈ℒK,m,umax as(33)x1TKTKKTm⋆mTm−umax2x1≤0.
It is worth to note that the conditionεP,γ⊂ℒK,m,umax (constraint (23)) is nothing than the implication (32) ⟹ (33). Therefore, by using the S-procedure lemma, this implication condition is equivalent to the existence of a positive scalar η such that the matrix inequality (26) holds.
As noted in the beginning of the previous section, the zero statex=0 is not the equilibrium point. Then, under the affine state-feedback control law (19) and for a certain and undisturbed dynamics (that is, for the nominal dynamics (13)), the zero-equilibrium point should verify the following condition:(34)Bm+Cf0+Dg0=0.
PosingC^=Cf0+Dg0, then relation (34) is recast as follows:(35)Bm+C^=0.
It is worth to note that, in condition (35), the term Bm+C^ is a column vector. Then, such relation in (35) can be written in a (scalar) matrix form like so:(36)Bm+C^TBm+C^=0.
For a small enough positive scalarεm, equality (36) can be transformed into condition (27). This completes the proof of Theorem 1.
It is worth mentioning that the three matrix inequalities (25)–(27) in Theorem 1 are BMIs, which are hardly tractable numerically. Thus, our objective is to convert these BMIs (25)–(27) into LMIs, which is the objective of the next section.Remark 3.
Recall that, in the previous development of the BMI stability conditions, we have considered a single control inputu and then nu=1. Hence, we used one row matrix gain K and one scalar gain m. This choice has led to the design of only one condition (26) satisfying the saturation condition of the controller u. For a general case and then for nu>1, K∈ℝnu×nx, m∈ℝnu, and umax∈ℝnu, the stability condition (26) becomes(37)KTeiTeiK−ηiPKTeiTeim⋆ηiγ+mTeiTeim−umaxTeiTeiumax<0,∀i=1,…,nu,where ei=0,…,0,1︷ith,0,…,0︸nucomponents∈ℝ1×nu and ηi>0 for all i=1,…,nu.


## 3.2. LMI-Based Stability Conditions

In this part, we aim at linearizing the three BMIs (25)–(27) in Theorem 1. Thus, we first linearize the BMI (25). After that, we linearize the BMI (26) and the BMI (27).


### 3.2.1. Linearization of the BMI (25)

Let us put firstS=P−1. We premultiply and postmultiply the BMI (25) by the matrix diagS,1. Then, we obtain(38)AS+BR+⋆+μS+A3Bm⋆−μγ+ξ<0,where R=KS and A3=ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT+ε4−1δ¯a2SFFTS+ε5−1δ¯b2RTR.Based on the Schur complement lemma, matrix inequality (38) is equivalent to(39)A4+μSBmA5O⋆−μγOA7⋆⋆−A6O⋆⋆⋆−A8<0,where A4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT, A5=δ¯aSFδ¯bRT, A6=diagε4,ε5, A7=111δ¯bmTδ¯cδ¯dδ¯eρ, and A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.


### 3.2.2. Linearization of the BMI (26)

We premultiply and postmultiply the BMI (26) by the matrix diagS,1. As a result, we obtain(40)RTR−ηSRTm⋆−umax2+mTm+ηγ<0.Let us introduce a new change of variables by posingG=RTO, L=m1, M=1001/ηγ, and α=−umax2. Then, we have RTR=GGT=GMGT, RTm=GL, and −umax2+mTm+ηγ=α+LTM−1L.Then, by applying the Schur lemma to inequality (40), we obtain(41)−ηS+GMGT−GLψ−1LTGT<0,(42)ψ=α+LTM−1L<0.Relying on the matrix inversion lemma (see Lemma3), its follows that(43)ψ−1=α−1−α−2LTH−1L,with(44)H=M+α−1LLT.Then, by substituting expression (43) in inequality (41), this last inequality (41) can be rewritten like so:(45)−ηS+GMGT−α−1GLLTGT+α−2GLLTH−1LLTGT<0.Through expression (44), it is straightforward to show that(46)α−1GLLT=GH−GM.Therefore, based on this relation, we can note the following:(47)α−1GLLTGT=GHGT−GMGT,(48)α−2GLLTH−1LLTGT=GHGT−2GMGT+GMH−1MGT.Thus, asGM=G, substituting relations (47) and (48) in inequality (45) yields(49)ηS−GH−1GT>0.Accordingly, inequality (41) is transformed into inequality (49).Our concern now is the auxiliary condition (42). As M>0, then through the Schur lemma, inequality (42) is equivalent to(50)αLT⋆−M<0.By applying again the Schur lemma on inequality (50), we obtain(51)−M−α−1LLT<0.Hence, it follows thatH>0. Thus, by multiplying inequality (49) by η−1 and relying on the Schur lemma, we obtain the following matrix inequality:(52)SG⋆ηH>0.Sinceα=−umax2, then we can write the following:(53)ηH=ηM−ηumax2LLT,with ηM=η001/γ.Therefore, by taking into account relation (53), the Schur complement lemma states that inequality (52) is equivalent to(54)SGO⋆ηML⋆⋆umax2η>0.Hence, the BMI (26) is converted into the LMI (54), where the two parameters γ and η should be fixed a priori.


### 3.2.3. Linearization of the BMI (27)

We now consider the BMI (27). It can be rewritten like so:(55)mTBTBm+2CTBm+C^TC^<εm.By applying the Schur lemma, this matrix inequality (55) is equivalent to the following LMI:(56)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0.


### 3.2.4. LMI Conditions

According to the previous linearization results, we state the following new theorem.Theorem 2.
Assume that, for some positive parametersγ, μ, η, εm, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e fixed a priori, there exist a symmetric matrix S, a matrix R, a scalar m, and some scalars ε1, ε2, ε3, ε4, ε5, ε6, ε7, ε8, and ε9 so that the following set of LMIs is feasible:(57)A4+μSBmA5O⋆−μγOA7⋆⋆−A6O⋆⋆⋆−A8<0,(58)SRTOO⋆ηOm⋆⋆1γ1⋆⋆⋆umax2η>0,(59)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0,where(60)A4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT,A5=δ¯aSFδ¯bRT,A6=diagε4,ε5,A7=111δ¯bmTδ¯cδ¯dδ¯eρ,A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.Then, the nonlinear system (15) with norm-bounded parametric uncertainties (17) and bounded disturbance (14) is robustly stabilizable by means of the saturated affine state-feedback control law (19) subject to constraint (18). Moreover, the matrix gain K of the control law (19) is(61)K=RS−1.Proof.
The proof of this theorem was already achieved in the previous sections.
As noted in Theorem2, the parameters γ, μ, η, εm, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e should be fixed a priori to obtain a solution of LMIs (57)–(59). Thus, the computation complexity for solving these three LMIs is reduced. However, in this way, the design conservatism will increase and obtaining a solution becomes more difficult. Indeed, we should test several times the feasibility of these LMIs (57)–(59) with different values of all these fixed parameters. Therefore, in order to reduce the conservatism of the designed LMIs, we should consider these parameters as decision variables, which should be optimized numerically. To this end, we next present improved LMI conditions. We note that the parameter εm can be selected as a decision variable to be minimized. Nevertheless, we will consider it as a predefined constant scalar, which will be fixed to be very small.Remark 4.
As noted in Remark3, for a general case where nu>1, we obtain the BMI (37) instead of the BMI (26). The linearization of the BMI stability condition (37) leads to the following LMI:(62)S⋆OOeiRηiOeim⋆⋆1γ1⋆⋆⋆1ηiumaxTeiTeiumax>0,∀i=1,2,…,nu.


## 3.2.1. Linearization of the BMI (25)

Let us put firstS=P−1. We premultiply and postmultiply the BMI (25) by the matrix diagS,1. Then, we obtain(38)AS+BR+⋆+μS+A3Bm⋆−μγ+ξ<0,where R=KS and A3=ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT+ε4−1δ¯a2SFFTS+ε5−1δ¯b2RTR.Based on the Schur complement lemma, matrix inequality (38) is equivalent to(39)A4+μSBmA5O⋆−μγOA7⋆⋆−A6O⋆⋆⋆−A8<0,where A4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT, A5=δ¯aSFδ¯bRT, A6=diagε4,ε5, A7=111δ¯bmTδ¯cδ¯dδ¯eρ, and A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.


## 3.2.2. Linearization of the BMI (26)

We premultiply and postmultiply the BMI (26) by the matrix diagS,1. As a result, we obtain(40)RTR−ηSRTm⋆−umax2+mTm+ηγ<0.Let us introduce a new change of variables by posingG=RTO, L=m1, M=1001/ηγ, and α=−umax2. Then, we have RTR=GGT=GMGT, RTm=GL, and −umax2+mTm+ηγ=α+LTM−1L.Then, by applying the Schur lemma to inequality (40), we obtain(41)−ηS+GMGT−GLψ−1LTGT<0,(42)ψ=α+LTM−1L<0.Relying on the matrix inversion lemma (see Lemma3), its follows that(43)ψ−1=α−1−α−2LTH−1L,with(44)H=M+α−1LLT.Then, by substituting expression (43) in inequality (41), this last inequality (41) can be rewritten like so:(45)−ηS+GMGT−α−1GLLTGT+α−2GLLTH−1LLTGT<0.Through expression (44), it is straightforward to show that(46)α−1GLLT=GH−GM.Therefore, based on this relation, we can note the following:(47)α−1GLLTGT=GHGT−GMGT,(48)α−2GLLTH−1LLTGT=GHGT−2GMGT+GMH−1MGT.Thus, asGM=G, substituting relations (47) and (48) in inequality (45) yields(49)ηS−GH−1GT>0.Accordingly, inequality (41) is transformed into inequality (49).Our concern now is the auxiliary condition (42). As M>0, then through the Schur lemma, inequality (42) is equivalent to(50)αLT⋆−M<0.By applying again the Schur lemma on inequality (50), we obtain(51)−M−α−1LLT<0.Hence, it follows thatH>0. Thus, by multiplying inequality (49) by η−1 and relying on the Schur lemma, we obtain the following matrix inequality:(52)SG⋆ηH>0.Sinceα=−umax2, then we can write the following:(53)ηH=ηM−ηumax2LLT,with ηM=η001/γ.Therefore, by taking into account relation (53), the Schur complement lemma states that inequality (52) is equivalent to(54)SGO⋆ηML⋆⋆umax2η>0.Hence, the BMI (26) is converted into the LMI (54), where the two parameters γ and η should be fixed a priori.


## 3.2.3. Linearization of the BMI (27)

We now consider the BMI (27). It can be rewritten like so:(55)mTBTBm+2CTBm+C^TC^<εm.By applying the Schur lemma, this matrix inequality (55) is equivalent to the following LMI:(56)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0.


## 3.2.4. LMI Conditions

According to the previous linearization results, we state the following new theorem.Theorem 2.
Assume that, for some positive parametersγ, μ, η, εm, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e fixed a priori, there exist a symmetric matrix S, a matrix R, a scalar m, and some scalars ε1, ε2, ε3, ε4, ε5, ε6, ε7, ε8, and ε9 so that the following set of LMIs is feasible:(57)A4+μSBmA5O⋆−μγOA7⋆⋆−A6O⋆⋆⋆−A8<0,(58)SRTOO⋆ηOm⋆⋆1γ1⋆⋆⋆umax2η>0,(59)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0,where(60)A4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+ε4+ε5+ε6+ε7+ε8+ε9FFT,A5=δ¯aSFδ¯bRT,A6=diagε4,ε5,A7=111δ¯bmTδ¯cδ¯dδ¯eρ,A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.Then, the nonlinear system (15) with norm-bounded parametric uncertainties (17) and bounded disturbance (14) is robustly stabilizable by means of the saturated affine state-feedback control law (19) subject to constraint (18). Moreover, the matrix gain K of the control law (19) is(61)K=RS−1.Proof.
The proof of this theorem was already achieved in the previous sections.
As noted in Theorem2, the parameters γ, μ, η, εm, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e should be fixed a priori to obtain a solution of LMIs (57)–(59). Thus, the computation complexity for solving these three LMIs is reduced. However, in this way, the design conservatism will increase and obtaining a solution becomes more difficult. Indeed, we should test several times the feasibility of these LMIs (57)–(59) with different values of all these fixed parameters. Therefore, in order to reduce the conservatism of the designed LMIs, we should consider these parameters as decision variables, which should be optimized numerically. To this end, we next present improved LMI conditions. We note that the parameter εm can be selected as a decision variable to be minimized. Nevertheless, we will consider it as a predefined constant scalar, which will be fixed to be very small.Remark 4.
As noted in Remark3, for a general case where nu>1, we obtain the BMI (37) instead of the BMI (26). The linearization of the BMI stability condition (37) leads to the following LMI:(62)S⋆OOeiRηiOeim⋆⋆1γ1⋆⋆⋆1ηiumaxTeiTeiumax>0,∀i=1,2,…,nu.


## 3.3. Enhanced LMI Stability Conditions

Our immediate concern is to convert the constant parametersγ, μ, η, ρ, δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e into decision variables. First of all, it is evident that the two parameters δ¯c and δ¯d presented in the LMI (57) are independent of other unknown variables. Then, they can be considered as decision variables. However, the two main difficulties lie on the one hand in the parameters γ, μ, and η and on the other hand in the three parameters δ¯a, δ¯b, δ¯e, and ρ. Transformation of δ¯a, δ¯e, and ρ is simple [12]. Furthermore, transformation of the parameters γ and δ¯b requires a certain judicious mathematical manipulation. Note that the parameter γ appears in both LMI (57) and LMI (58). However, transformation of the two remaining parameters μ and η is quite difficult and perhaps impossible. To simplify computation, we can pose μ=η. Hence, we state the following enhanced version of Theorem 2.Theorem 3.
Assume that, for some positive parametersλ and ϵm fixed a priori such that ϵm≪1, there exist a symmetric matrix S, a matrix R, a gain m, and some scalars γ, α>0, β>0, φ>0, ϵ1, ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8, ϵ9, δ¯c, and δ¯d so that the following optimization problem with LMI constraints is feasible:(63)minimizeε4+ε5+ε6+ε9−α+β+φ+δ¯c+δ¯d+ρsubject toA^4+λSBmA^5O⋆−λγOA^7⋆⋆−A6O⋆⋆⋆−A8<0,(64)SRTO⋆λm⋆⋆umax2λ−γ>0,(65)C^TC^+2C^TBm−εmmTBT⋆−ℐ<0,where(66)A^4=AS+BR+⋆+ε1CCT+ε2DDT+ε3EET+α+β+φ+ε7+ε8FFT,A^5=SFRT,A6=diagε4,ε5,A^7=111mTδ¯cδ¯dρ,A8=diagε1,ε2,ε3,ε6,ε7,ε8,ε9.Then, the uncertain disturbed nonlinear system (14) is robustly stabilizable via the saturated affine state-feedback control law (18), with K=RS−1, and(67)δ¯a=αε4,δ¯b=βε5+ε6,δ¯e=φε9.Proof.
We first consider expression (29). Thus, by applying the Young relation, we can obtain another related condition instead of inequality (30):(68)V˙x+μVx−γ≤μxTPx−μγ+2xTPA¯x+2xTPBm+ε1xTPCCTPx+ε1−1+ε2−1+ε3−1+ε2xTPDDTPx+ε3xTPEETPx+ε4δ¯a2+ε5+ε6δ¯b2+ε7+ε8+ε9δ¯e2xTPFFTPx+ε4−1xTFFTx+ε5−1xTKTKx+ε6−1mTm+ε7−1δ¯c2+ε8−1δ¯d2+ε9−1ρ2<0.
Let us introduce a new change of variables by posingα=ε4δ¯a2, β=ε5+ε6δ¯b2, and φ=ε9δ¯e2. Then, according to the previous section, we obtain the LMI (64), in which α, β, and φ are three decision variables. Hence, once the values of α, β, φ, ε4, ε5, ε6, and ε9 are obtained, we calculate δ¯a, δ¯b, and δ¯e according to the following relations: δ¯a=α/ε4, δ¯b=β/ε5+ε6, and δ¯e=φ/ε9.
The remaining problem is the parameterγ, which appears in both LMI (57) and LMI (58). We premultiply and postmultiply LMI (58) by the matrix ℐOOO⋆ℐOO⋆⋆Oℐ⋆⋆⋆O and then we obtain(69)SRTOO⋆ηmO⋆⋆umax2η1⋆⋆⋆1γ>0.
By applying the Schur lemma, inequality (69) is equivalent to the LMI (64). Now, the parameter γ is linear in the two LMIs (63) and (64).
In addition, the parameterμ appears in the LMI (57), whereas the parameter η appears in the LMI (58). As noted before, it is difficult to transform μ and η into two decision variables. Then, to simplify the computation procedure, we can pose μ=η=λ.
The objective is to look for maximum values of the uncertainty boundsδ¯a, δ¯b, δ¯c, δ¯d, and δ¯e and the disturbance bound ρ. The three parameters δ¯c, δ¯d, and ρ appear directly in the LMI (63) (matrix A^7). Then, these parameters should be maximized. However, the variables δ¯a, δ¯b, and δ¯e are incorporated, respectively, in the parameters α, β, and φ: α=ε4δ¯a2, β=ε5+ε6δ¯b2, and φ=ε9δ¯e2. Then, to maximize δ¯a, δ¯b, and δ¯e, we should maximize α, β, and φ. Moreover, as α (resp. β, φ) depends on ε4 (resp. ε5 and ε6, ε9), we should minimize ε4, ε5, ε6, and ε9 in order to ensure maximized values of δ¯a, δ¯b, and δ¯e. We emphasize that, in order to obtain a minimization objective function that can be handled with the LMI toolbox, we should transform the maximization problem of the variables α, β, φ, δ¯c, δ¯d, and ρ into a minimization problem of the quantity −α+β+φ+δ¯c+δ¯d+ρ. This completes the proof of Theorem 3.
It is worth to mention that Theorem 3 offers a wider choice of invariant ellipsoidsεP,γ (as γ varies) for optimization and will lead to less conservative estimation of the domain of attraction.
In addition, we stress that, in Theorem3, we imposed the two free parameters μ and η in Theorem 2 to be equal, i.e., μ=η=λ, in order to obtain an LMI with a single parameter λ. Nevertheless, this choice, which is the only solution to simplify computation, injects some degree of conservatism in the LMI conditions (63)–(65). Furthermore, the parameter λ is the only one that should be fixed a priori to obtain a solution of the optimization problem in Theorem 3.


## 4. Estimation of the Domain of Attraction

With all the ellipsoids satisfying the set invariance condition (21), we would like to choose from among them the largest one to get at least an estimation of the domain of attraction. Certain criteria are used in the literature to maximize the size of the attractive invariant ellipsoid εP,γ, such as maximization of its volume and maximization of the sum of its semiaxis [5, 9, 18, 21, 27, 47, 48, 52]. Actually, maximization of the volume of an ellipsoid is equivalent to minimization of the determinant of the Lyapunov matrix P, and maximization of the sum of its semiaxis is equivalent to minimization of the trace of the matrix P [9].In this part, our main goal is to find a robust saturated affine state-feedback control law (18) associated to the largest invariant ellipsoid ε⋆P⋆,γ⋆. Such largest invariant ellipsoid will be found for desired values of the maximum bounds of the parametric uncertainties δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e and also the maximum bound of the disturbance ρ. Thus, we will use LMI (63) (or systematically LMI (57)) for predefined values of these maximum bounds. However, we should also use the two LMIs (64) and (65) to look for a maximum value of the parameter γ and hence the associated gains K and m.Next, the size of the attractive invariant ellipsoidεP,γ is measured through the sum of its semiaxis. For this subject, we introduce the following theorem.Theorem 4.
The largest attractive invariant ellipsoid of the nonlinear system (15) with bounded disturbance (14) and norm-bounded parametric uncertainties (17), subject to the saturated affine state-feedback control law defined by expression (19) and constraint (18), is the ellipsoid εP,γ, for a fixed positive scalar εm≪1 and some positive parameter λ known a priori, where P=S−1 and γ, together with ε1, ε2, ε3, ε4, ε5, ε6, ε7, ε8, and ε9, are a solution of the following LMI-based optimization problem:(70)minimize−traceS−γ,A4+λSBmA5O⋆−λγOA7⋆⋆−A6O⋆⋆⋆−A8<0,SRTO⋆λm⋆⋆umax2λ−γ>0,C^TC^+2C^TBm−εmmTBT⋆−ℐ<0.Proof.
In expression (21), the attractive invariant ellipsoid ℰP,γ can be recast as(71)εPγ,1=x∈ℝ2:xTPγx≤1.
We emphasize that the trace of the matrixP/γ is equal to the sum of the semiaxis of the invariant ellipsoid εP/γ,1 for some fixed γ. Hence, the size (the length of the semiaxis) of εP/γ,1 increases as the trace of the matrix P/γ decreases. As P=S−1, P/γ=γS−1. Therefore, maximization of the size of εγS−1,1 is equivalent to the maximization of the trace of the matrix γS. As a result, to guarantee the largest attractive invariant ellipsoid εγS−1,1, we should maximize the parameter γ and also the trace of the matrix S. Notice that maximizing γ and traceS is equivalent to minimizing −traceS−γ. This completes the proof of Theorem 4.
It is worth to note that, for each fixed value of the parameterλ, the optimization problem in Theorem 4 provides a solution associated to an attractive invariant ellipsoid εP,γ with a (local) largest size, which is obtained for such λ. Therefore, in order to obtain the global maximum, we should vary the parameter λ, 0<λ<∞. To this end, the best manner is the use of the gridding method [68]. This method consists in making a new change of variable by defining κ=λ/1+λ and then λ=κ/1−κ. We know that λ>0 if and only if κ∈0,1. Then, we assign a uniform subdivision of the interval 0,1 and we solve the optimization problem in Theorem 4 for each value of this subdivision. Once this interval is covered, we look for the maximum value between all the obtained results. This maximum value corresponds to the largest invariant ellipsoid, which was obtained for some κ⋆ and then for some optimal values of the parameter λ, saying λ⋆.
In fact, finding the optimal valueλ⋆ and then the largest invariant attractive ellipsoid ε⋆ takes a lot of time. Indeed, suppose that we uniformly subdivide the interval 0,1, in which the parameter κ varies, and then we take Nκ values. Thus, for each value of κ, we run the optimization algorithm in Theorem 4. Let us assume that a single run takes tκ. Hence, the total simulation time in order to obtain the largest ellipsoid ε⋆ is about tc≈Nκ×tκ.Remark 5.
In several research works presented in the literature, the computation of the largest attractive invariant ellipsoidε⋆P⋆,γ⋆ is achieved mainly in two steps:(1)
Hunt for an invariant ellipsoidεP,1 by obtaining the matrix P(2)
With suchP, maximize the value of the parameter γ
Actually, these two steps are realized via two optimization algorithms. Compared with this hunting procedure, our proposed design of the saturated feedback control law guarantees the determination of the largest invariant ellipsoidε⋆P⋆,γ⋆ in only one step and then with only one optimization algorithm.


## 5. Simulation Results

In this section, we provide some simulation results to demonstrate the effectiveness of our developed method for the synthesis of the saturated affine state-feedback control law (19) for the robust stabilization of the disturbed uncertain nonlinear dynamics (15) of the pitch model of the helicopter. Moreover, we aim at identifying and analyzing the largest attractive invariant ellipsoid ε⋆P⋆,γ⋆ in order to obtain an estimation of the domain of attraction. Thus, in the sequel, we take two values of the saturation level in (18) of the control law u: umax=1 and umax=10. Moreover, we fix the parameter εm=10−5.The values of the parameters figured in the nonlinear dynamics (12) or of the pitch model of the helicopter are as follows [55–57]: Iyy=0.0283kg/m2, mheli=0.9941Kg, lcgx=0.0134m, lcgz=0.0289m, FvM=0.0041Nm/rad/s, g=9.81m/s2, and s=0.0057m/s2.Remark 6.
The choice of the parameters of the maximum value of the control inputu, umax, is based on two facts: (1) in the literature, the common choice of the saturation limit is umax=1 according to the invariance sets’ concept and (2) in the present work, we have considered a general case of the actuator saturation limit in the development of LMI constraints and hence in the design of the state-feedback controller in order to show the effect of the saturation limit umax on the largeness of the invariant attractive ellipsoid.
Thus, in our investigation, we selected two values of the saturation limit:umax=1 (which is the classical choice) and umax=10. We can also choose another value of such saturation limit such as umax=24, and we will obtain similar results.
Actually, the saturation limits are the intrinsic requirement of the physical actuators. Thus, the adopted different values of theumax will be used in the sequel just to show the feasibility and the conservativeness of the proposed design method of the two gains of the affine state-feedback controller and also for the computation of the largest domain of attraction in different situations.Remark 7.
It is worth to mention that for the computation of the solutions of the LMIs in Theorem2 and the LMI-based optimization problem in Theorems 3 and 4, we use the LMI toolbox of MATLAB. We can also use the toolbox YALMIP [10] in order to solve these LMI problems. Moreover, we can use it to (locally) solve some BMI problems using the bisection method.


### 5.1. Numerical Results

The optimization problem subject to LMI constraints in Theorem3 provides numerical results illustrated in Table 1 for the saturation level umax=1 and for different values of the fixed parameter λ. In this table, we provide the two gains, K and m, of the saturated affine state-feedback control law u, the maximum bounds of the parametric uncertainties, i.e., δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e, the maximum bound of the disturbance w, i.e., ρ, and the parameter γ. It is obvious that the three uncertainty bounds, δ¯a, δ¯b, and δ¯e, are very small, around 10−4. The two parameters, δ¯c and δ¯d, are found to be equal and decrease slightly as λ increases. However, the value of the allowable disturbance bound ρ is found to be very high, about 6×103.Table 1
Numerical simulation results obtained using the optimization algorithm in Theorem 3 for different values of the parameterλ and for umax=1.
λ
=
0.01
λ
=
0.1
λ
=
1
λ
=
10
λ
=
100
K
T
10
−
3
×
−
0.0372
−
0.2566
10
−
3
×
−
0.3786
−
2.6152
−
0.0204
−
0.0437
−
2.3360
−
0.5053
−
236.7854
−
5.1249
m
0.013321
0.013321
0.013321
0.013321
0.013321
δ
¯
a
10
−
4
×
4.5975
10
−
4
×
4.6197
10
−
4
×
3.0634
10
−
4
×
2.7950
10
−
4
×
2.7613
δ
¯
b
10
−
4
×
3.2525
10
−
4
×
3.2666
10
−
4
×
2.1641
10
−
4
×
1.9764
10
−
4
×
1.9526
δ
¯
c
=
δ
¯
d
3.6314
3.6515
2.2531
1.9868
1.9407
δ
¯
e
10
−
4
×
4.5944
10
−
4
×
4.6215
10
−
4
×
3.0796
10
−
4
×
2.7990
10
−
4
×
2.7613
ρ
10
3
×
6.4069
10
3
×
6.4673
10
3
×
6.0719
10
3
×
5.4419
10
3
×
5.7484
γ
81.4039
8.1419
0.8527
0.0863
0.0087We emphasize that the obtained results for the allowable maximum bounds of the parametric uncertainties,δ¯a, δ¯b, and δ¯e, are dissatisfying (they are found to be very small). Moreover, the maximum disturbance bound is found to be very high. It is worth noting that the optimization algorithm in Theorem 3 is characterized by an objective function that depends on 10 parameters ε4, ε5, ε6, ε9, α, β, φ, δ¯c, δ¯d, and ρ, which are optimized together. Then, in order to improve the obtained results, we will fix in the next the parameter ρ and the remaining ones will be optimized. As discussed in the beginning of this paper, in the second section, the external disturbance w can represent the turbulent moments on the helicopter. Without loss of generality, we will fix ρ=10.Thus, forρ=10, the optimization algorithm subject to LMI constraints in Theorem 3 gives numerical results illustrated in Table 2 for the case umax=1 and in Table 3 for umax=10. As in Table 1, we provide, for different values of the parameter λ, the gain matrix K, the gain constant m, the maximum bounds of the parametric uncertainties δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e, and the parameter γ.Table 2
Numerical simulation results using Theorem 3 for different values of the parameterλ and for ρ=10 and umax=1.
λ
=
0.01
λ
=
0.1
λ
=
1
λ
=
10
λ
=
100
λ
=
200
K
T
−
0.00089
−
0.00503
−
0.00298
−
0.02858
−
0.09235
−
0.11705
−
3.6488
−
0.6523
−
298.7673
−
5.8965
10
3
×
−
1.1796
−
0.0117
m
0.013275
0.013283
0.013268
0.013243
0.013251
0.013262
δ
¯
a
0.0067
0.0181
0.7729
1.9138
5.4867
7.7133
δ
¯
b
0.0800
0.1020
2.2687
2.6586
2.6496
2.6447
δ
¯
c
=
δ
¯
d
0.0048
0.0029
0.0043
0.0011
0.0022
0.0038
δ
¯
e
0.0112
0.0155
0.4577
0.7132
0.7459
0.7475
γ
99.1571
9.9099
0.9893
0.0986
0.0099
0.0049Table 3
Numerical simulation results using Theorem 3 for different values of the parameterλ and for ρ=10 and umax=10.
λ
=
0.01
λ
=
0.1
λ
=
1
λ
=
10
λ
=
100
λ
=
200
K
T
−
0.00048
−
0.04438
−
0.00581
−
0.05694
−
0.0917
−
0.1166
−
3.6420
−
0.6515
−
297.9499
−
5.8866
10
3
×
−
1.1761
−
0.0117
m
0.01327
0.013319
0.013316
0.013320
0.013321
0.013321
δ
¯
a
0.0321
0.8835
1.2411
2.4613
6.8305
9.3191
δ
¯
b
0.1500
3.6262
3.6585
3.4313
3.3309
3.3217
δ
¯
c
=
δ
¯
d
9.9995
14.5636
14.3332
14.3665
14.4657
14.7111
δ
¯
e
0.0360
2.1881
4.8950
6.0531
6.2291
6.1668
γ
9990.7
997.292
99.0173
9.8937
0.9851
0.4922We note from Tables2 and 3 that the optimization problem provides identical values for the two uncertainty bounds δ¯c and δ¯d. Moreover, for all values of λ, the value of the gain m is about 0.0133 as in Table 1. In addition, it is evident that, for the same value of parameter λ, we have almost the same gain matrix K. Furthermore, as λ increases, the size of the gain matrix K and the uncertainty bounds, δ¯a, δ¯b, and δ¯e, increase. For umax=1, δ¯c is found to be around 10−3, whereas for umax=10, δ¯c is about 14. In addition, it is evident that the values of the maximum bounds of the parametric uncertainties δ¯a, δ¯b, and δ¯e are bigger than those obtained in Table 1 since the parameter ρ is fixed here.We can also observe from Tables2 and 3 that, as the parameter λ increases, the value of the parameter γ decreases. Actually, we can note that γ≈umax2/λ. This shows that the size of the attractive invariant ellipsoid, which depends on γ, varies with respect to the value of the saturation level umax.


### 5.2. Computation of the Largest Attractive Invariant Ellipsoid

To obtain the largest attractive invariant ellipsoidε⋆P⋆,γ⋆, we solve the optimization problem in Theorem 4. Thus, in order to obtain the global largest ellipsoid ε⋆P⋆,γ⋆, we should vary the free parameter λ according to the gridding method as noted before. Moreover, in this optimization problem, we fix the bounds of the parametric uncertainties and we take a common value, δ¯, for all these bounds like so: for umax=1, we take δ¯a=⋯=δ¯e=δ¯=1, whereas for umax=10, we fix δ¯a=⋯=δ¯e=δ¯=10. Actually, with δ¯=10 and umax=1, the optimization problem in Theorem 4 is unfeasible. It was found unfeasible also for all δ¯≥2 and umax=1. Then, we minimized the uncertainty bound to obtain a feasible solution. This will be demonstrated and discussed next.Figure2 reveals evolution of the trace of the matrix γS, or systematically the sum of the semiaxis of the invariant ellipsoid εS−1,γ, as the parameter λ varies. It is obvious from Figure 2 that, by sweeping through λ, we obtain the maximum trace of the matrix γS for the two cases umax=1 and umax=10. Such maximum trace corresponds to the largest invariant ellipsoid ε⋆P⋆,γ⋆. Obtained results are summarized in Table 4. In this table, we provide the parameters, P⋆ and γ⋆, of the largest attractive invariant ellipsoid ε⋆P⋆,γ⋆, the associated feedback gains, K⋆ and m⋆, of the saturated affine state-feedback control law u, the nominal value, λ⋆, of the free parameter λ, and the size of the attractive invariant ellipsoid ε⋆P⋆,γ⋆. According to Figure 2 and Table 4, in the first case umax=1, the size of the largest invariant ellipsoid is about 1.7906. However, in the second case umax=10, such ellipsoid has a size about 46.5870. We recall that, for umax=1, we have chosen δ¯=1 and for umax=10, we have adopted δ¯=10.Figure 2
Variation of the trace of the matrixγS with respect to the parameter λ and for two different values of the saturation level: umax=1 and umax=10. Here, for umax=1 (resp. umax=10), we have adopted δ¯=1 (resp. δ¯=10). Moreover, in the two cases, we have fixed ρ=10.Table 4
Numerical results using Theorem 4 corresponding to the largest attractive invariant ellipsoidε⋆P⋆,γ⋆ for umax=1 and umax=10.
u
max
=
1 and δ¯=1
u
max
=
10 and δ¯=10
P
⋆
10.1505
0.5523
0.5523
0.0628
64.4531
2.1626
2.1626
0.1963
K
⋆
−
9.0604
−
1.0242
−
20.3362
−
1.8458
m
⋆
0.013235
0.013303
λ
⋆
16.9211
17.3824
γ
⋆
0.0583
5.7446
trace
γ
⋆
S
⋆
1.7906
46.5870Our objective now is to see the effect of the maximum bound of the different parametric uncertaintiesδ¯a, δ¯b, δ¯c, δ¯d, and δ¯e and also the saturation limit umax on the size of the attractive invariant ellipsoid εP,γ. To do this, we used the optimization algorithm in Theorem 4 and we varied the parameter λ as mentioned previously. This task was achieved for a single value of the maximum uncertainty bound, saying δmax=δ¯a=⋯=δ¯e. Thus, we obtained the largest ellipsoid ε⋆P⋆,γ⋆. Now, we will repeat the same procedure by also varying the parameter δmax and fixing the saturation level umax. Figure 3 reveals obtained results for umax=1 and umax=10. We stress first that, for umax=1, the allowable maximum bound of the parametric uncertainty allowing to obtain an invariant ellipsoid ε⋆P⋆,γ⋆ is δmax=1.06. However, for umax=10, this value of δmax increases considerably and reaches the limit 12.05.Figure 3
Variation of the size of the largest attractive invariant ellipsoidε⋆P⋆,γ⋆ with respect to the variation of the allowable maximum bound of the parametric uncertainties for the two cases: umax=1 and umax=10. Here, we have fixed δ¯a=⋯=δ¯e=δmax, with always ρ=10.Moreover, it is obvious that, for each value ofumax, the size of the largest ellipsoid increases rapidly as δmax decreases. A tiny decrease of the uncertainty bound δmax leads to a very important increase in the size of the ellipsoid ε⋆P⋆,γ⋆. For example, for umax=1 and for δmax=0.5, the size is about 103 and for δmax=0.1, the size is around 109. Similarly, for the saturation limit umax=10 and for δmax=0.5, the size is around 108 and for δmax=1.5, the size is around 105.Furthermore, it is clear that the size of the largest invariant ellipsoid decreases asδmax increases. The ellipsoid becomes smaller as the uncertainty bound δmax approaches its feasible limit (i.e., the value 1.06 for umax=1 and 12.05 for umax=10).In addition, we remark that, for the same value of uncertainty boundδmax, the size of the largest ellipsoid ε⋆P⋆,γ⋆ increases as the saturation limit umax increases (as noted previously for δmax=0.5).Using numerical results illustrated in Table4, we plotted in Figure 4 the corresponding largest attractive invariant ellipsoids ε⋆P⋆,γ⋆ for the two cases umax=1 and umax=10. In Figure 4, the two inclined lines (colored in pink and green) correspond to the bounds of the actuator saturation (20), which are defined as Kx+m=±umax. The lower line (colored in pink) is depicted for Kx+m=umax.Figure 4
Largest attractive invariant ellipsoidsε⋆P⋆,γ⋆ computed for two different values of umax.
(a)
(b)It is obvious from Figure4 that, in the two cases, the largest invariant ellipsoid ε⋆P⋆,γ⋆ is contained in the region ℒK,m,umax, where Kx+m≤umax. Therefore, the constraints on the affine state-feedback control law given in (18) are well respected.In the end of this part, we can note the two following statements:(1)
For a constant saturation limitumax, an increase of the maximum bound of the parametric uncertainties leads to the decrease of the largeness of the attractive invariant ellipsoid.(2)
For a constant maximum bound of the uncertainty, an increase of the saturation limitumax causes the attractive invariant ellipsoid to become larger.Remark 8.
In some research papers, the definition of the attractive invariant ellipsoid was chosen to be as follows:(72)ε´P=x∈ℝnx:xTPx≤1.
This setε´P is equivalent to that adopted in the present work in (21), that is, εP,1. Hence, we have γ=1. Our immediate concern is to show that the adopted ellipsoid (21) with a free parameter γ gives less conservative results compared with the ellipsoid (72). We use the same optimization algorithm in Theorem 4 where in the present case, we fix the parameter γ as a constant one and then we take γ=1. Table 5 shows the numerical results obtained for a fixed parameter γ=1. Notice that the symbol ! in Table 5 means that the LMI-based optimization problem in Theorem 4 is unfeasible. To realize a comparison from conservatism point of view, we adopt the same parameters in Table 4. As noted previously, the maximum bound of the disturbance is fixed to ρ=10. It is clear from Table 5 that, for the first case umax=1 and δ¯=1, the optimization problem in Theorem 4 was found to be unfeasible, with γ=1. It was found to be feasible for all δ¯≤0.648. Recall that, for a free parameter γ, the optimization problem was found to be feasible for all δ¯<2. This fact shows that, using a free parameter γ in the definition of the attractive invariant ellipsoid εP,γ, that is, the set (21), instead of fixing γ=1, gives less conservative stability conditions. Moreover, from Table 5 and for the case umax=10 and δ¯=10, the optimization problem with γ=1 is found to be feasible, as in Table 4. Nevertheless, the largest ellipsoid ε⋆′P⋆ obtained for γ=1, is too small compared with that obtained using a free parameter γ, that is, the ellipsoid ε⋆P⋆,γ⋆. Recall that the largeness of the attractive invariant ellipsoid is measured by the “trace” function as noted in Tables 4 and 5. As noted in Table 4, for a free parameter γ, the volume of the largest ellipsoid was found to be about 46.5870. However, for a fixed parameter γ=1, the volume of the largest ellipsoid ε⋆′P⋆ is about 7.3840. The difference between the two sizes of the ellipsoid is evident. Another attractive result that can be observed from Table 5 is that the obtained matrix gain K⋆ is too large compared with that obtained in Table 4.
Accordingly, we emphasize that the parameterγ in the definition of the attractive ellipsoid (21) leads to less conservative results and then less restrictive LMI stability conditions. In addition, it contributes in obtaining a largest ellipsoid with a controller gain having a small size. Hence, the choice of γ≠0 reduces the conservatism of the LMI-based optimization problem in Theorem 4.Table 5
Numerical results using Theorem 4 with a fixed parameterγ=1 and for umax=1 and umax=10.
u
max
=
1 and δ¯=1
u
max
=
10 and δ¯=10
P
⋆
!
10
3
×
2.1069
0.0255
0.0255
0.0004
K
⋆
!
10
2
×
−
2.8612
−
0.0633
m
⋆
!
0.013317
λ
⋆
!
99.6036
trace
S
⋆
!
7.3840Remark 9.
We noted in the end of Section4, just before Remark 5, that the computation time for solving the LMI-based optimization problem in Theorem 4 and then for the identification of the largest invariant attractive ellipsoid ε⋆P⋆,γ⋆ depends on the number Nκ of uniformly distributed points κ in the interval 0,1 and the simulation time tκ for each value of the parameter κ. The total simulation time tc can be computed to be about tc≈Nκ×tκ. Tables 6 and 7 present the simulation results for different values of the parameters Nκ for umax=1 (Table 6) and for umax=10 (Table 7). Recall that, for umax=1, we have δ¯=1, whereas for umax=10, we fixed δ¯=10. Moreover, the maximum bound ρ of the disturbance is fixed to be ρ=10 in the two cases. Actually, in order to obtain these results in Tables 6 and 7, we have fixed the interval in which the parameter κ varies as follows: 0.01:δκ:0.99, where we have selected four cases: δκ=0.01, δκ=0.001, δκ=0.0005, and δκ=0.0002. For these values of δκ, it corresponds, respectively, the number Nκ=99, Nκ=981, Nκ=1961, and Nκ=4901. Thus, using these intervals and these values of the parameters Nκ, we simulated the optimization problem under LMI constraints in Theorem 4 and then we obtained the results in Tables 6 and 7. In these Tables, we give the two optimal gains, K⋆ and m⋆, of the controller, λ⋆ (notice that λ=κ/1−κ), γ⋆, the size of the ellipsoid (traceγ⋆S⋆), the simulation/computation time tc, and the number of feasible solutions Nfs. It is important to note that not all the Nκ-cases are feasible. From the results in Tables 6 and 7, it reveals that the number of feasible solutions is only about 4.8% for umax=1 and about 6% for umax=10. The computation time tc for finding the largest ellipsoid increases significantly as Nκ increases too. We also stress that as Nκ increases, the optimal solution converges to that already obtained in Table 4. We found the same values for Nκ=4901. Nevertheless, in this case, the computation time is very high, it is computed to be about 4.5 minutes for umax=1 and it is about 2.2 minutes for the second case umax=10.Table 6
Computation time for solving the LMI-based optimization algorithm in Theorem 4 for the caseumax=1.
N
κ
99
981
1961
4901
K
⋆
T
−
10.5679
−
1.1050
−
9.2061
−
1.0321
−
9.1147
−
1.0271
−
9.0604
−
1.0242
m
⋆
0.013238
0.013237
0.013239
0.013235
λ
⋆
19.0000
17.1818
17.0180
16.9211
γ
⋆
0.0519
0.0574
0.0580
0.0583
trace
γ
⋆
S
⋆
1.6557
1.7874
1.7894
1.7906
t
c
s
6.1988
54.9516
106.1182
264.1296
N
fs
5
48
95
238Table 7
Computation time for solving the LMI-based optimization algorithm in Theorem 4 for the caseumax=10.
N
κ
99
981
1961
4901
K
⋆
T
−
18.6484
−
1.8069
−
19.7871
−
1.8309
−
20.3059
−
1.8453
−
20.3362
−
1.8458
m
⋆
0.013306
0.013311
0.013306
0.013303
λ
⋆
15.6667
16.8571
17.3486
17.3824
γ
⋆
6.3740
5.9236
5.7557
5.7446
trace
γ
⋆
S
⋆
45.7911
46.5153
46.5625
46.5870
t
c
s
3.1946
27.3301
53.4916
132.4823
N
fs
6
59
117
293


### 5.3. Simulations for the Robust Stabilization of the Pitch Dynamics

In this part, we would show the robustness and effectiveness of the designed saturated affine state-feedback control law (19) in the robust stabilization of the uncertain disturbed nonlinear dynamics (15) of the pitch model of the helicopter. The control gains K and m are provided in Table 4. We recall that, for umax=1, we have δ¯a=δ¯b=δ¯c=δ¯d=δ¯e=1, whereas for umax=10, we take δ¯a=δ¯b=δ¯c=δ¯d=δ¯e=10. Moreover, we recall that the external disturbance w in the nonlinear dynamics (15) is bounded according to condition (14) with ρ=10. In addition, the uncertain parameters δa, δb, δc, δd, and δe in (15) vary randomly over time so that the boundedness condition (17) is satisfied. Furthermore, for the simulation of the controlled nonlinear system (15), we will take first an initial condition, x0, located outside the largest attractive ellipsoid ε⋆P⋆,γ⋆: for the case umax=1, we take x0=0.1−1T, whereas for the case umax=10, we take x0=0.3−6T. Each initial condition is located near the boundary of the corresponding ellipsoid (see Figure 4). Furthermore, we will take an initial condition located outside the invariant attractive ellipsoid: x0=5−10T, for the two saturation limits.First of all, we will study the dynamics of the helicopter model under the saturated control lawu in the nominal case, that is, without parametric uncertainties and disturbance. Figure 5(a) shows temporal evolution of the states x1 and x2 of the certain undisturbed controlled nonlinear system (15), and Figure 5(b) reveals the saturated affine state-feedback control law u for the saturation limit umax=1. It is obvious that the state of the pitch dynamics converges to zero, which is our main objective in this paper. Moreover, the control law u converges to a constant value, which is found to be equal to the gain m. Actually, such constant value of the control input will ensure that the pitch dynamics of the helicopter stays at the zero state. In addition, we note that the reached maximum value of the control law u is about 0.23, which is less than the desired saturation limit umax=1.Figure 5
Evolution of the two states,x1 and x2, of the nonlinear system (15) and the saturated affine state-feedback control law u for umax=1 in the nominal case. Here, the system is without uncertainties (δa,b,c,d,e=0) and disturbance (w=0).
(a)
(b)We have also analyzed the nominal pitch dynamics of the helicopter model for the caseumax=10. We have observed almost the same behavior in Figure 5. The reached maximum value of the controller u is around 5. The controller converges to the constant value m≈0.01336.Now, we will show simulation results for the controlled nonlinear system subject to both parametric uncertainties and external disturbance. Simulation results are shown in Figure6 for umax=1 and Figure 7 for umax=10. Figure 6(a) (resp. Figure 7(a)) presents temporal evolution of the two states, x1 and x2, of the nonlinear system (15) under uncertainties and disturbance. Figure 6(b) (resp. Figure 7(b)) reveals evolution of the saturated control law u for umax=1 (resp. umax=10). As noted previously, the initial condition is located inside the largest attractive invariant ellipsoid. Moreover, the disturbance w is injected into the system at t=2s and during 3s. We recall that the external disturbance w satisfies condition (14) with ρ=10.Figure 6
Temporal evolution of the statesx1 and x2 of the nonlinear system (15) and the saturated affine state-feedback control law u for the saturation level umax=1. Here, the system is subject to randomly time-varying uncertainties and also a randomly time-varying external disturbing torque w. Moreover, δ¯a,b,c,d,e=1.
(a)
(b)Figure 7
Evolution of the two statesx1 and x2 of the nonlinear system (15) (a) and the saturated affine state-feedback control law u (b) for the saturation level umax=10. In this case, we have δ¯a,b,c,d,e=10.
(a)
(b)It is obvious from Figure6(a) that the motion of the pitch dynamics of the helicopter model experiences some fluctuations around the desired zero state when the disturbance is applied. Once the effect of the external disturbing torque w vanishes, the controlled system stabilizes again around its desired position with some very weak perturbations provoked by the injected parametric uncertainties. Moreover, in Figure 6(b), the control signal u varies around the value m=0.0133. The control signal varies between two very small values: ±0.1. We stress that the effects of the parametric uncertainties and the external disturbance are compensated. This shows the robustness of the control law u towards the parametric uncertainties and the external disturbance with high amplitude.However, forumax=10, the effect of the parametric uncertainties and the external disturbance w is clear in Figure 7. Figure 7(a) is almost identical to Figure 6(a). Moreover, when the disturbance is injected, the control signal u undergoes some fluctuations, which vary between ±1.5. This happens because the maximum bound of the uncertainties is very large (δ¯a,…,e=10) in this case compared with the first case, i.e., for umax=1.It is worth to note that, in the previous three cases, the saturation level±umax of the controller was well respected.We have tested another case by taking an initial condition located outside the largest attractive invariant ellipsoidε⋆P⋆,γ⋆. We take here only the case umax=1, and then the corresponding largest ellipsoid is given by Figure 4(a). As noted previously, the initial condition is chosen to be x0=5−10T. Obtained results are illustrated in Figure 8. We observe first from Figure 8(a) that the state x1 of the helicopter model experiences some smooth oscillations before its convergence and stabilization at the zero position despite the presence of the external disturbance and the parametric uncertainties. In fact, only weak fluctuations are observed as in Figure 7(a). However, the interesting phenomenon observed here is the saturation of the control law u depicted in Figure 8(b).Figure 8
Temporal evolution of the two statesx1 and x2 of the uncertain disturbed nonlinear system (15) and the saturated affine state-feedback control law u for the saturation level umax=1. Here, the initial condition is selected to be outside the largest attractive invariant ellipsoid ε⋆P⋆,γ⋆ in Figure 4(a).
(a)
(b)Remark 10.
It is worth to note that, for a predefined set of the system parameters, the previous established optimization problems under LMI constraints are (should be) simulated offline in order to compute the largest attractive ellipsoid and then the associated feedback gainsK and m of the input-saturated affine state-feedback controller u before its application into a real helicopter system in practice. Thus, for a prescribed saturation level umax, the possible maximum allowable bounds of the parametric uncertainties δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e and the possible maximum bound of the external disturbance signal w, i.e., ρ, we look, as described previously, for the largest attractive invariant ellipsoid ε⋆P⋆,γ⋆. Thus, once this set is found, the corresponding feedback gains K and m will be used in order to stabilize the pitch dynamics of the simple helicopter model in a real-world application.


## 5.1. Numerical Results

The optimization problem subject to LMI constraints in Theorem3 provides numerical results illustrated in Table 1 for the saturation level umax=1 and for different values of the fixed parameter λ. In this table, we provide the two gains, K and m, of the saturated affine state-feedback control law u, the maximum bounds of the parametric uncertainties, i.e., δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e, the maximum bound of the disturbance w, i.e., ρ, and the parameter γ. It is obvious that the three uncertainty bounds, δ¯a, δ¯b, and δ¯e, are very small, around 10−4. The two parameters, δ¯c and δ¯d, are found to be equal and decrease slightly as λ increases. However, the value of the allowable disturbance bound ρ is found to be very high, about 6×103.Table 1
Numerical simulation results obtained using the optimization algorithm in Theorem 3 for different values of the parameterλ and for umax=1.
λ
=
0.01
λ
=
0.1
λ
=
1
λ
=
10
λ
=
100
K
T
10
−
3
×
−
0.0372
−
0.2566
10
−
3
×
−
0.3786
−
2.6152
−
0.0204
−
0.0437
−
2.3360
−
0.5053
−
236.7854
−
5.1249
m
0.013321
0.013321
0.013321
0.013321
0.013321
δ
¯
a
10
−
4
×
4.5975
10
−
4
×
4.6197
10
−
4
×
3.0634
10
−
4
×
2.7950
10
−
4
×
2.7613
δ
¯
b
10
−
4
×
3.2525
10
−
4
×
3.2666
10
−
4
×
2.1641
10
−
4
×
1.9764
10
−
4
×
1.9526
δ
¯
c
=
δ
¯
d
3.6314
3.6515
2.2531
1.9868
1.9407
δ
¯
e
10
−
4
×
4.5944
10
−
4
×
4.6215
10
−
4
×
3.0796
10
−
4
×
2.7990
10
−
4
×
2.7613
ρ
10
3
×
6.4069
10
3
×
6.4673
10
3
×
6.0719
10
3
×
5.4419
10
3
×
5.7484
γ
81.4039
8.1419
0.8527
0.0863
0.0087We emphasize that the obtained results for the allowable maximum bounds of the parametric uncertainties,δ¯a, δ¯b, and δ¯e, are dissatisfying (they are found to be very small). Moreover, the maximum disturbance bound is found to be very high. It is worth noting that the optimization algorithm in Theorem 3 is characterized by an objective function that depends on 10 parameters ε4, ε5, ε6, ε9, α, β, φ, δ¯c, δ¯d, and ρ, which are optimized together. Then, in order to improve the obtained results, we will fix in the next the parameter ρ and the remaining ones will be optimized. As discussed in the beginning of this paper, in the second section, the external disturbance w can represent the turbulent moments on the helicopter. Without loss of generality, we will fix ρ=10.Thus, forρ=10, the optimization algorithm subject to LMI constraints in Theorem 3 gives numerical results illustrated in Table 2 for the case umax=1 and in Table 3 for umax=10. As in Table 1, we provide, for different values of the parameter λ, the gain matrix K, the gain constant m, the maximum bounds of the parametric uncertainties δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e, and the parameter γ.Table 2
Numerical simulation results using Theorem 3 for different values of the parameterλ and for ρ=10 and umax=1.
λ
=
0.01
λ
=
0.1
λ
=
1
λ
=
10
λ
=
100
λ
=
200
K
T
−
0.00089
−
0.00503
−
0.00298
−
0.02858
−
0.09235
−
0.11705
−
3.6488
−
0.6523
−
298.7673
−
5.8965
10
3
×
−
1.1796
−
0.0117
m
0.013275
0.013283
0.013268
0.013243
0.013251
0.013262
δ
¯
a
0.0067
0.0181
0.7729
1.9138
5.4867
7.7133
δ
¯
b
0.0800
0.1020
2.2687
2.6586
2.6496
2.6447
δ
¯
c
=
δ
¯
d
0.0048
0.0029
0.0043
0.0011
0.0022
0.0038
δ
¯
e
0.0112
0.0155
0.4577
0.7132
0.7459
0.7475
γ
99.1571
9.9099
0.9893
0.0986
0.0099
0.0049Table 3
Numerical simulation results using Theorem 3 for different values of the parameterλ and for ρ=10 and umax=10.
λ
=
0.01
λ
=
0.1
λ
=
1
λ
=
10
λ
=
100
λ
=
200
K
T
−
0.00048
−
0.04438
−
0.00581
−
0.05694
−
0.0917
−
0.1166
−
3.6420
−
0.6515
−
297.9499
−
5.8866
10
3
×
−
1.1761
−
0.0117
m
0.01327
0.013319
0.013316
0.013320
0.013321
0.013321
δ
¯
a
0.0321
0.8835
1.2411
2.4613
6.8305
9.3191
δ
¯
b
0.1500
3.6262
3.6585
3.4313
3.3309
3.3217
δ
¯
c
=
δ
¯
d
9.9995
14.5636
14.3332
14.3665
14.4657
14.7111
δ
¯
e
0.0360
2.1881
4.8950
6.0531
6.2291
6.1668
γ
9990.7
997.292
99.0173
9.8937
0.9851
0.4922We note from Tables2 and 3 that the optimization problem provides identical values for the two uncertainty bounds δ¯c and δ¯d. Moreover, for all values of λ, the value of the gain m is about 0.0133 as in Table 1. In addition, it is evident that, for the same value of parameter λ, we have almost the same gain matrix K. Furthermore, as λ increases, the size of the gain matrix K and the uncertainty bounds, δ¯a, δ¯b, and δ¯e, increase. For umax=1, δ¯c is found to be around 10−3, whereas for umax=10, δ¯c is about 14. In addition, it is evident that the values of the maximum bounds of the parametric uncertainties δ¯a, δ¯b, and δ¯e are bigger than those obtained in Table 1 since the parameter ρ is fixed here.We can also observe from Tables2 and 3 that, as the parameter λ increases, the value of the parameter γ decreases. Actually, we can note that γ≈umax2/λ. This shows that the size of the attractive invariant ellipsoid, which depends on γ, varies with respect to the value of the saturation level umax.


## 5.2. Computation of the Largest Attractive Invariant Ellipsoid

To obtain the largest attractive invariant ellipsoidε⋆P⋆,γ⋆, we solve the optimization problem in Theorem 4. Thus, in order to obtain the global largest ellipsoid ε⋆P⋆,γ⋆, we should vary the free parameter λ according to the gridding method as noted before. Moreover, in this optimization problem, we fix the bounds of the parametric uncertainties and we take a common value, δ¯, for all these bounds like so: for umax=1, we take δ¯a=⋯=δ¯e=δ¯=1, whereas for umax=10, we fix δ¯a=⋯=δ¯e=δ¯=10. Actually, with δ¯=10 and umax=1, the optimization problem in Theorem 4 is unfeasible. It was found unfeasible also for all δ¯≥2 and umax=1. Then, we minimized the uncertainty bound to obtain a feasible solution. This will be demonstrated and discussed next.Figure2 reveals evolution of the trace of the matrix γS, or systematically the sum of the semiaxis of the invariant ellipsoid εS−1,γ, as the parameter λ varies. It is obvious from Figure 2 that, by sweeping through λ, we obtain the maximum trace of the matrix γS for the two cases umax=1 and umax=10. Such maximum trace corresponds to the largest invariant ellipsoid ε⋆P⋆,γ⋆. Obtained results are summarized in Table 4. In this table, we provide the parameters, P⋆ and γ⋆, of the largest attractive invariant ellipsoid ε⋆P⋆,γ⋆, the associated feedback gains, K⋆ and m⋆, of the saturated affine state-feedback control law u, the nominal value, λ⋆, of the free parameter λ, and the size of the attractive invariant ellipsoid ε⋆P⋆,γ⋆. According to Figure 2 and Table 4, in the first case umax=1, the size of the largest invariant ellipsoid is about 1.7906. However, in the second case umax=10, such ellipsoid has a size about 46.5870. We recall that, for umax=1, we have chosen δ¯=1 and for umax=10, we have adopted δ¯=10.Figure 2
Variation of the trace of the matrixγS with respect to the parameter λ and for two different values of the saturation level: umax=1 and umax=10. Here, for umax=1 (resp. umax=10), we have adopted δ¯=1 (resp. δ¯=10). Moreover, in the two cases, we have fixed ρ=10.Table 4
Numerical results using Theorem 4 corresponding to the largest attractive invariant ellipsoidε⋆P⋆,γ⋆ for umax=1 and umax=10.
u
max
=
1 and δ¯=1
u
max
=
10 and δ¯=10
P
⋆
10.1505
0.5523
0.5523
0.0628
64.4531
2.1626
2.1626
0.1963
K
⋆
−
9.0604
−
1.0242
−
20.3362
−
1.8458
m
⋆
0.013235
0.013303
λ
⋆
16.9211
17.3824
γ
⋆
0.0583
5.7446
trace
γ
⋆
S
⋆
1.7906
46.5870Our objective now is to see the effect of the maximum bound of the different parametric uncertaintiesδ¯a, δ¯b, δ¯c, δ¯d, and δ¯e and also the saturation limit umax on the size of the attractive invariant ellipsoid εP,γ. To do this, we used the optimization algorithm in Theorem 4 and we varied the parameter λ as mentioned previously. This task was achieved for a single value of the maximum uncertainty bound, saying δmax=δ¯a=⋯=δ¯e. Thus, we obtained the largest ellipsoid ε⋆P⋆,γ⋆. Now, we will repeat the same procedure by also varying the parameter δmax and fixing the saturation level umax. Figure 3 reveals obtained results for umax=1 and umax=10. We stress first that, for umax=1, the allowable maximum bound of the parametric uncertainty allowing to obtain an invariant ellipsoid ε⋆P⋆,γ⋆ is δmax=1.06. However, for umax=10, this value of δmax increases considerably and reaches the limit 12.05.Figure 3
Variation of the size of the largest attractive invariant ellipsoidε⋆P⋆,γ⋆ with respect to the variation of the allowable maximum bound of the parametric uncertainties for the two cases: umax=1 and umax=10. Here, we have fixed δ¯a=⋯=δ¯e=δmax, with always ρ=10.Moreover, it is obvious that, for each value ofumax, the size of the largest ellipsoid increases rapidly as δmax decreases. A tiny decrease of the uncertainty bound δmax leads to a very important increase in the size of the ellipsoid ε⋆P⋆,γ⋆. For example, for umax=1 and for δmax=0.5, the size is about 103 and for δmax=0.1, the size is around 109. Similarly, for the saturation limit umax=10 and for δmax=0.5, the size is around 108 and for δmax=1.5, the size is around 105.Furthermore, it is clear that the size of the largest invariant ellipsoid decreases asδmax increases. The ellipsoid becomes smaller as the uncertainty bound δmax approaches its feasible limit (i.e., the value 1.06 for umax=1 and 12.05 for umax=10).In addition, we remark that, for the same value of uncertainty boundδmax, the size of the largest ellipsoid ε⋆P⋆,γ⋆ increases as the saturation limit umax increases (as noted previously for δmax=0.5).Using numerical results illustrated in Table4, we plotted in Figure 4 the corresponding largest attractive invariant ellipsoids ε⋆P⋆,γ⋆ for the two cases umax=1 and umax=10. In Figure 4, the two inclined lines (colored in pink and green) correspond to the bounds of the actuator saturation (20), which are defined as Kx+m=±umax. The lower line (colored in pink) is depicted for Kx+m=umax.Figure 4
Largest attractive invariant ellipsoidsε⋆P⋆,γ⋆ computed for two different values of umax.
(a)
(b)It is obvious from Figure4 that, in the two cases, the largest invariant ellipsoid ε⋆P⋆,γ⋆ is contained in the region ℒK,m,umax, where Kx+m≤umax. Therefore, the constraints on the affine state-feedback control law given in (18) are well respected.In the end of this part, we can note the two following statements:(1)
For a constant saturation limitumax, an increase of the maximum bound of the parametric uncertainties leads to the decrease of the largeness of the attractive invariant ellipsoid.(2)
For a constant maximum bound of the uncertainty, an increase of the saturation limitumax causes the attractive invariant ellipsoid to become larger.Remark 8.
In some research papers, the definition of the attractive invariant ellipsoid was chosen to be as follows:(72)ε´P=x∈ℝnx:xTPx≤1.
This setε´P is equivalent to that adopted in the present work in (21), that is, εP,1. Hence, we have γ=1. Our immediate concern is to show that the adopted ellipsoid (21) with a free parameter γ gives less conservative results compared with the ellipsoid (72). We use the same optimization algorithm in Theorem 4 where in the present case, we fix the parameter γ as a constant one and then we take γ=1. Table 5 shows the numerical results obtained for a fixed parameter γ=1. Notice that the symbol ! in Table 5 means that the LMI-based optimization problem in Theorem 4 is unfeasible. To realize a comparison from conservatism point of view, we adopt the same parameters in Table 4. As noted previously, the maximum bound of the disturbance is fixed to ρ=10. It is clear from Table 5 that, for the first case umax=1 and δ¯=1, the optimization problem in Theorem 4 was found to be unfeasible, with γ=1. It was found to be feasible for all δ¯≤0.648. Recall that, for a free parameter γ, the optimization problem was found to be feasible for all δ¯<2. This fact shows that, using a free parameter γ in the definition of the attractive invariant ellipsoid εP,γ, that is, the set (21), instead of fixing γ=1, gives less conservative stability conditions. Moreover, from Table 5 and for the case umax=10 and δ¯=10, the optimization problem with γ=1 is found to be feasible, as in Table 4. Nevertheless, the largest ellipsoid ε⋆′P⋆ obtained for γ=1, is too small compared with that obtained using a free parameter γ, that is, the ellipsoid ε⋆P⋆,γ⋆. Recall that the largeness of the attractive invariant ellipsoid is measured by the “trace” function as noted in Tables 4 and 5. As noted in Table 4, for a free parameter γ, the volume of the largest ellipsoid was found to be about 46.5870. However, for a fixed parameter γ=1, the volume of the largest ellipsoid ε⋆′P⋆ is about 7.3840. The difference between the two sizes of the ellipsoid is evident. Another attractive result that can be observed from Table 5 is that the obtained matrix gain K⋆ is too large compared with that obtained in Table 4.
Accordingly, we emphasize that the parameterγ in the definition of the attractive ellipsoid (21) leads to less conservative results and then less restrictive LMI stability conditions. In addition, it contributes in obtaining a largest ellipsoid with a controller gain having a small size. Hence, the choice of γ≠0 reduces the conservatism of the LMI-based optimization problem in Theorem 4.Table 5
Numerical results using Theorem 4 with a fixed parameterγ=1 and for umax=1 and umax=10.
u
max
=
1 and δ¯=1
u
max
=
10 and δ¯=10
P
⋆
!
10
3
×
2.1069
0.0255
0.0255
0.0004
K
⋆
!
10
2
×
−
2.8612
−
0.0633
m
⋆
!
0.013317
λ
⋆
!
99.6036
trace
S
⋆
!
7.3840Remark 9.
We noted in the end of Section4, just before Remark 5, that the computation time for solving the LMI-based optimization problem in Theorem 4 and then for the identification of the largest invariant attractive ellipsoid ε⋆P⋆,γ⋆ depends on the number Nκ of uniformly distributed points κ in the interval 0,1 and the simulation time tκ for each value of the parameter κ. The total simulation time tc can be computed to be about tc≈Nκ×tκ. Tables 6 and 7 present the simulation results for different values of the parameters Nκ for umax=1 (Table 6) and for umax=10 (Table 7). Recall that, for umax=1, we have δ¯=1, whereas for umax=10, we fixed δ¯=10. Moreover, the maximum bound ρ of the disturbance is fixed to be ρ=10 in the two cases. Actually, in order to obtain these results in Tables 6 and 7, we have fixed the interval in which the parameter κ varies as follows: 0.01:δκ:0.99, where we have selected four cases: δκ=0.01, δκ=0.001, δκ=0.0005, and δκ=0.0002. For these values of δκ, it corresponds, respectively, the number Nκ=99, Nκ=981, Nκ=1961, and Nκ=4901. Thus, using these intervals and these values of the parameters Nκ, we simulated the optimization problem under LMI constraints in Theorem 4 and then we obtained the results in Tables 6 and 7. In these Tables, we give the two optimal gains, K⋆ and m⋆, of the controller, λ⋆ (notice that λ=κ/1−κ), γ⋆, the size of the ellipsoid (traceγ⋆S⋆), the simulation/computation time tc, and the number of feasible solutions Nfs. It is important to note that not all the Nκ-cases are feasible. From the results in Tables 6 and 7, it reveals that the number of feasible solutions is only about 4.8% for umax=1 and about 6% for umax=10. The computation time tc for finding the largest ellipsoid increases significantly as Nκ increases too. We also stress that as Nκ increases, the optimal solution converges to that already obtained in Table 4. We found the same values for Nκ=4901. Nevertheless, in this case, the computation time is very high, it is computed to be about 4.5 minutes for umax=1 and it is about 2.2 minutes for the second case umax=10.Table 6
Computation time for solving the LMI-based optimization algorithm in Theorem 4 for the caseumax=1.
N
κ
99
981
1961
4901
K
⋆
T
−
10.5679
−
1.1050
−
9.2061
−
1.0321
−
9.1147
−
1.0271
−
9.0604
−
1.0242
m
⋆
0.013238
0.013237
0.013239
0.013235
λ
⋆
19.0000
17.1818
17.0180
16.9211
γ
⋆
0.0519
0.0574
0.0580
0.0583
trace
γ
⋆
S
⋆
1.6557
1.7874
1.7894
1.7906
t
c
s
6.1988
54.9516
106.1182
264.1296
N
fs
5
48
95
238Table 7
Computation time for solving the LMI-based optimization algorithm in Theorem 4 for the caseumax=10.
N
κ
99
981
1961
4901
K
⋆
T
−
18.6484
−
1.8069
−
19.7871
−
1.8309
−
20.3059
−
1.8453
−
20.3362
−
1.8458
m
⋆
0.013306
0.013311
0.013306
0.013303
λ
⋆
15.6667
16.8571
17.3486
17.3824
γ
⋆
6.3740
5.9236
5.7557
5.7446
trace
γ
⋆
S
⋆
45.7911
46.5153
46.5625
46.5870
t
c
s
3.1946
27.3301
53.4916
132.4823
N
fs
6
59
117
293


## 5.3. Simulations for the Robust Stabilization of the Pitch Dynamics

In this part, we would show the robustness and effectiveness of the designed saturated affine state-feedback control law (19) in the robust stabilization of the uncertain disturbed nonlinear dynamics (15) of the pitch model of the helicopter. The control gains K and m are provided in Table 4. We recall that, for umax=1, we have δ¯a=δ¯b=δ¯c=δ¯d=δ¯e=1, whereas for umax=10, we take δ¯a=δ¯b=δ¯c=δ¯d=δ¯e=10. Moreover, we recall that the external disturbance w in the nonlinear dynamics (15) is bounded according to condition (14) with ρ=10. In addition, the uncertain parameters δa, δb, δc, δd, and δe in (15) vary randomly over time so that the boundedness condition (17) is satisfied. Furthermore, for the simulation of the controlled nonlinear system (15), we will take first an initial condition, x0, located outside the largest attractive ellipsoid ε⋆P⋆,γ⋆: for the case umax=1, we take x0=0.1−1T, whereas for the case umax=10, we take x0=0.3−6T. Each initial condition is located near the boundary of the corresponding ellipsoid (see Figure 4). Furthermore, we will take an initial condition located outside the invariant attractive ellipsoid: x0=5−10T, for the two saturation limits.First of all, we will study the dynamics of the helicopter model under the saturated control lawu in the nominal case, that is, without parametric uncertainties and disturbance. Figure 5(a) shows temporal evolution of the states x1 and x2 of the certain undisturbed controlled nonlinear system (15), and Figure 5(b) reveals the saturated affine state-feedback control law u for the saturation limit umax=1. It is obvious that the state of the pitch dynamics converges to zero, which is our main objective in this paper. Moreover, the control law u converges to a constant value, which is found to be equal to the gain m. Actually, such constant value of the control input will ensure that the pitch dynamics of the helicopter stays at the zero state. In addition, we note that the reached maximum value of the control law u is about 0.23, which is less than the desired saturation limit umax=1.Figure 5
Evolution of the two states,x1 and x2, of the nonlinear system (15) and the saturated affine state-feedback control law u for umax=1 in the nominal case. Here, the system is without uncertainties (δa,b,c,d,e=0) and disturbance (w=0).
(a)
(b)We have also analyzed the nominal pitch dynamics of the helicopter model for the caseumax=10. We have observed almost the same behavior in Figure 5. The reached maximum value of the controller u is around 5. The controller converges to the constant value m≈0.01336.Now, we will show simulation results for the controlled nonlinear system subject to both parametric uncertainties and external disturbance. Simulation results are shown in Figure6 for umax=1 and Figure 7 for umax=10. Figure 6(a) (resp. Figure 7(a)) presents temporal evolution of the two states, x1 and x2, of the nonlinear system (15) under uncertainties and disturbance. Figure 6(b) (resp. Figure 7(b)) reveals evolution of the saturated control law u for umax=1 (resp. umax=10). As noted previously, the initial condition is located inside the largest attractive invariant ellipsoid. Moreover, the disturbance w is injected into the system at t=2s and during 3s. We recall that the external disturbance w satisfies condition (14) with ρ=10.Figure 6
Temporal evolution of the statesx1 and x2 of the nonlinear system (15) and the saturated affine state-feedback control law u for the saturation level umax=1. Here, the system is subject to randomly time-varying uncertainties and also a randomly time-varying external disturbing torque w. Moreover, δ¯a,b,c,d,e=1.
(a)
(b)Figure 7
Evolution of the two statesx1 and x2 of the nonlinear system (15) (a) and the saturated affine state-feedback control law u (b) for the saturation level umax=10. In this case, we have δ¯a,b,c,d,e=10.
(a)
(b)It is obvious from Figure6(a) that the motion of the pitch dynamics of the helicopter model experiences some fluctuations around the desired zero state when the disturbance is applied. Once the effect of the external disturbing torque w vanishes, the controlled system stabilizes again around its desired position with some very weak perturbations provoked by the injected parametric uncertainties. Moreover, in Figure 6(b), the control signal u varies around the value m=0.0133. The control signal varies between two very small values: ±0.1. We stress that the effects of the parametric uncertainties and the external disturbance are compensated. This shows the robustness of the control law u towards the parametric uncertainties and the external disturbance with high amplitude.However, forumax=10, the effect of the parametric uncertainties and the external disturbance w is clear in Figure 7. Figure 7(a) is almost identical to Figure 6(a). Moreover, when the disturbance is injected, the control signal u undergoes some fluctuations, which vary between ±1.5. This happens because the maximum bound of the uncertainties is very large (δ¯a,…,e=10) in this case compared with the first case, i.e., for umax=1.It is worth to note that, in the previous three cases, the saturation level±umax of the controller was well respected.We have tested another case by taking an initial condition located outside the largest attractive invariant ellipsoidε⋆P⋆,γ⋆. We take here only the case umax=1, and then the corresponding largest ellipsoid is given by Figure 4(a). As noted previously, the initial condition is chosen to be x0=5−10T. Obtained results are illustrated in Figure 8. We observe first from Figure 8(a) that the state x1 of the helicopter model experiences some smooth oscillations before its convergence and stabilization at the zero position despite the presence of the external disturbance and the parametric uncertainties. In fact, only weak fluctuations are observed as in Figure 7(a). However, the interesting phenomenon observed here is the saturation of the control law u depicted in Figure 8(b).Figure 8
Temporal evolution of the two statesx1 and x2 of the uncertain disturbed nonlinear system (15) and the saturated affine state-feedback control law u for the saturation level umax=1. Here, the initial condition is selected to be outside the largest attractive invariant ellipsoid ε⋆P⋆,γ⋆ in Figure 4(a).
(a)
(b)Remark 10.
It is worth to note that, for a predefined set of the system parameters, the previous established optimization problems under LMI constraints are (should be) simulated offline in order to compute the largest attractive ellipsoid and then the associated feedback gainsK and m of the input-saturated affine state-feedback controller u before its application into a real helicopter system in practice. Thus, for a prescribed saturation level umax, the possible maximum allowable bounds of the parametric uncertainties δ¯a, δ¯b, δ¯c, δ¯d, and δ¯e and the possible maximum bound of the external disturbance signal w, i.e., ρ, we look, as described previously, for the largest attractive invariant ellipsoid ε⋆P⋆,γ⋆. Thus, once this set is found, the corresponding feedback gains K and m will be used in order to stabilize the pitch dynamics of the simple helicopter model in a real-world application.


## 6. Conclusion and Future Works

In this paper, an LMI-based approach for designing a robust affine state-feedback control law to stabilize the pitch dynamics of a helicopter model was proposed. The nonlinear dynamics of the helicopter was subject to an external disturbance and norm-bounded parametric uncertainties. Moreover, the problem of the actuator saturation in the design of the control law was as well addressed. We showed that the stabilization problem is represented as a solving problem of BMI constraints. Furthermore, with a judicious utilization of the Schur lemma and the matrix inversion lemma, these BMIs were transformed into LMIs. We have also developed an optimization problem with enhanced LMI constraints permitting to compute the maximum bounds of the parametric uncertainties.In addition, we have proposed an LMI-based approach for the maximization of the attractive invariant ellipsoid for the uncertain disturbed nonlinear dynamics of the helicopter model under the saturated affine state-feedback control law. The largeness of the ellipsoid was measured by means of the length of its semiaxis. We have showed that the optimization problem computes efficiently the largest ellipsoid in only one step.Finally, we have showed through numerical simulations the performance of the synthesized saturated controller in the robust stabilization of the nonlinear dynamics under the norm-bounded parametric uncertainties and the external disturbing torque.In the literature, the estimation of the domain of attraction of the closed-loop system subject to actuation saturations is one of important issues and several methods have been achieved for both linear and nonlinear systems. The difference between these methods lies in the approach by which the saturation nonlinearity was handled [51]. Its treatment can be classified into three main approaches: (1) the first one is to treat the actuator saturation problem as achieved in the present work, (2) the second approach is to treat it as a locally/generalized sector bounded nonlinearity [17, 69, 70], (3) while the third method is to represent the saturation nonlinearity as a (polytopic or linear) differential inclusion (see [36, 51] and references insides). As noted in Remark 5, the solution presented in this work leads to the computation of the largest attractive invariant ellipsoid in only one step, compared to some related approaches that used a two-step method. Thus, the approach presented in this paper is more simpler and is less restrictive.In addition, it worth to note that, in this paper, a saturated affine state-feedback controller was designed. To the best of our knowledge, such problem has not been considered in the literature. Generally, a saturated linear feedback controller has been considered to stabilize linear and nonlinear systems. Then, in order to realize a comparison and hence to show the possible efficiency of our design approach, a future work of this paper is the design of a saturated affine feedback controller using the sector-bounded nonlinearity method and the differential inclusion approach.Furthermore, we aim at extending the present methodology of the saturation affine feedback controller for more complex nonlinear systems with different Lipschitzian conditions [65], and with measurement delays [71, 72] and also for the design of observer-based feedback controllers for Lipschitz nonlinear systems [3, 8, 64]. Moreover, we hope to extend this work for impulsive hybrid nonlinear systems, such as the biped robots [73] and the impact mechanical oscillators subject to multiple rigid constraints [74–76].In the present work, we considered the simple pitch dynamics of a helicopter model as an application. Moreover, the zero state is not the equilibrium point of such model. Thus, we have designed an affine feedback controller under saturation to achieve the robust stabilization at the zero state. Another important application that has such feature is the robot manipulators [77]. Indeed, because of the gravitational matrix, the dynamics of robot manipulators has an equilibrium point different to the zero state. Thus, several control approaches have been adopted for this subject [77]. Our objective is then to extend the design method of the saturated affine state-feedback controller for the case of manipulator robots.



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