On the Stabilization of the Inverted-Cart Pendulum Using the Saturation Function Approach

Abstract

A simple stabilizing controller for the cart-pendulum system is designed in this paper. Our control strategy describes the underactuated system as a chain of integrators with a high-order smooth nonlinear perturbation and assumes initialization of the system in the upper half plane. The design procedure involves two sequentially associated control actions: one linear and one bounded quasilinear. The first control action brings the nonactuated coordinate near to the upright position and keeps it inside of a well-characterized small vicinity, whereas the second control action asymptotically brings the whole state of the system to the origin. The corresponding closed-loop stability analysis uses standard linear stability arguments as well as the traditional Lyapunov method and the LaSalle's theorem. Our proposed control law ensures global stability of the system in the upper half plane. We illustrate the effectiveness of the proposed control strategy via numerical simulations.

1. Introduction

Because of its control-related challenging features, the well-known cart-pendulum system has been extensively studied in recent times by the control community (see, for instance [1] and the references therein). This control benchmark consists of a free vertical rotating pendulum with a pivot point mounted on a cart horizontally moved by a horizontal force (which corresponds to the system input). The control problem comprises to swinging up the pendulum from its stable hanging position, in order to bring and maintain it to its unstable upright position. What makes this simple mechanical system an interesting control benchmark is the fact that the pendulum angular acceleration cannot be controlled, that is, the cart-pendulum system is a two degrees-of-freedom mechanical underactuated system. Hence, many common stabilizing control techniques developed for fullyactuated systems cannot be directly applied to this system. It must be pointed out that the cart-pendulum system is not input-output (statically or dynamically) feedback linearizable (see for instance [2]). Moreover, the cart-pendulum system loses controllability and other control-related geometric properties when the pendulum moves through the horizontal plane (see [1, 3]). Since the system is locally controllable around the unstable equilibrium point, closed-loop stabilization by linear pole placement can be used (see, for instance, [4]).

Stabilizing the cart-pendulum system involves two main aspects: (i) swinging up the pendulum from the stable hanging position to the unstable upright vertical position (see, for instance, [5–12]) and (ii) stabilizing the system around its unstable equilibrium point. For this second aspect it is commonly assumed that the free endpoint of the pendulum is initially located above the horizontal plane or lies inside a well-characterized open vicinity of zero (the vicinity defines the closed-loop stability domain). We focus our attention on the former problem. Let us now review some remarkable works on this aspect. In [13] a nonlinear controller, based on the backstepping procedure, is used to solve the stabilization problem in the unstable equilibrium point; the proposed controller ensures full state convergence. A controller based on nested saturation functions is proposed in [14]. A similar work is discussed in [15], where a chain of integrators is considered as a model for the cart-pendulum system. In [9] a stabilization technique using switching and saturation functions (in addition to the Lyapunov method) is introduced. A control strategy based on controlled Lagrangians is presented in [16], and a proposal using similar tools is exposed in [17]. A feedback control scheme based on matching conditions is described in [18], while a simple matching condition is used in [19] to solve the cart-pendulum regulation problem. A very interesting nonlinear control strategy based on energy shaping techniques combined with input-to-state stability methods is presented in [20]. A solution which exploits power-based passivity properties of the cart-pendulum system is proposed in [21]. A nonlinear controller based on both the fixed point backstepping procedure and saturation functions is proposed in [22]. This list of published works is by no means exhaustive. Let us conclude it by mentioning the work published in [23], where the challenging nature of the cart-pendulum problem is underscored to the nonlinear control community.

In this work we consider the Inverted Pendulum model introduced in [22], to propose a simple control strategy, which combines a linear action with a quasilinear action. This strategy is justified by using the traditional Lyapunov method and applying the theorem of LaSalle. We point out that the proposed controller here differs from the obtained in [15, 22]. Because, contrary to these works, we do not need to use sophisticated nonlinear tools to carry out the stability analysis. For instance, we do not use the fixed point control scheme, nor the modified backstepping procedure. Also, our control law has a simple structure than the ones proposed in these two works. Roughly speaking, the linear control action confines, both, the angular position and the angular velocity in a small compact set, which defines the closed-loop stability domain (completely characterized by the cart-pendulum parameters and the controller parameters). The bounded quasilinear control action guarantees, then, full state convergence. We must emphasize that our solution avoids the necessity of solving partial differential equations, nonlinear differential equations, or fixed point control equations.

The paper is organized as follows. Section 2 is concerned with modeling issues as well as the problem statement. We present our proposal in Section 3, which we illustrate with a simulated control scheme. We conclude with some final remarks in Section 4.

2. Problem Statement

2.1. The Cart-Pendulum Model

Consider the cart-pendulum system (see Figure 1), described by the following set of normalized differential equations (see for instance [4]): ̈̈()̈̈𝜃̇cos𝜃𝑞+𝜃−sin𝜃=0,1+𝛿𝑞+cos𝜃𝜃−²sin𝜃=𝑓, (2.1) where 𝑞 is the normalized displacement of the cart; 𝜃 is the actual angle that the pendulum forms with the vertical; 𝑓 is the horizontal normalized force applied to the cart (i.e., the system input), and 𝛿>0 is a real constant that depends directly on both, the cart and the pendulum masses. In the nonforced case corresponding to 𝑓=0 and 𝜃∈(−𝜋/2,𝜋/2) the above system has only one unstable equilibrium point given by 𝑥=(𝜃=0,𝜃̇=0,𝑞=𝑞,𝑞̇=0), with 𝑞 being constant. Some simple algebra allows us to derive a new control variable 𝑢: ̈1𝑞=𝛿+sin²𝜃𝑓+𝜃̇²sin𝜃−cos𝜃sin𝜃≜𝑢. (2.2) Thus, system (2.1) can be written in a very simple way as̈̈𝜃=sin𝜃−cos𝜃𝑢,𝑞=𝑢. (2.3) Now, we proceed to express system (2.3) as if it were a four-order chain of integrators plus an additional nonlinear perturbation. For this end we define the new coordinates: 𝑧₁=𝑞+2tanh⁻¹𝜃tan2,𝑧₃𝑧=tan𝜃,₂=𝑞̇+𝜃̇sec𝜃,𝑧₄=𝜃̇sec²𝜃. (2.4) Then, system (2.3) can be written as 𝑧̇₁=𝑧₂,𝑧̇₂=𝑧₃+𝛼𝑧₃𝑧²₄,𝑧̇₃=𝑧₄,𝑧̇₄=𝑣, (2.5) where the term 𝛼(𝑧₃) is given by 𝛼𝑧₃=𝑧₃1+𝑧²₃^3/2 (2.6) and 𝑣 is now the new control variable defined as 𝑣≜sec²𝜃(−𝑢cos𝜃+sin𝜃)+2𝜃̇²sec²𝜃tan𝜃. (2.7)

Figure 1 

Cart-pendulum system.

Notice that the above set of transformations are well defined for all −𝜋/2<𝜃<𝜋/2. That is, the pendulum moves inside the upper half plane. On the other hand, it is easy to verify that function |𝛼(𝑧)|≤𝜅=2/3^1.5.

Remark 2.1. We emphasize that model (2.5) was proposed and resolved using the stability analysis tool known as the fixed point controller in combination with a version of the backstepping procedure, by Olfati-Saber [22]. On the other hand, it is our opinion that the main contribution of our control strategy is that we solved the stability problem using the Lyapunov method combined with the theorem of LaSalle.

We can now formulate our control problem.

Problem Formulation
Given the cart-pendulum system described as in (2.5), we want to bring the pendulum to the upright position and, simultaneously, bring the cart to the origin or any other fixed desired position.

We introduce the following useful definitions, used in forthcoming developments.

Definition 2.2. Let 𝑥∈ℝ. The classical linear saturation function is defined as 𝜎_𝑚(𝑥)=|𝑥|𝑚𝑥𝑥if≤𝑚,|𝑥|if|𝑥|>𝑚, (2.8) where constant 𝑚 is strictly positive.

Definition 2.3. By a sigmoidal function 𝑠_𝑚(𝑥), we mean a smooth function that is bounded, strictly increasing with the property that 𝑠_𝑚(0)=0; 𝑥𝑠_𝑚(𝑥)≥0 and |𝑠_𝑚(𝑥)|≤𝑚, for all 𝑥∈ℝ.

We now proceed to propose our control strategy.

3. Regulation of the Cart-Pendulum System

3.1. Linear Transformation

Inspired by what is presented in [24], we first introduce the following linear transformation:𝑥𝑥≜¹𝑥₂𝑥₃𝑥₄=1331𝑧012100110001¹𝑧₂𝑧₃𝑧₄, (3.1) which leads to, when applied to (2.5),𝑥̇₁=𝑥₂+𝑥₃+𝑥₄+3𝛼𝑧₃𝑥²₄𝑥̇+𝑣;₂=𝑥₃+𝑥₄+𝛼𝑧₃𝑥²₄𝑥̇+𝑣;₃=𝑥₄𝑥̇+𝑣;₄=𝑣. (3.2) In the following section we split the new control input 𝑣 into two control actions. One part of this control, namely, 𝑣₁, brings both the state 𝑥₃ and the state 𝑥₄ to a small compact set defining the closed-loop stability domain and, consequently, renders the nonlinear terms of system (3.2) to an arbitrarily small vicinity of zero. Simultaneously to the linear control action, a bounded quasilinear control action, namely, 𝑣₂, stabilizes the missing states 𝑥₁ and 𝑥₂.

3.2. Stabilization of the States 𝑥₃ and 𝑥₄

In order to guarantee that the states 𝑥₃ and 𝑥₄ are bounded, we split 𝑣 as𝑣=−𝑥₃−𝑥^{𝑣10𝑥0𝑓𝑒37}₄+𝑣₂, (3.3) where |𝑣₂|≤𝜀, with 𝜀>0. Thus, after substituting (3.3) into (3.2), we have that𝑥̇₁=𝑥₂+3𝛼𝑧₃𝑥²₄+𝑣₂;𝑥̇₂=𝛼𝑧₃𝑥²₄+𝑣₂;𝑥̇₃=−𝑥₃+𝑣₂;𝑥̇₄=−𝑥₃−𝑥₄+𝑣₂. (3.4) We emphasize that, if |𝑣₂|≤𝜀, with 𝜀 small enough, then the states 𝑥₃ and 𝑥₄ converge toward the vicinity 𝐵(𝑥₃₄)≤𝛿_𝜀 (for simplicity 𝐵(𝑥₃₄)=𝑥²₃+𝑥²₄.), where the bound 𝛿_𝜀 can be made small by minimizing 𝜀 and, consequently, all the nonlinear terms in (3.4) can be arbitrarily approximated to zero. This results in the dominance of the linear dynamics over their, respectively, nonlinear dynamics. That is, signal 𝑣₂ is then selected as a bounding function, where 𝜀 is given latter in our discussion and 𝛿 is a small positive constant fixed as needed.

In order to analyze the boundedness of both the state 𝑥₃ and the state 𝑥₄, we consider the definite positive function given by𝑉₁𝑥₄,𝑥₃=12𝑥₃−𝑥₄²+12𝑥²₄. (3.5) Differentiating (3.5) and taking into account (3.4) we have that𝑉̇₁𝑥₄,𝑥₃=−2𝑥²₄+𝑥₄𝑣₂. (3.6) Now, given the assumption |𝑣₂|≤𝜀 we have that 𝑉̇₁ is in fact bounded as follows: 𝑉̇₁𝑥₄,𝑥₃≤−𝑥₄−𝜀+2𝑥₄. (3.7) (Evidently, when 𝑣₂≈0, the subsystem (3.4) converges asymptotically to zero, because its time derivative, given by (3.6), is equal to −2𝑥²₄, semidefinite negative and, applying the theorem of LaSalle, we have that, both variables, 𝑥₃ and 𝑥₄, converge to zero.) Note that if |𝑥₄|>𝜀/2+𝛿, with 𝛿>0 an arbitrarily small number, then, from (3.7), we have that 𝑉̇₁<0. Consequently, there exists a finite time 𝑇₁ after which we have𝑥₄(𝑡≤𝜀)2+𝛿;∀𝑡>𝑇₁, (3.8) that is 𝑥₄ is bounded. Even more, because 𝑉₁(𝑥₄,𝑥₃) is a nonincreasing function, provided that |𝑣₂|≤𝜀, then the state 𝑥₃ is also bounded. However, in order to compute the corresponding bound of 𝑥₃, we propose a positive function, 𝑊(𝑥₃)=𝑥²₃/2, whose time derivative along the of the dynamics of the state 𝑥₃, in (3.4), satisfies𝑊̇𝑥₃=−𝑥²₃+𝑥₃𝑣₂≤−𝑥₃𝑥₃−𝜀. (3.9) It implies that there exists a finite time 𝑇₂>𝑇₁ such that𝑥₃(𝑡)≤𝜀+𝛿;∀𝑡>𝑇₂. (3.10)

Remark 3.1. Summarizing, after some period of time 𝑡>𝑇₂, we have that variables, 𝑥₃ and 𝑥₄, are bounded. Besides, it holds that 𝑉̇₁𝑥₄,𝑥₃<−32𝑥²₄+𝑣²₂2. (3.11)

3.3. Stabilization of Both the State 𝑥₂ and the State 𝑥₁

In order to stabilize the missing state variables we propose the bounding control action 𝑣₂ as follows: 𝑣₂=−𝜎_𝑚𝑥₂−𝑘_𝑖𝜎_𝑚𝑥₁, (3.12) with the control parameter 𝑘_𝑖 being characterized by 0<𝑘_𝑖<1. Observe that in order to simplify the corresponding bounded analysis, we used the linear saturation function 𝜎_𝑚(𝑥₁). However, this function can be substituted by any other nonlinear saturation function. On the other hand, we have that 𝑣₂≤𝜀≜𝑚𝑘_𝑖+1. (3.13) Then, after substituting the above controller into the second equation of (3.4) we get 𝑥̇₂=−𝜎_𝑚𝑥₂−𝑘_𝑖𝜎_𝑚𝑥₁+𝛼𝑧₃𝑥²₄. (3.14) We introduce now the following positive definite function: 𝑉₂=^𝑥2₀𝜎_𝑚(𝑠)𝑑𝑠 (3.15) in order to verify the boundedness of the state 𝑥₂. Differentiating 𝑉₂ and using (3.14) we have that𝑉̇₂=𝜎_𝑚𝑥₂−𝜎_𝑚𝑥₂−𝑘_𝑖𝜎_𝑚𝑥₁−𝛼𝑧₃𝑥²₄. (3.16) Selecting 𝑚>𝑘_𝑖𝑚+𝜅₀𝜀²/4 we can assure that 𝑉̇₂<0, if |𝑥₂|>(𝑘_𝑖+𝜅₀𝜀²/4𝑚)+𝛿. Therefore, there is a finite time 𝑇₃>𝑇₂>0 such that 𝑥₂(𝑡)≤𝑘_𝑚2≜𝑘_𝑖+𝜅₀𝜀²4𝑚+𝛿;∀𝑡>𝑇₃. (3.17) We emphasize that the restriction 𝑚>𝛿+𝑘_𝑖𝑚+𝜅₀𝜀²/4 can be always satisfied. Indeed, from the definition of 𝜀 given in (3.13) we evidently have 1>𝑘_𝑖+𝜅₀𝑚41+𝑘_𝑖²+𝛿𝑚 (3.18) (just to illustrate how this inequality holds take, for instance, 𝑘_𝑖=2/3,𝑚=1, and 𝛿=10⁻³, for a given 𝜅₀=0.39). Finally, once the state 𝑥₂ is confined to move inside the region defined by 𝑘_𝑚2, the linear saturation function no longer acts over this state; that is, 𝜎_𝑚(𝑥₂)=𝑥₂. Therefore, 𝑣₂ turns out to be 𝑣₂=−𝑥₂−𝑘_𝑖𝜎_𝑚𝑥₁. (3.19) In the same way, after 𝑡>𝑇₃, we can claim that the model in (3.4) leads to𝑥̇₁=−𝑘_𝑖𝜎_𝑚𝑥₁+3𝛼𝑧₃𝑥²₄,𝑥̇₂=−𝑥₂−𝑘_𝑖𝜎_𝑚𝑥₁+𝛼𝑧₃𝑥²₄,𝑥̇₃=−𝑥₃−𝜎_𝑚𝑥₂−𝑘_𝑖𝜎_𝑚𝑥₁,𝑥̇₄=−𝑥₄−𝑥₃−𝜎_𝑚𝑥₂−𝑘_𝑖𝜎_𝑚𝑥₁. (3.20) Now, in instead of showing that the state 𝑥₁ is bounded, we show in what follows that, after a finite period of time 𝑡>𝑇₃, all the states asymptotically converge to zero. Let us first introduce the following useful lemma.

Lemma 3.2. Consider the first two equations in (3.20) and the following positive definite function: 𝑉_𝑚𝑥₂,𝑥₁=^𝑥2₀𝜎_𝑚(𝑠)𝑑𝑠+𝑘_𝑖^𝑥1₀𝜎_𝑚(𝑠)𝑑𝑠. (3.21) After a finite period of time 𝑡>𝑇₃, the following inequality holds: 𝑉̇_𝑚𝑥₂,𝑥₁≤𝐾_𝑚𝑥²₄−12𝑥²₂+𝑘²_𝑖𝜎²_𝑚𝑥₁−12𝑣²₂, (3.22) where 𝐾_𝑚≜𝑚𝜅₀(3𝑘_𝑖+1)+𝛿.

The proof of this Lemma is given in Appendix A.

3.4. Asymptotic Convergence to the Origin of the Whole State

From the above discussion we conclude that, after the finite time 𝑡>𝑇₃>0, the states 𝑥₁, 𝑥₂, and 𝑥₃ are bounded in some compact set, which defines the closed-loop stability domain. To guarantee that all the states asymptotically converge to zero we propose the following candidate Lyapunov function:𝑉_𝑇(𝑥)=𝑉₁𝑥₄,𝑥₃+𝑉_𝑚𝑥₄,𝑥₃, (3.23) where 𝑉₁ and 𝑉_𝑚 were previously defined in (3.5) and (3.21), respectively. Since functions 𝑉₁(∗) and 𝑉_𝑚(∗) are strictly positive definite function, with their respective arguments, we can claim that 𝑉_𝑇(𝑥) qualifies as a candidate Lyapunov function. So, in case that 𝑡>𝑇₃ we have that the time derivative of 𝑉_𝑇 satisfies the following inequality (see Lemma 3.2 and Remark 3.1): 𝑉̇_𝑇(𝑥)≤−32−𝐾_𝑚𝑥²₄−12𝑥²₂+𝑘²_𝑖𝑠²_𝑚𝑥₁. (3.24) Selecting 𝐾_𝑚<3/2 we have that 𝑉̇_𝑇(𝑥)≤0 (e.g., 𝑘_𝑖=2/3and𝑚=1, for a given 𝜅₀=0.39.) From Lyapunov's direct method we ensure the stability of the whole state in the Lyapunov sense. In order to prove now asymptotic stability, we use the well-known LaSalle's theorem [25]. In the region defined as𝑆=𝑥∈ℝ⁴∶𝑉̇_𝑇(𝑥)=0 (3.25) we have that 𝑥₄(𝑡)=0,𝑥₂(𝑡)=0, and 𝑥₁(𝑡)=0. Thus, in the set 𝑆, we also have 𝑣₂=0. Now, from the four chained integrators model (3.4) we have 𝑥₃(𝑡)=0, in the set 𝑆. Therefore, the largest invariant set 𝑀⊂𝑆 is given by 𝑥=0. Thus, according to LaSalle's theorem all the trajectories of system (3.20) asymptotically converge towards to the largest invariant set 𝑀={𝑥=0}.

We summarize our previous discussion with the next proposition, which corresponds to our main result.

Proposition 3.3. Consider the closed-loop cart-pendulum system as described by model (2.5) with: 𝑣=−𝑧₃−2𝑧₄−𝜎_𝑚𝑧₂+2𝑧₃+𝑧₄−𝑘_𝑖𝜎_𝑚𝑧₁+3𝑧₂+3𝑧₃+𝑧₄. (3.26) Then the closed-loop system is globally asymptotically stable and locally exponentially stable, provided that the parameters 𝑚 and 𝑘_𝑖 satisfy the inequalities 1>𝑘_𝑖+𝜅₀𝑚4𝑘_𝑖+1²+𝛿𝑚,𝑚𝜅₀3𝑘_𝑖3+1+𝛿<2. (3.27)

Remark 3.4. In order to simplify as much as possible the previous stability analysis we used the proposed 𝑣₂, which is formed using a linear saturation function. However, nonlinear saturation functions can also be used, as we showed in Appendix B.

3.5. An Illustrative Example

In order to show the effectiveness of the proposed nonlinear control strategy we developed an experiment that allows us to compare the behavior of the strategy, in the presence and the absence of a damping force. The damping force was added to the left-hand side of the first equation in (2.1), as 0.6𝜃̇. We chose the controller parameter values to be 𝑚=1,𝑘_𝑖=0.666. As far as the initial conditions are concerned we take (𝜃,𝜃̇,𝑞,𝑞)̇=(1.15[rad],0,1,0.25). Figure 2 shows the results coming out from the numerical simulations. As can be seen we have, as expected, a quite effective performance for the controller. Even when the sustained damping force is present, the closed-loop response is still quite well. Observe that in order to compensate the damping force effect the cart has to make larger displacements. This numerical result is of importance, because, contrarily to our strategy, many others developments for the same porpoises are very sensitive to the presence of an unmatched damping force in the nonactuated coordinate.

(a)

(b)

(c)

(d)

Figure 2 

Comparison between the closed-loop responses when the damping force is presented. The dotted lines correspond to the system response when the damping force perturbation is present, while the solid lines represent the case when this force is absent.

4. Concluding Remarks

In this paper a new control strategy is proposed in order to solve the well-known cart-pendulum regulation problem, assuming that the pendulum is initialized in the upper half plane. The control strategy used a control-oriented model of the considered system (a model consisted of a nonlinearly perturbed chain of four integrators), previously introduced in [22]. The model choice lets us design a simple stabilizer consisted of two parts. The first part characterizes a linear controller, devoted to bring the nonactuated coordinate (i.e., both the angular position and the angular velocity) near to the unstable vertical position and keep it inside of a small vicinity which defines the closed-loop stability domain. The other part is a bounded controller which, in conjunction with the linear part, ensures that the closed-loop whole state asymptotically converges to the origin. The combined control law ensures then the regulation of the system. Our stability analysis was carried out using standard arguments from linear systems theory in conjunction with the traditional Lyapunov method and the famous LaSalle's theorem. We strongly believe that many other nonlinear underactuated dynamical systems can be stabilized using our simple control approach. We must point out that a main advantage of this work is that we did not need to solve PDE, nonlinear differential equations, and nested saturation functions. Finally, the numerical experiments carried out with an academic example illustrated how effective is our control strategy when an unknown damping force is present.

Appendices

A. Proof of Lemma 3.2

Proof. We must remark that the time derivative of 𝑉_𝑚 (3.21) around the trajectories defined by the first two equations of (3.20) is given by 𝑉̇_𝑚=𝛼𝑧₃𝑥²₄3𝑘_𝑖𝜎_𝑚𝑥^{𝜛0(𝑥)0𝑥0𝑓𝑒37}₁+𝜎_𝑚𝑥₂+𝑘_𝑖𝜎_𝑚𝑥₁𝑥^{𝜛1𝑥,𝑣20𝑥0𝑓𝑒37}₂−𝑣²₂. (A.1) Then after 𝑡>𝑇₃ we must have 𝑣₂=−𝑥₂−𝑘_𝑖𝜎_𝑚(𝑥₁). Therefore, 𝜛₁(𝑥,𝑣₂) can be expressed as 𝜛₁𝑥,𝑣₂1=−2𝑥²₂+𝑘²_𝑖𝜎²_𝑚𝑥₁−12𝑣²₂, (A.2) and evidently 𝜛₀(𝑥) can be bounded by 𝜛₀(𝑥)≤𝐾_𝑚𝑥²₄≜𝑚𝜅₀3𝑘_𝑖+1𝑥²₄. (A.3) Substituting (A.2) and (A.3) into (A.1), we get inequality (3.22), which concludes this proof.

B. Proof of Remark 3.4

Proof. For simplicity we use as nonlinear saturation function the following: 𝑚tanh(𝑥). That is, 𝑣 is formed as: 𝑣=−𝑥₃−𝑥^{𝑣10𝑥0𝑓𝑒37}₄+−𝑚tanh𝑥₂−𝑘^{𝑣20𝑥0𝑓𝑒37}_𝑖𝑚tanh𝑥₁. (B.1) Selecting 𝑣₁ and the bound for 𝑣₂ as discussed in Section 3 and taking into account expressions (3.3) and (3.13), we guarantee that there exists a time 𝑡>𝑇₂, such that 𝑥₄(𝑡<𝜀)2𝑚𝑘+𝛿=^𝑖+12+𝛿;∀𝑡>𝑇₂>𝑇₁. (B.2) Therefore, the first and the second equations of (3.20) become 𝑥̇₁=𝑥₂−𝑚tanh𝑥₂−𝑘_𝑖𝑚tanh𝑥₁+3𝛼𝑧₃𝑥²₄,𝑥̇₂=−𝑚tanh𝑥₂−𝑘_𝑖𝑚tanh𝑥₁+𝛼𝑧₃𝑥²₄. (B.3) To analyze the boundedness of 𝑥₂, we use the positive definite function 𝐸₂=𝑥²₂/2, whose time derivative can be bounded as 𝐸̇₂=−𝑚𝑥₂tanh𝑥₂−𝑘_𝑖𝑚𝑥₂tanh𝑥₁+𝑥₂𝛼𝑧₃𝑥²₄≤−𝑚𝑥₂tanh𝑥₂−𝑘_𝑖−𝜅₀𝑚𝑘_𝑖+1²4. (B.4) Hence, selecting 𝑚 and 𝑘_𝑖, such that 𝛿+𝑘_𝑖+𝜅₀𝑚𝑘_𝑖+1²4≜𝜂_𝑚𝑘𝑖<1, (B.5) therefore, there is a time 𝑡>𝑇₃ such that 𝑥₂<tanh⁻¹𝜂_𝑚𝑘𝑖≜𝑥_𝑚𝑘𝑖,∀𝑡>𝑇₃. (B.6) Indeed, it follows because, if tanh(𝑥₂)>𝜂_𝑚𝑘𝑖, then 𝐸̇₂<0. Notice that the state 𝑥₂ can be confined to move inside of a compact set relying on the bound 𝑥_𝑚𝑘𝑖. Note that this bound can be manipulated, almost, as desired. Then we can select, for instance, 𝑥_𝑚𝑘𝑖<1.9 to make |𝑥−tanh(𝑥)|<|tanh(𝑥)|;∀|𝑥|<1.9 (B.7) hold. Simple geometric arguments can be applied to prove this inequality. Until now we have only provided sufficient conditions to guarantee that 𝑥₂, 𝑥₃, and 𝑥₄ are bounded (with the corresponding bounds being freely fixed). Now we are in conditions to proof that the whole state asymptotically converges to the origin.
We first choose a positive function (similar to the one used in Lemma 3.2) defined as 𝐸_𝑚𝑥₂,𝑥₁=^𝑥2₀𝑠_𝑚(𝑠)𝑑𝑠+𝑘_𝑖^𝑥1₀𝑠_𝑚(𝑠)𝑑𝑠. (B.8) Differentiating the above equation with respect to (B.3), we have, after using simple algebra as in Lemma 3.2, the following inequality: 𝐸̇_𝑚𝑥₂,𝑥₁≤𝐾_𝑚𝑥²₄+𝑘_𝑖tanh𝑥₁𝑥₂−1^{𝜛(𝑥)0𝑥0𝑓𝑒37}2tanh𝑥₂+𝑘_𝑖tanh𝑥₁²−12𝑣²₂. (B.9) Notice that 𝜛(𝑥) can be expressed as 𝜛(𝑤)1=−2tanh²𝑥₂−𝑘²_𝑖2tanh²𝑥₁+𝑘_𝑖tanh𝑥₁𝑥₂−tanh𝑥₂. (B.10) Now, under the assumption 𝑡>𝑇₃, selecting 𝑥_𝑚𝑘𝑖<1.9, and taking into account (B.7) in the above expression, we have 𝜛(𝑤)1≤−2tanh²𝑥₂−𝑘²_𝑖2tanh²𝑥₁+𝑘_𝑖tanh𝑥₁tanh𝑥₂1≤−2tanh𝑥₂+𝑘_𝑖tanh𝑥₁². (B.11) Thus, 𝐸̇_𝑚 can be bounded as 𝐸̇_𝑚𝑥₂,𝑥₁≤𝐾_𝑚𝑥²₄−12tanh𝑥₂+𝑘_𝑖tanh𝑥₁²−12𝑣²₂. (B.12) We built now the candidate Lyapunov function 𝐸_𝑇=𝐸_𝑚+𝑉₁, with 𝑉₁ defined as in (3.5). Then, using some simple algebra and Remark 3.1 it is easy to show that 𝐸̇_𝑇 can be bounded as 𝐸̇_𝑇(𝑥)≤−32−𝐾_𝑚𝑥²₄−12tanh𝑥₂+𝑘_𝑖tanh𝑥₁². (B.13) Selecting 𝐾_𝑚<3/2 (as in Proposition 3.3), we have that 𝐸̇ is semidefinite negative. Hence all the states are bounded. Finally, invoking the LaSalle's theorem and following standard arguments, we can show that the whole state of the closed-loop system asymptotically converges to the origin. This concludes the proof.