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

Authors: Carlos Aguilar-Ibañez; Juan C. Martínez-García; Alberto Soria-López; José de Jesús Rubio
Journal: Mathematical Problems in Engineering
 (2011)

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## 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.


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## Body

## 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. Section2 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 Figure1), described by the following set of normalized differential equations (see for instance [4]): (2.1)cos⁡θq̈+θ̈-sin⁡θ=0,(1+δ)q̈+cos⁡θθ̈-θ̇2sin⁡θ=f,
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: (2.2)q̈=1δ+sin⁡2θ(f+θ̇2sin⁡θ-cos⁡θsin⁡θ)≜u.
Thus, system (2.1) can be written in a very simple way as(2.3)θ̈=sin⁡θ-cos⁡θu,q̈=u.
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: (2.4)z1=q+2tanh-1(tanθ2),z3=tanθ,z2=q.+θ.secθ,z4=θ.sec2θ.
Then, system (2.3) can be written as (2.5)z.1=z2,z.2=z3+α(z3)z42,z.3=z4,z.4=v,
where the term α(z3) is given by (2.6)α(z3)=z3(1+z32)3/2
and v is now the new control variable defined as (2.7)v≜sec2θ(-ucos⁡θ+sin⁡θ)+2θ̇2sec2θtanθ.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=2/31.5.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.
Letx∈ℝ. The classical linear saturation function is defined as
(2.8)σm(x)={xif|x|≤m,mx|x|if|x|>m,
where constant m is strictly positive.Definition 2.3.
By a sigmoidal functionsm(x), we mean a smooth function that is bounded, strictly increasing with the property that sm(0)=0; xsm(x)≥0 and |sm(x)|≤m, for all x∈ℝ.We now proceed to propose our control strategy.


## 2.1. The Cart-Pendulum Model

Consider the cart-pendulum system (see Figure1), described by the following set of normalized differential equations (see for instance [4]): (2.1)cos⁡θq̈+θ̈-sin⁡θ=0,(1+δ)q̈+cos⁡θθ̈-θ̇2sin⁡θ=f,
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: (2.2)q̈=1δ+sin⁡2θ(f+θ̇2sin⁡θ-cos⁡θsin⁡θ)≜u.
Thus, system (2.1) can be written in a very simple way as(2.3)θ̈=sin⁡θ-cos⁡θu,q̈=u.
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: (2.4)z1=q+2tanh-1(tanθ2),z3=tanθ,z2=q.+θ.secθ,z4=θ.sec2θ.
Then, system (2.3) can be written as (2.5)z.1=z2,z.2=z3+α(z3)z42,z.3=z4,z.4=v,
where the term α(z3) is given by (2.6)α(z3)=z3(1+z32)3/2
and v is now the new control variable defined as (2.7)v≜sec2θ(-ucos⁡θ+sin⁡θ)+2θ̇2sec2θtanθ.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=2/31.5.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.
Letx∈ℝ. The classical linear saturation function is defined as
(2.8)σm(x)={xif|x|≤m,mx|x|if|x|>m,
where constant m is strictly positive.Definition 2.3.
By a sigmoidal functionsm(x), we mean a smooth function that is bounded, strictly increasing with the property that sm(0)=0; xsm(x)≥0 and |sm(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:(3.1)x≜[x1x2x3x4]=[1331012100110001][z1z2z3z4],
which leads to, when applied to (2.5),(3.2)x.1=x2+x3+x4+3α(z3)x42+v;x.2=x3+x4+α(z3)x42+v;x.3=x4+v;x.4=v.
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 Statesx3 and x4

In order to guarantee that the statesx3 and x4 are bounded, we split v as(3.3)v=-x3-x4︷v1+v2,
where |v2|≤ε, with ε>0. Thus, after substituting (3.3) into (3.2), we have that(3.4)x.1=x2+3α(z3)x42+v2;x.2=α(z3)x42+v2;x.3=-x3+v2;x.4=-x3-x4+v2.
We emphasize that, if |v2|≤ε, with ε small enough, then the states x3 and x4 converge toward the vicinity B(x34)≤δε (for simplicity B(x34)=x32+x42.), where the bound δε 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 statex3 and the state x4, we consider the definite positive function given by(3.5)V1(x4,x3)=12(x3-x4)2+12x42.
Differentiating (3.5) and taking into account (3.4) we have that(3.6)V.1(x4,x3)=-2x42+x4v2.
Now, given the assumption |v2|≤ε we have that V.1 is in fact bounded as follows:
(3.7)V.1(x4,x3)≤-|x4|(-ε+2|x4|).
(Evidently, when v2≈0, the subsystem (3.4) converges asymptotically to zero, because its time derivative, given by (3.6), is equal to -2x42, 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 V.1<0. Consequently, there exists a finite time T1 after which we have(3.8)|x4(t)|≤ε2+δ;∀t>T1,
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)=x32/2, whose time derivative along the of the dynamics of the state x3, in (3.4), satisfies(3.9)W.(x3)=-x32+x3v2≤-|x3|(|x3|-ε).
It implies that there exists a finite time T2>T1 such that(3.10)|x3(t)|≤ε+δ;∀t>T2.Remark 3.1.
Summarizing, after some period of timet>T2, we have that variables, x3 and x4, are bounded. Besides, it holds that
(3.11)V.1(x4,x3)<-32x42+v222.


### 3.3. Stabilization of Both the Statex2 and the State x1

In order to stabilize the missing state variables we propose the bounding control actionv2 as follows:
(3.12)v2=-σm(x2)-kiσm(x1),
with the control parameter ki being characterized by 0<ki<1. Observe that in order to simplify the corresponding bounded analysis, we used the linear saturation function σm(x1). However, this function can be substituted by any other nonlinear saturation function. On the other hand, we have that
(3.13)|v2|≤ε≜m(ki+1).
Then, after substituting the above controller into the second equation of (3.4) we get
(3.14)x.2=-σm(x2)-kiσm(x1)+α(z3)x42.
We introduce now the following positive definite function:  (3.15)V2=∫0x2σm(s)ds
in order to verify the boundedness of the state x2. Differentiating V2 and using (3.14) we have that(3.16)V.2=σm(x2)(-σm(x2)-kiσm(x1)-α(z3)x42).
Selecting m>kim+κ0ε2/4 we can assure that V.2<0, if |x2|>(ki+κ0ε2/4m)+δ. Therefore, there is a finite time T3>T2>0 such that
(3.17)|x2(t)|≤km2≜ki+κ0ε24m+δ;∀t>T3.
We emphasize that the restriction m>δ+kim+κ0ε2/4 can be always satisfied. Indeed, from the definition of ε given in (3.13) we evidently have
(3.18)1>ki+κ0m4(1+ki)2+δm
(just to illustrate how this inequality holds take, for instance, ki=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 km2, the linear saturation function no longer acts over this state; that is, σm(x2)=x2. Therefore, v2 turns out to be
(3.19)v2=-x2-kiσm(x1).
In the same way, after t>T3, we can claim that the model in (3.4) leads to(3.20)x.1=-kiσm(x1)+3α(z3)x42,x.2=-x2-kiσm(x1)+α(z3)x42,x.3=-x3-σm(x2)-kiσm(x1),x.4=-x4-x3-σm(x2)-kiσm(x1).
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:
(3.21)Vm(x2,x1)=∫0x2σm(s)ds+ki∫0x1σm(s)ds.
After a finite period of time t>T3, the following inequality holds:
(3.22)V.m(x2,x1)≤Kmx42-12(x22+ki2σm2(x1))-12v22,
where Km≜mκ0(3ki+1)+δ.The proof of this Lemma is given in AppendixA.


### 3.4. Asymptotic Convergence to the Origin of the Whole State

From the above discussion we conclude that, after the finite timet>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:(3.23)VT(x)=V1(x4,x3)+Vm(x4,x3),
where V1 and Vm were previously defined in (3.5) and (3.21), respectively. Since functions V1(*) and Vm(*) are strictly positive definite function, with their respective arguments, we can claim that VT(x) qualifies as a candidate Lyapunov function. So, in case that t>T3 we have that the time derivative of VT  satisfies the following inequality (see Lemma 3.2 and Remark 3.1): (3.24)V.T(x)≤-(32-Km)x42-12(x22+ki2sm2(x1)).
Selecting Km<3/2 we have that V.T(x)≤0 (e.g., ki=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 as(3.25)S={x∈R4:V.T(x)=0}
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:
(3.26)v=-z3-2z4-σm(z2+2z3+z4)-kiσm(z1+3z2+3z3+z4).
Then the closed-loop system is globally asymptotically stable and locally exponentially stable, provided that the parameters m and ki satisfy the inequalities
(3.27)1>ki+κ0m4(ki+1)2+δm,mκ0(3ki+1)+δ<32.Remark 3.4.
In order to simplify as much as possible the previous stability analysis we used the proposedv2, 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,ki=0.666. As far as the initial conditions are concerned we take (θ,θ.,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.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.
(a)(b)(c)(d)


## 3.1. Linear Transformation

Inspired by what is presented in [24], we first introduce the following linear transformation:(3.1)x≜[x1x2x3x4]=[1331012100110001][z1z2z3z4],
which leads to, when applied to (2.5),(3.2)x.1=x2+x3+x4+3α(z3)x42+v;x.2=x3+x4+α(z3)x42+v;x.3=x4+v;x.4=v.
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 Statesx3 and x4

In order to guarantee that the statesx3 and x4 are bounded, we split v as(3.3)v=-x3-x4︷v1+v2,
where |v2|≤ε, with ε>0. Thus, after substituting (3.3) into (3.2), we have that(3.4)x.1=x2+3α(z3)x42+v2;x.2=α(z3)x42+v2;x.3=-x3+v2;x.4=-x3-x4+v2.
We emphasize that, if |v2|≤ε, with ε small enough, then the states x3 and x4 converge toward the vicinity B(x34)≤δε (for simplicity B(x34)=x32+x42.), where the bound δε 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 statex3 and the state x4, we consider the definite positive function given by(3.5)V1(x4,x3)=12(x3-x4)2+12x42.
Differentiating (3.5) and taking into account (3.4) we have that(3.6)V.1(x4,x3)=-2x42+x4v2.
Now, given the assumption |v2|≤ε we have that V.1 is in fact bounded as follows:
(3.7)V.1(x4,x3)≤-|x4|(-ε+2|x4|).
(Evidently, when v2≈0, the subsystem (3.4) converges asymptotically to zero, because its time derivative, given by (3.6), is equal to -2x42, 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 V.1<0. Consequently, there exists a finite time T1 after which we have(3.8)|x4(t)|≤ε2+δ;∀t>T1,
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)=x32/2, whose time derivative along the of the dynamics of the state x3, in (3.4), satisfies(3.9)W.(x3)=-x32+x3v2≤-|x3|(|x3|-ε).
It implies that there exists a finite time T2>T1 such that(3.10)|x3(t)|≤ε+δ;∀t>T2.Remark 3.1.
Summarizing, after some period of timet>T2, we have that variables, x3 and x4, are bounded. Besides, it holds that
(3.11)V.1(x4,x3)<-32x42+v222.


## 3.3. Stabilization of Both the Statex2 and the State x1

In order to stabilize the missing state variables we propose the bounding control actionv2 as follows:
(3.12)v2=-σm(x2)-kiσm(x1),
with the control parameter ki being characterized by 0<ki<1. Observe that in order to simplify the corresponding bounded analysis, we used the linear saturation function σm(x1). However, this function can be substituted by any other nonlinear saturation function. On the other hand, we have that
(3.13)|v2|≤ε≜m(ki+1).
Then, after substituting the above controller into the second equation of (3.4) we get
(3.14)x.2=-σm(x2)-kiσm(x1)+α(z3)x42.
We introduce now the following positive definite function:  (3.15)V2=∫0x2σm(s)ds
in order to verify the boundedness of the state x2. Differentiating V2 and using (3.14) we have that(3.16)V.2=σm(x2)(-σm(x2)-kiσm(x1)-α(z3)x42).
Selecting m>kim+κ0ε2/4 we can assure that V.2<0, if |x2|>(ki+κ0ε2/4m)+δ. Therefore, there is a finite time T3>T2>0 such that
(3.17)|x2(t)|≤km2≜ki+κ0ε24m+δ;∀t>T3.
We emphasize that the restriction m>δ+kim+κ0ε2/4 can be always satisfied. Indeed, from the definition of ε given in (3.13) we evidently have
(3.18)1>ki+κ0m4(1+ki)2+δm
(just to illustrate how this inequality holds take, for instance, ki=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 km2, the linear saturation function no longer acts over this state; that is, σm(x2)=x2. Therefore, v2 turns out to be
(3.19)v2=-x2-kiσm(x1).
In the same way, after t>T3, we can claim that the model in (3.4) leads to(3.20)x.1=-kiσm(x1)+3α(z3)x42,x.2=-x2-kiσm(x1)+α(z3)x42,x.3=-x3-σm(x2)-kiσm(x1),x.4=-x4-x3-σm(x2)-kiσm(x1).
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:
(3.21)Vm(x2,x1)=∫0x2σm(s)ds+ki∫0x1σm(s)ds.
After a finite period of time t>T3, the following inequality holds:
(3.22)V.m(x2,x1)≤Kmx42-12(x22+ki2σm2(x1))-12v22,
where Km≜mκ0(3ki+1)+δ.The proof of this Lemma is given in AppendixA.


## 3.4. Asymptotic Convergence to the Origin of the Whole State

From the above discussion we conclude that, after the finite timet>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:(3.23)VT(x)=V1(x4,x3)+Vm(x4,x3),
where V1 and Vm were previously defined in (3.5) and (3.21), respectively. Since functions V1(*) and Vm(*) are strictly positive definite function, with their respective arguments, we can claim that VT(x) qualifies as a candidate Lyapunov function. So, in case that t>T3 we have that the time derivative of VT  satisfies the following inequality (see Lemma 3.2 and Remark 3.1): (3.24)V.T(x)≤-(32-Km)x42-12(x22+ki2sm2(x1)).
Selecting Km<3/2 we have that V.T(x)≤0 (e.g., ki=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 as(3.25)S={x∈R4:V.T(x)=0}
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:
(3.26)v=-z3-2z4-σm(z2+2z3+z4)-kiσm(z1+3z2+3z3+z4).
Then the closed-loop system is globally asymptotically stable and locally exponentially stable, provided that the parameters m and ki satisfy the inequalities
(3.27)1>ki+κ0m4(ki+1)2+δm,mκ0(3ki+1)+δ<32.Remark 3.4.
In order to simplify as much as possible the previous stability analysis we used the proposedv2, 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,ki=0.666. As far as the initial conditions are concerned we take (θ,θ.,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.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.
(a)(b)(c)(d)


## 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.



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*Source: 856015-2011-10-17.xml*