Robust Adaptive Output Feedback Control for a Class of UnderactuatedAerial Vehicleswith Input andOutput Constraints

*is paper addresses the problem of nonlinear robust adaptive output feedback controller for a class of underactuated aerial vehicles with input and output constraints. To solve the problem, the modular design strategy is proposed for the control design. By using the neural networks (NNs) to approximate system uncertainties and observers to reconstruct system states, robust adaptive output feedback controllers are developed. By using a combination of saturation functions and barrier functions, input and output constraints are simultaneously dealt with. *e design methodology shows that a cascaded system of an input-to-state stable (ISS) subsystem driven by an ultimate bounded (UB) subsystem enjoys ultimate boundedness property. In addition, the tracking error converges to adjustable neighbourhoods of the origin.


Introduction
In recent years, control of underactuated systems has attracted much attention. Unlike the fully actuated system in which the number of independent control inputs is equal to degrees of freedom (DOFs), the underactuated system has fewer independent control inputs than DOFs [1,2]. Because of the underactuated property, several undesired properties such as higher relative degree and non-minimum phase behavior are manifested. erefore, powerful techniques developed for fully actuated systems may not be directly applied. For these reasons, control system design for underactuated systems is not an easy task. In fact, based on recent surveys [3][4][5][6][7], control of general underactuated systems is currently a major open problem.
In the control design for underactuated systems, all DOFs cannot be controlled; instead, a portion of DOFs is controlled to have desired behaviours while remainders are required to be bounded. is requirement leads the system in the question to a non-minimum phase system [8]. Finding the general solution of non-minimum phase systems is still a quite challenging research topic [9]. In some special cases, underactuated systems with collocated partial feedback linearization, i.e., actuated configuration variables of the underactuated system with noninteracting inputs, can be globally linearized by the use of an invertible change of control as shown in [10]. An alternative approach is the reduction of the order of the system; i.e., the underactuated system is separated into fully actuated subsystems by adding virtual controls, and these subsystems are constructed as a cascaded structure for the control design [11]. However, this approach is only applied in some classes of systems because control of cascaded systems is not solvable in general [12,13].
Constraints appear in most mechanical control systems due to different reasons such as physical restrictions and/or control tasks. ere are different kinds of constraints, for example, input constraint, output constraint, and mixed constraints. Optimal control is one of the powerful tools in dealing with constraints [14,15]. However, the computation is expensive. With the rapid development of control techniques, model predictive control (MPC) has effectiveness to handle constraints [16]. e computation is cheaper in comparison with optimal control because of the receding horizon control. In particular, when constraints have the form of saturations, a combination of the saturation function, barrier function, and direct Lyapunov method is a good choice to seek an analytic solution.
e excellent work dealing with bounded control by using the nested saturation function was proposed in [17]. Based on this work, several bounded controllers have been developed [18,19]. Concerning with output constraints, different approaches have been proposed [20][21][22]. Among these approaches, the barrier Lyapunov technique is one of the most powerful tools with less computation [23][24][25].
e model of underactuated systems is usually uncertain and affected by disturbances. ese factors cause the performance degradation and may destroy the stability of systems. Moreover, when only outputs are available for the feedback, the state feedback control cannot be implemented. In this case, several robust adaptive output feedback control schemata have been proposed to solve the problem [26,27]. However, these approaches have effectiveness for fully actuated systems. Obstacles arise when systems under consideration are underactuated systems and constraints are taken into account. In this situation, the control system design becomes much more challenging because it is necessary to simultaneously deal with underactuated property, uncertain dynamics, disturbance rejection, state estimation, and constraint satisfaction.
Motivated by the above consideration, in this paper, we are interested in designing the robust adaptive output feedback control for a class of underactuated aerial vehicles with input and output constraints. In particular, we focus on the class of underactuated aerial vehicles with a single thrust direction and full torque actuation. To solve the problem, the modular design strategy is proposed for the control system design. Namely, by constructing the underactuated aerial vehicle as a cascaded structure consisting of fully actuated subsystems, the robust adaptive output feedback control can be separately designed for each subsystem. In this setting, the observers are employed to reconstruct system states for the feedback and the radial basis function NNs (RBF NNs) are used to approximate the system uncertainties. To deal with the input and output constraints, a combination of saturation and barrier functions is integrated into the Lyapunov control design. Our design methodology yields the ultimate boundedness of all the states in the closed-loop system, while the tracking error converges to the small neighbourhood of the origin which can be made arbitrary small. e main contributions in this paper include the following aspects: (i) e modular design strategy is proposed to design the robust adaptive output feedback control for the class of underactuated aerial vehicles with taking into account multiple constraints. (ii) e assumption on limitations of orientation angles in the existing studies is removed by dealing with the constraints on orientation angles. It follows that the problem of the gimbal lock is solved, and the vehicle will be never overturned. (iii) e proposed methodology yields the new result for the control cascaded system design. Namely, we show that an ISS subsystem driven by a subsystem enjoying UB property results in the UB property of the overall cascaded subsystem. e rest of the paper is organized as follows. In Section 2, the control problem is formulated. Section 3 presents the control design while the stability of the overall system is analysed in Section 4. In Section 5, the simulations are provided to illustrate the effectiveness of the proposed control. e conclusion of this paper is presented in Section 6.

Notation.
roughout this paper, R n denotes the Euclidean space with n− dimension. ‖•‖ is the Euclidean norm of vector •. a ⟶ 0 denotes that a converges to 0. For integer indices i and j, the symbol σ i ∈ R n denotes the vector saturation function with elements σ ij , j � 1, · · · , n. A ∈ R n×n denotes the n × n− matrix, and λ min (A) and λ max (A) denote the minimum and maximum eigenvalues of matrix A, respectively. Given two vector a, b ∈ R n , then a ≤ b denotes a i ≤ b i , i � 1, · · · , n.

Problem Formulation.
e system considered in this paper is the class of underactuated aerial vehicles with a single thrust direction and full torque actuation [2]. In particular, the dynamics of the considered system can be described by the following Euler-Lagrange equations: where q � [x, y, z] T ∈ R 3 denotes the position of the vehicle in the reference frame; η � [ϕ, θ, ψ] ∈ R 3 denotes the orientation angles with respect to the roll, pitch, and yaw; u ∈ R is the total thrust; τ � [τ ϕ , τ θ , τ ψ ] T ∈ R 3 is the torque; m is the mass of the rigid body; g is the acceleration due to gravity; e 3 � [0, 0, 1] T is the unit vector; J(η) ∈ R 3×3 is the effective inertial matrix; C(η, _ η) ∈ R 3×3 is the Coriolis and centrifugal force matrix; R ∈ SO(3) is the rotation matrix having the form and d q and d η ∈ R 3 are unknown disturbances with the known bounds where d * q and d * η are the known positive constants. In this paper, we are interested in the input constraints as follows: 2 Mathematical Problems in Engineering |τ| ≤ τ, (6) and we are also interested in the following output constraints: with u ∈ R and τ ∈ R 3 being the positive constants. (5) and (6) ensure the final values of the control inputs. Constraint (7) ensures that the vehicle will be never overturned [28][29][30].

Remark 1. Constraints
We notice that the considered system has four control inputs [u, τ T ] T ∈ R 4 and six DOFs [q T , η T ] T ∈ R 6 . Accordingly, it is impossible for such systems to track an arbitrary reference trajectory. Instead, we select four outputs [q T , ψ] T for the tracking purpose as usual in [2,29]. erefore, the control problem considered in this paper can be stated as follows.
Problem 1. Consider the class of underactuated vehicles described by (1) and (2). Let [q d (t) T , ψ d (t)] T ∈ R 4 be a given smooth time-varying desired trajectory with its time derivatives bounded up to the second order. Under conditions that the velocities _ q and _ η are not available for the feedback control, the terms J(η), C(η, _ η), and m are uncertain and the disturbances d q and d τ are bounded; design a robust adaptive output feedback control U � [u, τ T ] T for system (1) and (2) such that the following criteria are met: (i) Constraints (5)-(7) are satisfied (ii) All the signals in the closed-loop system are bounded (iii) e output [q T , ψ] T follows an adjustable neighborhood of its desired trajectory [q T d , ψ d ] T As the problem is formulated, the velocities _ q and _ η are unavailable for the feedback while the uncertain terms J(η), C(η, _ η), and m, constraints (5)-(7), and disturbances (4) are taken into account. erefore, direct design of the controls u and τ is not an easy task. To overcome this difficulty, the control strategy based on the modular design is proposed as follows. First, we decouple and construct systems (1) and (2) as the cascaded structure in which position subsystem (1) is driven by orientation subsystem (2). Second, the robust output feedback control is designed for position subsystem (1) such that subsystem (1) is ISS while constraint (5) is satisfied. ird, the robust adaptive output feedback control is designed to achieve the stability of orientation subsystem (2) and simultaneously tackle constraints (6) and (7).

Remark 2.
We notice that by dealing with constraint (7), the following two issues are ensured. First, the assumption on limitations of orientation angles (|ϕ|, |θ| < π/2) in the existing studies [1,31] is rejected, and the so-called problem gimbal lock is solved. Second, the vehicle will be never overturned.

Preliminaries.
is section briefly recalls the notions of RBF NN, high-gain observer, and saturation function which are used in the control design in the next section.

RBF Neural Networks.
e RBF neural networks consist of two layers, in which the hidden layer completes a fixed nonlinear transformation. en, the output layer linearly combines the outputs of the hidden layer. erefore, the networks can be simply represented by [32] f nn (X) � W T G(X), (8) with the input vector X � [x 1 , x 2 , · · · , x m ] T ∈ Ω X ⊂ R m , weight vector W ∈ R ℓ , node number ℓ > 1, and basis function G(X) � [g 1 (X), g 2 (X), · · · , g ℓ (X)] T ∈ R ℓ . Here, g j (X) is the Gaussian function as follows: where κ j � [κ j1 , κ j2 , · · · , κ jℓ ] T and h j are the center and the width, respectively. It is shown that by universe approximation results that any continuous function f(X) defined on a compact set Ω X ⊂ R m can be approximated by W T G(X) if ℓ is chosen sufficiently large. is is described as where ε(X) is the function approximation error and W * is the ideal weigh vector. e following assumption for the function approximation is made.
Assumption 1. (see [32]) Over the compact set Ω X ⊂ R m , the neural network ideal weight vector W * , and the function approximation error ε(X) of neural networks is bounded by where W m and ε m are the positive constants.

High-Gain
Observer. Since only the positions and orientation angles are measurable and the linear and angular velocities are not available, we need to estimate the velocities _ q and _ η for the output feedback control. e following highgain observer is used later.

Mathematical Problems in Engineering
Lemma 2. (see [33]) Assume that the output y(t) of a system and its first n − 1 derivatives are bounded, that is, |y (k) | < Y k , k � 1, · · · , n − 1, with positive constants Y k . Consider the following linear system: with ε being any small positive constant, where ς i , i � 1, · · · , n, are the state variables of the observer. Choosing the parameters ] 1 , · · · , ] n such that the polynomial s n + ] 1 s n− 1 + · · · + ] n− 1 s + 1 is Hurwitz, then we have the following properties:

Saturation Function.
To deal with input constraint (5), the following saturation function is used.
Definition 1 (see [17]). Given a positive constant σ, a function σ: R ⟶ R is said to be a linear saturation with the bound σ if it is a continuous, nondecreasing function sat-

Control Design
is section presents the control design for systems (1) and (2) according to the proposed control strategy in the previous section.

Position Control by Bounded Robust Output Feedback.
e goal of this section is to decouple subsystem (1) from orientation subsystem (2) and design the robust output feedback control for (1) such that the position q tracks its desired trajectory q d , constraint (5) is satisfied, and subsystem (1) is ISS. To this end, we assume that the uncertain mass m in (1) can be written as where m is the known nominal term and Δ m is an uncertain term with the known bound as Denoting the states as and using the relationship 1/m � 1/m − Δ m /mm from (14), system (1) can be rewritten as where d p � d q − uRe 3 /mm is bounded by with the known bound Remark 3. e upper bound d * p is determined by (19) only when the control u is bounded by u, i.e., constraint (5) must be satisfied. e satisfaction of this constraint will be guaranteed in the control design later.

Position Observer Design.
Because the velocity q 2 is not available, we will estimate q 2 for the output feedback control by using the following observer: where q 1 and q 2 are the estimations of q 1 and q 2 , respectively, and L 1 and L 2 ∈ R 3×3 are the positive constant observer gain matrices. By defining the observer error and subtracting (20) from (17), the dynamic equation of the observer error is given by where Lemma 3. For system (22), if L 1 and L 2 are chosen such that L is Hurwitz, then there exists a time instant T o such that the observer error satisfies □ Remark 4. It is shown from (24) that the observer error q exponentially decays with the convergent rate ρ o . Moreover, the observer error is uniformly ultimately bounded with the ultimate bound δ o which can be adjusted by the observer design matrices L 1 and L 2 .

Bounded Robust Output Feedback Control Design.
Having position observer (20), we proceed to decouple (1) from (2) and design the control u for position subsystem (1) to solve the tracking problem. To this end, defining the tracking errors and using the relationship q 2 � q 2 + q 2 , the position tracking error system is obtained as We now decouple (26) from (2) by adding and subtracting the right-hand side of p 2 − equation in (26) by the virtual control F ∈ R 3 of the following form: where R d has the same form with R except that the angles ϕ, θ, and ψ are replaced with their desired ϕ d , θ d , and ψ d , respectively. Accordingly, p 2 − equation in (26) can be further rewritten as Having (28), we are able to freely design the control F for (26). Furthermore, solving (27) with respect to ϕ d , θ d , and u [29], we obtain the real control u and the desired orientation angles as where we use the denotations S ψ d � sin ψ d and C ψ d � cos ψ d .
In view of (30), u satisfies constraint (5) if F is bounded with the known bound. For this reason, the control F is designed as where σ 1 and σ 2 ∈ R 3 are the saturation function vectors in Definition 1.

Assumption 2. e desired trajectory q d is bounded by
where σ ij is the bound of σ ij for i � 1, 2 and j � 1, 2, 3. From (28), (26), and (31) we obtain the closed loop position tracking error system as _ Theorem 1. Consider system (34) under Assumption 2 and Lemma 3. If condition (33) is satisfied, then for any initial condition p(0), the following statements hold: where d * p is defined in (19) and ϖ * is a positive constant satisfying then all the system states are ultimately bounded, i.e., there exist positive definite functions V 1 (p 1 , p 2 , q) and W 1 (p 1 , p 2 , q), a positive number C 1 , and a time instant  (2). On the other hand, we deal with the underactuated property of the considered system. Indeed, in view of (26), the one-dimensional control u ∈ R 1 cannot be used to control the three-dimensional output p 2 ∈ R 3 . In view of (34), the tracking errors (p 1 , p 2 ) are driven by q and ϖ. e observer error q can be made small by adjusting the observer gain matrix L as shown in Lemma 3. e remainder is how to make ϖ small. In view of (29), R ⟶ R d implies ϖ ⟶ 0, or we need to have η ⟶ η d , where η d is the desired orientation angles. is goal will be achieved in the next section.

Orientation Control by Bounded Robust Adaptive Output
Feedback.
e goal of this section is the design of the control τ for orientation subsystem (2) to solve the orientation tracking problem, i.e., η tracks its desired η d and constraints (6) and (7) are simultaneously tackled. To this end, we denote the states as and rewrite system (2) in the form

Bounded Robust Adaptive Output Feedback Control
Design. Having high-gain observer (40), we proceed to design the control τ for orientation subsystem (39). In the control design, constraints (6) and (7) are simultaneously dealt with by the use of the saturation and barrier functions. e RBF NNs are used to approximate the unknown terms J(η 1 ) and C(η 1 , η 2 ) in (39). e control design is based on the backstepping method. For more detail, the design schema consists of following two steps: Step 1 Define the tracking errors as where α is the virtual control. Let ζ 2 be the error between ζ 2 and ζ 2 , involving (42); ζ 2 is bounded by and dynamic equation of To deal with output constraint (7), let us consider the barrier Lyapunov function candidate [24] as follows: where b i for i � 1, 2, 3 is defined as with ψ d being the bound of ψ d . e time derivative of (46) along (45) is given by where Substituting (45) into (48) yields 6 Mathematical Problems in Engineering Choosing where A 0 and A 1 ∈ R 3×3 are positive matrices, and substituting (51) into (50), yield Using Lemma 2 in [34], in the set |ζ 1i | < b i , i � 1, 2, 3, we have and the first term in (52) satisfies e first two terms in (55) are negative; the last term will be cancelled in the next step.
Step 2 Taking the time derivative ζ 2 in (43) along (2) yields Consider the Lyapunov function candidate as and its time derivative along (56) is given by us, the real control τ appears in (58). We need to design the control τ to make _ V 3 ≤ 0. To this end, we notice that the term ζ T 2 ( _ J(η 1 ) − 2C(η 1 , η 2 ))ζ 2 � 0 according to the skew symmetric property. e unknown terms C(η 1 , η 2 )α and J(η 1 ) _ α are estimated by using the RBF NNs approximation function from (10) as where Z � [η T 1 , η T 2 ] T is the input vector of the NNs and W * is the ideal unknown constant weight matrix, which has the following form: with w * i ∈ R ℓ , i � 1, 2, 3, and W * is bounded by where W m is a positive constant; ℓ is the node number of the corresponding NN; and Φ(Z) ∈ R 3ℓ is the basis function vector of the following form: with Φ i (Z) ∈ R ℓ ; ϵ(Z) is the error vector with the known bound as where ϵ * is a positive constant. It was mentioned that the optimal weight W * is unavailable for the approximation purpose. Furthermore, the state η 2 is also unavailable for the feedback and hence cannot be the input of the NNs. erefore, approximation (59) cannot be implemented. To deal with this situation, consider the relation between η 2 and η 2 in Lemma 4; we use the estimated state η 2 as the input of the NNs for the approximation of the left-hand side of (59). e possibility of the use of the estimated state η 2 is shown in Appendix C. Accordingly, the input of the NNs is Z � [η T 1 , η T 2 ] T , and hence, the basis function vector is Φ(Z).
Next, we need to deal with input constraint (6). To do this, the control τ is designed in the following form: for i � ϕ, θ, ψ, with τ o � [τ oϕ , τ oθ , τ oψ ] T being the free control to be designed. Denoting the control input error as and substituting (55), (59), and (65) into (58), we obtain In view of (66), Δτ ≠ 0 when the saturation appears. We need to reject Δτ. is can be accomplished by using the following auxiliary system [35]: Mathematical Problems in Engineering where υ ∈ R 3 is the auxiliary vector; K υ is a positive definite matrix; k 0 is a positive number; and ϱ is a positive constant that should be chosen as an appropriate value in accordance with the requirement of the tracking performance. us, we have transformed the control τ under input constraint (6) to the free control τ o by adding the auxiliary variable υ. Consequently, in order to obtain the control τ o based on the Lyapunov method, we consider the following Lyapunov function candidate: where w i � w * i − w i , for i � 1, 2, 3, is the estimation error and w i is the estimation of w * i . e goal now is to select the control τ o to make the time derivative _ V 4 negative. e process of selecting τ o is detailed in Appendix C. At this point, in order to state the result of this section, we propose the control τ o and the update law w i in advance as follows: where A 2 is a positive matrix and μ i is a positive constant. In control (69), the first term is proposed to cancel the second term in (55). e second term is used for the stabilization of ζ 2 and dealing with d τ and ϵ. e third term is the estimation of (59). e last term is used to tackle the appearance of the input saturation.
(ii) Constraints (6) and (7) are satisfied; in particular, the tracking error ζ 1 is bounded by

(iii) e tracking error ζ(t) converges to a small neighbourhood of the origin which can be made arbitrarily small.
Proof. See Appendix C. □ Remark 7. It is noticed that the second term of V 3 in (57) is the function of the state ζ 2 instead of ζ 2 . is allows us to know the behavior of the state ζ 2 even when ζ 2 is unavailable for the feedback.

Remark 8.
In order to deal with constraint (7), we use the Euler angles in the representation of the attitude. is makes the control design become much easier because the use of the unit quaternion may lead to inconsistent behavior, i.e., the unwinding phenomenon [2]. Although the use of Euler angles has drawback which is the so-called gimbal lock [36], i.e., the rotation matrix is singular when ϕ � π/2 or θ � π/2, this drawback is tackled by the satisfaction of constraint (7).

Stability Analysis
is section investigates the stability of the overall closedloop system under the controllers designed in the previous section. e overall error system consists of the closed-loop position error subsystem from (34) as follows: _ Systems (73) and (74) have the cascaded structure in which (73) is driven by (74). e result proposed in [11,37] cannot be directly applied because of constraint (7) and disturbances. In the following, we show that the controllers designed in the previous section achieve the ultimate boundedness property of the overall systems (73) and (74).

Theorem 3. Consider the closed-loop overall tracking error system described by (73) and (74) under Assumption 2. If the initial condition is such that |ζ
where (18), (23), and (29), ϖ and d q are bounded by (77) Inequality (76) can be further written as where c 6 � max i�1,2,3 2c i and c 7 � (c 4 2u/m + c 5 d * p )/ c 6 . Taking the integral (78) over the time [0, t] yields Inequality (80) shows that for the finite time t, V 1 (t) will not approach infinity, and hence, the solution (p 1 , p 2 , q) does not escape in the finite time. (ii) Ultimate boundedness property of the overall system: we need to prove that the time derivative of the total Lyapunov function _ V 1 + _ V 4 is negative outside some compact set. It is noticed that _ V 4 satisfies (71) from eorem 2. e remainder is _ V 1 . We shall show that _ V 1 satisfies (37) in eorem 1, or in other words, we need to verify condition (36) of eorem 1. To this end, by direct computation from (29), we have where k ϖ � 3 � 2 √ (u/m) > 0. From (72) of eorem 2, the solution ζ 1 (t) is bounded by To evaluate the behavior of V 4 , solving (71) with respect to V 4 , we obtain Inequality (83) shows that V 4 (t) exponentially decays and the size of the final value is C 4 /κ which can be made arbitrarily small. Accordingly, from (82) and (83), there exists a time instant T ϖ such that with ϖ * being the constant in (36). Involving (81), we have Condition (85) implies the satisfaction of condition (36) in eorem 1. According to eorem 1, there exists a time instant T 2 > max T 1 , T ϖ such that inequality (37) in eorem 1 is achieved or we have Now let us consider the Lyapunov function candidate from (86) and (71); the time derivative _ V c along closed-loop system (73) and (74) is given by where W c � W 1 + κV 4 and C c � C 1 + C 4 . e foregoing inequality (89) shows that the solution of the overall closed loop system is ultimately bounded with the ultimate bound C c which can be made arbitrarily small by making C 1 and C 4 small. e proof is completed. □ Remark 9. It is noticed from eorem 3 that cascaded systems (73) and (74), in which ISS system (73) with the bounded input is driven by system (74) with ultimate boundedness property, enjoy ultimate boundedness property. is fact is a novelty and gives us a flexible tool in the control design for the cascaded system. In comparison with the existing studies [11,37], this fact allows us not only to deal with constraints, disturbances, and uncertainties of the system but also take observers for state estimation into account.

Theorem 4. Consider the class of underactuated vehicles described by (1) and (2) with constraints (5)-(7), the velocities estimated by observers
Proof. Constraint (5) is satisfied from eorem 1, and constraints (6) and (7) are satisfied from eorem 2. All the signals in the closed loop are ultimately bounded, and the tracking error converges to the adjustable neighbourhoods of the origin from eorem 3.
For the comparison purpose, we carry out the simulation with the full-state feedback control proposed in [1].

Conclusion
is paper studies the robust adaptive output feedback control for the class of underactuated aerial vehicles with multiple constraints. By constructing the system as the cascaded structure, the modular design strategy is proposed to separately design the controller for each subsystem. By integrating observers, saturations, and barrier functions and NN approximation into the control system design, the controller has ability to simultaneously deal with system uncertainties, unavailable velocities, and constraints. e overall system enjoys the ultimate boundedness property. e simulation on the quadrotor helicopter shows that the control objectives are completed.

A. Proof of Lemma 3
L is Hurwitz; then, there exists a symmetric positive definite matrix P o ∈ R 3×3 satisfying the Lyapunov equation [39, eorem 4.6]: where Q o ∈ R 3×3 is a positive definite matrix.
Consider the Lyapunov function candidate as follows: and its time derivative along (22) is given by To use the term − λ min (Q o )‖q‖ 2 to dominate +2d * q λ max (P o )‖q‖, let 0 < c o < 1 and rewrite the above inequality as Using the fact that λ min (P o )‖q‖ 2 ≤ V o (q) ≤ λ max (P o )‖q‖ 2 , the observer error satisfies Using condition (33), we conclude that constraint (5) is ensured.

B.2. Ultimate Boundedness of State.
Consider the Lyapunov function candidate V 1 (p 1 , p 2 , q) of the following form: where p 1i , p 2i , σ 1i , and ς i for i � 1, · · · , 3 are the elements of p 1 , p 2 , σ, and ς, respectively. We need to show that the time derivative _ V 1 along the trajectory of system (34) is negative outside a compact set. To do this, taking the time derivative of V 1 along (34) yields For convenience, we use short denotations as follows: Denoting c 1 � ����������� 1 + λ max 2 (L 2 ) and using the following inequalities inequality (B.5) can be further written as where we use p T 2 σ 2 (p 2 ) ≥ p T 2 σ 2 (p 2 )/2 + ‖σ 2 (p 2 )‖ 2 /2. e last two terms in (B.3) satisfy where we use the result (A.3) in the proof of Lemma 3. Substituting (B.6)-(B.8) into (B.3), we obtain In view of (B.10), the term p T 2 σ 2 /2 will become p T 2 p 2 /2 if p 2 is in the linear region of saturation function σ 2 . In the following, we show that if the state p 2 reaches the saturation, then p 2 will enter the linear region of saturation function σ 2 after a final time and stays therein. To do this, we consider the following two cases.

Case 1.
e state p 2 reaches the saturation. In this case, we have p T 2 σ 2 (p 2 ) � |p 21 |σ 21 + |p 22 |σ 22 + |p 23 |σ 23 � |p T 2 |σ 2 and (B.10) can be rewritten as where |p 2, min | � min i�1,2,3 |p 2i | > 0. Choosing matrices P o , Q o , and L 2 such that and using result (24) from Lemma 3, then there exists a time instant T 0 > 0 such that and the last term in (B.11) is negative. Next, defining the set and using condition (36), if ∀(p 1 , p 2 , q) ∉ Ω 1 and t > T 0 , then we have Inequality (B.16) shows that there exists a time instant T 1 > T 0 > 0 such that the solution (p 1 , p 2 , q) starting outside Ω 1 enters Ω 1 after the finite time T 1 and stays therein for all t > T 1 . In the other words, the state p 2 enters the linear region of the saturation function σ 2 .

Mathematical Problems in Engineering 13
Case 2.
e state p 2 is in the linear region of σ 2 . In this case, we have p T 2 σ 2 (p 2 ) � p T 2 p 2 � ‖p 2 ‖ 2 and (B.10) is rewritten as Since ‖ϖ‖ ≤ ϖ * and ‖d q ‖ ≤ d * q , and inequality (B.17) can be further rewritten as

(B.21)
In view of (B.20), _ V < 0, outside a compact set as follows: According to an extension of Lyapunov theorem [32], we conclude that all the system states are ultimately bounded.