Stabilization of the Ball on the Beam System by Means of the Inverse Lyapunov Approach

Abstract

A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.

1. Introduction

The ball and the beam system (BBS) is a popular and important nonlinear system due to its simplicity and easiness to understand and implement in the laboratory. It is also an unstable system, and for this reason it has been widely used not only as a test bed for the effectiveness of control design techniques offered by modern control theory [1, 2] but also to avoid the danger that usually accompanies real unstable systems when brought to the laboratory. In fact, the dynamics of this system are very similar to those found in aerospace systems.

The BBS system consists of a beam, which is made to rotate in a vertical plane by applying a torque at the center of rotation, and a ball that is free to roll along the beam. Since the system does not have a well-defined relative degree at the origin, the exact input-output linearization approach cannot be directly applied to stabilize it around the origin; that is, this system is not feedback linearizable by means of static or dynamic state feedback. This obstacle makes it difficult to design either a stabilizable or a tracking controller [2, 3]. Fortunately, the system is locally controllable around the origin. Hence, it is possible to control it, if it is initialized close enough to the origin by using the direct pole placement method.

Due to its importance several works related to the control of the BBS can be found in the literature. A control strategy based on an approximate feedback linearization was proposed by Hauser et al. in [2]. The main idea consists of discarding certain terms to avoid singularities. The drawback of this strategy is that the closed-loop system behaves properly in a small region, but it fails in a large one. In the same spirit, combined with suitable intelligent switches, we mention the works of [4, 5]. In the first work the authors present a control scheme that switches between exact and approximate input-output linearization control laws; in the other work the use of exact input-output linearization in combination with fuzzy dynamic control is proposed. A constructive approach based on the Lyapunov theory was developed in [6], where a numerical approximation for solving one PDE was considered. In the similar works of [7–10], energy matching conditions were used for the stabilization of the BBS. They also used some numerical approximations in order to solve approximately two matching conditions required to derive the candidate Lyapunov function. A major contribution, rather similar to the matching energy-based approach, was considered in [11, 12]. In these works, the authors solved the two matching conditions related with the potential and kinetic energies of the closed-loop system. In [13], a nested saturation design was proposed in order to bring the ball and the beam to the unstable equilibrium position. Following the same idea, a global asymptotic stabilization was developed with state-dependent saturation levels [14]. A novel work based on a modified nonlinear PD control strategy, tested in the laboratory, was presented in [15]. Finally, many control strategies for the stabilization of the BBS can be found in the literature, but most of them manage the physical model by introducing some nonlinear approximations or switching through singularities (see [1, 3]).

In this paper we propose a novel inverse Lyapunov-based procedure in combination with the energy shaping method to stabilize the BBS. Intuitively, the Lyapunov function is found as if it were an inverse stability problem; that is, we first choose the dissipation rate function of the time derivative of the unknown candidate Lyapunov function. For that purpose, we shape a suitable candidate Lyapunov function, which is locally strictly positive definite inside an admissible set of attraction. Afterwards, the control is proposed in such a way that the time derivative of the obtained Lyapunov function is forced to be equal to the proposed dissipation rate function. The proposed Lyapunov function is formed by adding a kinetic energy function and a particular function, which can be considered as the corresponding potential energy function. To carry this out, we found two restriction equations related to the potential and kinetic energies. The main characteristic of our control strategy is that we do not need to force the closed-loop system to follow another stable Euler-Lagrange or Hamiltonian system, contrary to what was previously proposed in [7–11, 16, 17].

The rest of this paper is organized as follows. In Section 2 we present the control model of the BBS. In Section 3 we briefly introduce the inverse Lyapunov method for solving the stabilization of the BBS; we also discuss the asymptotic convergence of the closed-loop system. In Section 4 we present some numerical simulations to assess the effectiveness of our control strategy. In Section 5 some conclusions are given.

2. System Dynamics

Consider the BBS shown in Figure 1, which consists of a beam that rotates in a vertical plane when a torque is applied to its rotational center and a ball that freely moves forwards and backwards along the beam with only a single degree of freedom. The BBS nonlinear model is described by the following set of differential equations (see [1, 2]): 𝐽𝑚+^𝐵𝑅³𝑟̈−𝑚𝑟𝜃̇²+𝑚𝑔sin𝜃+𝛽𝑟̇=0,𝑚𝑟²+𝐽_𝐵+𝐽𝜃̈+2𝑚𝑟𝑟𝜃̇̇+𝑚𝑔𝑟cos𝜃=𝜏, (2.1) where 𝑟 is the ball position along the beam, 𝜃 is the beam angle, 𝐽 is the moment of inertia of the beam around the rotating pivot, 𝐽_𝐵 is the moment of inertia of the ball with respect to its center, 𝑅 is the radius ball, 𝑚 is the ball mass, 𝛽>0 is the friction coefficient, and 𝜏 is the torque of the system. After applying the following feedback 𝜏=𝑢𝑚𝑟²+𝐽_𝐵+𝐽+2𝑀𝑟𝑟𝜃̇̇+𝑚𝑔𝑟cos𝜃 (2.2) into system (2.1), it can be rewritten as 𝑟̈=𝑑𝑟𝜃̇²̈−𝑛sin𝜃−𝑏𝑟,̇𝜃=𝑢, (2.3) where 𝛽𝑏=𝑚+𝐽_𝐵/𝑅³𝑚,𝑑=𝑚+𝐽_𝐵/𝑅³,𝑛=𝑚𝑔𝑚+𝐽_𝐵/𝑅³. (2.4) Note that 𝑢 can be seen as a virtual controller that acts directly on the actuated coordinate 𝜃. Naturally, the latter system equations can be written as 𝑞̈=𝑆(𝑥)+𝐹𝑢, (2.5) where 𝑞^𝑇=(𝑟,𝜃) and 𝑥^𝑇=(𝑞,𝑞)̇.¹

Figure 1 

The ball and beam system.

3. Control Strategy

The control objective is to bring all the states of system (2.3) to the unstable equilibrium point 𝑥=0, restricting both the beam angle and the ball position to inside the admissible set 𝑄∈𝑅², defined by 𝑄=𝑞=(𝑟,𝜃)∶|𝑟|≤𝐿∧|𝜃|≤𝜋𝜃<2, (3.1) where the positive constants 𝐿 and 𝜃 are known. To this end, a suitable candidate Lyapunov function is constructed by using the Inverse Lyapunov Approach.

3.1. Inverse Lyapunov Approach

A brief description of the inverse Lyapunov method, inspired in the previous work of Ortega and García-Canseco [18], is introduced next.

Let us propose a candidate Lyapunov function for the closed-loop system energy function, of the form 𝑉(𝑥)=12𝑞^𝑇̇𝐾_𝑐(𝑟)𝑞̇+𝑉_𝑝𝑞, (3.2) where the closed-loop inertia matrix 𝐾_𝑐(𝑟)=𝐾^𝑇_𝑐(𝑟)>0, and the closed-loop potential energy functions 𝑉_𝑝(𝑞)>0, will be defined in the forthcoming developments. A straightforward calculation shows that, along with the solutions of (2.5), 𝑉̇ is given by 𝑉̇(𝑥)=∇_𝑞𝑉^𝑇𝑞̇+∇_̇𝑞𝑉^𝑇(𝑆(𝑥)+𝐹𝑢(𝑥)). (3.3)

Comment 1. In fact, 𝑉_𝑝(𝑞) is selected such that ∇_𝑥𝑉_𝑝(𝑥)|_𝑥=0 and ∇²_𝑥𝑉_𝑝(𝑥)|_𝑥=0>0; that is, we require that 𝑉_𝑝 be strictly locally convex around the origin.

Fixing the following auxiliary variable as 𝜂(𝑥)=𝜃̇+𝛼(𝑟)𝑟̇, with 𝛼(𝑟)≠0, forall𝑟∈𝑄 (it is given in advance), we want to find 𝑢(𝑥)∈𝑅, 𝑉_𝑝(𝑞)∈𝑅⁺ and 𝐾_𝑐(𝑟)>0, with 𝑥∈𝐷⊂𝑄×𝑅².² Such that 𝑉̇ can be rewritten as 𝑉̇(𝑥)=𝜂(𝑥)𝛽(𝑥)+𝑢(𝑥)+𝑅(𝑥), (3.4) where 𝛽(𝑥) and 𝑅(𝑥) are continuous functions. Then, we propose the control law as 𝑢(𝑥)=−𝑘_𝑑𝜂(𝑥)−𝛽(𝑥), (3.5) for some 𝑘_𝑑>0, which evidently leads to 𝑉̇(𝑥)=−𝑘_𝑑𝜂²(𝑥)+𝑅(𝑥). (3.6) In order to guarantee that 𝑉̇ be semidefinite negative, we require that a 𝑘_𝑑 exists, such that −𝑘_𝑑𝜂²(𝑥)+𝑅(𝑥)≤0. (3.7)

Physically, we are choosing a convenient dissipation function, 𝜂(𝑥), of the unknown closed-loop energy function 𝑉(𝑥), with the property that 𝑛 divides (𝑉̇−𝑅)(𝑥). We must underscore that the fixed 𝜂 relies on the unactuated coordinate 𝑟, in agreement with the structure of the closed-loop energy function.

On the other hand, 𝑅(𝑥) is the work of the friction forces, which act over the unactuated coordinate.

Closed-Loop System Stability
If we are able to shape the candidate Lyapunov function (3.2), such that its time derivative, along the trajectories of system (2.3), can be expressed as (3.4) under the assumption that (3.7) holds, then 𝑉 qualifies as a Lyapunov function, because it is a nonincreasing and positive definite function in the neighborhood of the origin and proper on its sub-level (i.e., there exists a 𝑐>0, such that 𝑉(𝑥)≤𝑐 defines a compact set, with closed level curves). Consequently, 𝑥 is stable in the Lyapunov sense.

3.2. Solving the BBS Stabilization Problem by Applying the Inverse Lyapunov Approach

In this section we explain how to take the original expression of 𝑉̇, defined in (3.3), to the desired form (3.4) for the particular case of the BBS.

Defining, 𝐾_𝑐, as³𝐾_𝑐=𝑘₁𝑘₂𝑘₂𝑘₃, (3.8) and according to (3.3), we have that 𝑉̇ can be expressed as 𝑉̇(𝑥)=𝑞^𝑇̇𝜐_𝑝𝑞+𝜐_𝑑(𝑥)+𝐾_𝑐𝐹𝑢+𝑅_𝜐(𝑥), (3.9) where 𝜐_𝑑(𝑥)=𝑣_𝑑𝑖²_𝑖=1=12∇_𝑞𝑞^𝑇̇𝐾_𝑐𝑞̇+𝐾_𝑐𝑑𝑟𝜃̇²0𝜐_𝑝,𝑞=𝑣_𝑝𝑖²_𝑖=1=∇_𝑞𝑉_𝑝𝑞+𝐾_𝑐0−𝑛sin𝜃. (3.10)(3.11) Equating equation (3.9) with (3.3), we obtain, after some simple algebraic manipulations, the following: 𝑅^{𝑇00𝑥0𝑓𝑒37}_𝜐(𝑥)−𝑅(𝑥)+𝑞^𝑇̇𝐾_𝑐𝐹−𝜂^{𝑇10𝑥0𝑓𝑒37}(𝑥)𝑢+𝑞^𝑇̇𝜐_𝑝(𝑞)+𝜐_𝑑(𝑥)^{𝑇20𝑥0𝑓𝑒37}−𝜂(𝑥)𝛽(𝑥)=0. (3.12)

From the above we have that 𝑅_𝑣(𝑥)=𝑅(𝑥). Now, as the matrix 𝐾_𝑐 and functions 𝑣_𝑝(𝑞) and 𝑣_𝑑(𝑥) can be seen as free control parameters, we can select them, such that the following equalities hold: 𝑞^𝑇̇𝐾_𝑐𝐹=𝑛(𝑥),𝑞^𝑇̇𝜐_𝑝𝑞+𝜐_𝑑(𝑥)=𝜂(𝑥)𝛽(𝑥). (3.13)(3.14)

This implies that 𝑇_𝑖=0, with 𝑖=0,1,2. Indeed, it is justified because 𝐾_𝑐 is constituted by three free parameters.

Note that this equation has two unknown parameters, given by 𝑘₁ and 𝑉_𝑝. Hence, in order to solve it, we require that 𝑞^𝑇̇𝜐_𝑝𝑞=𝜁_𝑝𝑞𝑛(𝑥),𝑞^𝑇̇𝜐_𝑑(𝑥)=𝜁_𝑑(𝑥)𝑛(𝑥), (3.15) where the continuous functions 𝜁_𝑝 and 𝜁_𝑑 will be computed later by using simple polynomials factorization. Consequently, 𝛽(𝑥) is directly computed by: 𝛽(𝑥)=𝜁_𝑝𝑞+𝜁_𝑑(𝑥). (3.16) Finally, 𝜁_𝑝 and 𝜁_𝑑 are obtained according to the following remark.

Remark 3.1. Notice that 𝑞^𝑇̇𝜐_𝑝(𝑞) and 𝑞^𝑇̇𝜐_𝑑(𝑞) are polynomials with respect to variables (𝑟,̇𝜃̇). Consequently, thefollowing equalities 𝑞^𝑇̇𝜐_𝑝𝑞_{̇𝜃=−𝛼(𝑟)̇𝑟}=0,𝑞^𝑇̇𝜐_𝑑(𝑥)_{̇𝜃=−𝛼(𝑟)̇𝑟}=0 (3.17)(3.18) imply that functions 𝜁_𝑝(𝑞) and 𝜁_𝑑(𝑥) satisfy the restrictions in (3.15). In other words, the selected 𝜂 must divide the two scalar functions 𝑞^𝑇̇𝜐_𝑝(𝑞) and 𝑞^𝑇̇𝜐_𝑑(𝑞).

3.2.1. Computing the Needed Candidate Lyapunov Function

In this section we obtain the unknown control variables 𝐾_𝑐 and 𝑉_𝑝. We begin by solving the restriction equation (3.13). For simplicity, we set 𝛼(𝑟)=1 and 𝜂=−𝑟̇+𝜃̇. Therefore, from (3.13) and (3.8) we evidently have that 𝑞^𝑇̇𝐾_𝑐𝐹=𝑘₂𝑟̇+𝑘₃𝜃̇=−𝑟̇+𝜃̇, (3.19) which leads to 𝑘₂=−1 and 𝑘₃=1. Now, substituting the fixed values 𝑘₂ and 𝑘₃ (3.10), we have that 𝜐_𝑑1𝜐_𝑑2=12∇_𝑞𝑞^𝑇̇𝐾_𝑐𝑞̇+𝐾_𝑐𝑑𝑟𝜃̇²0=𝑘^′₁2𝑟²̇+𝑑𝑘₁𝑟𝜃̇²−𝑑𝑟𝜃̇². (3.20) Next, substituting the above 𝑣_𝑑1 and 𝑣_𝑑2 in (3.17), we obtain 𝑣_𝑑2(𝑥)+𝜐_𝑑1(𝑥)_{̇𝜃=̇𝑟}=𝑟²̇𝑘^′₁2+𝑑𝑟𝑘₁−1=0, (3.21) which produces the following equation; 𝑘^′₁=−2𝑟𝑑(𝑘₁−1) and whose solution is given by 𝑘₁=1+𝑘₁𝑒^−𝑑𝑟2, where 𝑘₁>0. Hence, matrix 𝐾_𝑐 can be taken as 𝐾_𝑐=1+𝑘₁𝑒^−𝑑𝑟2−1−11. (3.22) According with (3.22), we have that det(𝐾_𝑐)=𝑘₁𝑒^−𝑑𝑟2>0; that is, 𝐾_𝑐>0, when 𝑟 is finite.

Now, to obtain 𝑉_𝑝(𝑞), we proceed to substitute the obtained 𝐾_𝑐 into the relation (3.11), having 𝜐_𝑝1𝜐_𝑝2=∇_𝑞𝑉_𝑝+𝐾_𝑐0=−𝑛sin𝜃−1+𝑘₁𝑒^−𝑑𝑟2𝑛sin𝜃+𝜕𝑉_𝑝𝜕𝑟𝑛sin𝜃+𝜕𝑉_𝑝𝜕𝜃. (3.23) From (3.18), we have that 0=𝜐_𝑝2𝑞+𝜐_𝑝1𝑞=−𝑛𝑘₁𝑒^−𝑑𝑟2sin𝜃+𝜕𝑉_𝑝+𝜕𝑟𝜕𝑉_𝑝,𝜕𝜃 (3.24) whose solution is given by 𝑉_𝑝𝑞=𝑛𝑘₁^𝑟₀sin(𝜃−𝑟+𝑠)𝑒^−𝑑𝑠2𝑑𝑠+Ω(𝑟−𝜃), (3.25) where Ω(∗) must be selected such that 𝑉_𝑝 has a local minimum at the origin 𝑞=0. To assure this condition, it is enough to define Ω(𝑠)=𝑘_𝑝𝑠²/2, where 𝑘_𝑝>𝑛𝑘₁. Therefore, 𝑉_𝑝 reads as 𝑉_𝑝𝑞=𝑛𝑘₁^𝑟₀sin(𝜃−𝑟+𝑠)𝑒^−𝑑𝑠2𝑘𝑑𝑠+^𝑝2(𝑟−𝜃)², (3.26) where the integral term can be exactly computed as sin(^𝑟₀𝜃−𝑟+𝑠)𝑒^−𝑑𝑠2𝑑𝑠=cos(𝜃−𝑟)𝐼_sin𝑟+sin(𝜃−𝑟)𝐼_cos𝑟, (3.27) where 𝐼_cos𝑟=^𝑟₀cos(𝑠)𝑒^−𝑑𝑠2𝑑𝑠=𝛼₁𝜙_s(𝑟),𝐼_sin𝑟=^𝑟₀sin(𝑠)𝑒^−𝑑𝑠2𝑑𝑠=𝛼₁𝛼₀+𝜙_𝑐(𝑟), (3.28) and⁴𝛼₀𝑖=2Imerf2√𝑑,𝛼₁=√𝜋exp(−1/4𝑑)4√𝑑,𝜙_𝑠(𝑟)=2Reerf𝑖+2𝑑𝑟2√𝑑,𝜙_𝑐(𝑟)=−2Imerf𝑖+2𝑑𝑟2√𝑑. (3.29)

Remark 3.2. Notice that relation (3.7) can be rewritten as 𝑅_𝑑(𝑥)=𝑞^𝑇̇𝐻𝑞̇, where 𝐻=−𝑘_𝑑𝑘_𝑑+𝑏𝑘₁𝑘/2_𝑑+𝑏𝑘₁/2−𝑏𝑘₁−𝑘_𝑑𝑞,̇ (3.30) so that 𝑞^𝑇̇𝐻𝑞̇<0, if the parameter 𝑘_𝑑 is selected such that −𝑏+4(𝑘₁−1)𝑘_𝑑>0; recall that 𝑘₁>1.

Hence, the needed controller, defined by (3.5) and (3.16), is given that⁵𝑢=−𝑘_𝑑−𝑟̇+𝜃̇−𝑛sin𝜃+𝜕𝑉_𝑝−𝜃̇𝜕𝜃−𝑑𝑟²+𝑑𝑘₁𝑟𝑟̇𝑟̇+𝜃̇𝑒^−𝑑𝑟2, (3.31) where 𝑘₁>0 and 𝑘_𝑑>0.

We end this section introducing the following important remark.

Remark 3.3. Notice that we can always compute 𝑐̃=max_𝑐>0𝑞∈𝑄∶𝑉_𝑝𝑞=𝑐;suchthat𝑉_𝑝𝑞=𝑐isaclosedcurve. (3.32)

To illustrate the geometrical estimation of the bound “𝑐̃”, we fix the parameter values 𝑘_𝑝=2.5 and 𝑘_𝑝=1.3, the other physical parameter values were set as 𝑛𝑘₁=1 and 𝑑=0.015, and the admissible restricted set was chosen as |𝑟|≤0.6 [m] and |𝜃|≤0.5 [rad]. This setup allows us to give an estimated 𝑐, which evidently guaranties that the level curves of 𝑉_𝑝 are closed. By a thorough numerical inspection, we found that 𝑐=̃0.108 for 𝑘_𝑝=2.5 and 𝑐̃=0.05 for 𝑘_𝑝=1.3. Figure 2 shows the corresponding level curves.

(a)

(b)

Figure 2 

Level curves of the function Φ(𝑞) around the origin, for two sets of values: 𝜂𝑘=1, 𝛿=0.015 and 𝑘𝑝=2.5 (a) and 𝜂𝑘=1, 𝛿=0.015 and 𝑘𝑝=1.3 (b) with restrictions on |𝑟|≤0.6 (m) and |𝜃|≤0.5 (rad).

3.3. Asymptotic Convergence of the Closed-Loop System

Since the obtained 𝑉 is a non-increasing and positive definite function in some neighborhood that contains the origin, then the closed-loop system is, at least, locally stable in the Lyapunov sense. To assure that the trajectories of the closed-loop system asymptotically converge to the origin, restricted to 𝑞(𝑡)∈𝑄, for 𝑡>0, we must define the set Ω_̃𝑐∈𝑅⁴, where Ω_̃𝑐=𝑞,𝑞̇∶𝑞∈𝑄∧𝑉𝑞,𝑞̇<𝑐̃. (3.33) The set Ω_̃𝑐 defines a compact invariant set because, for any initial conditions 𝑥₀=(𝑞₀,𝑞₀̇), with 𝑞₀∈𝑄, provided that 𝑉(𝑥₀)<𝑐̃, then 𝑉(𝑥)<𝑐̃, with 𝑞∈𝑄.

The rest of the stability proof is based on LaSalle’s invariant theorem [19, 20]. To apply this theorem we need to define a compact (closed and bounded) set Ω_̃𝑐, which must satisfy that every solution of system (2.3), in closed-loop with (3.31), starting in Ω_̃𝑐 remains in Ω_̃𝑐, for all future time. Then, we define the following invariant set 𝑆, as: 𝑆=𝑥∈Ω_̃𝑐∶𝑉̇(𝑥)=0=𝑞,𝑞̇∈Ω_̃𝑐∶𝑅_𝑑(𝑥)=𝑞^𝑇̇𝐻𝑞̇=0, (3.34) where 𝐻<0. Let 𝑀 be the largest invariant set in 𝑆. Because the theorem of LaSalle claims that every solution starting in a compact set Ω_̃𝑐 approaches 𝑀, as 𝑡→∞, we compute the largest invariant set 𝑀⊂𝑆. Clearly, we have that 𝑟̇=0 and 𝜃̇=0, on the set 𝑆. Therefore, we must have that 𝑟̈=0 and 𝜃̈=0, on the set 𝑆. Similarly, we must have that 𝑟=𝑟_∗ and 𝜃=𝜃_∗, with 𝑟_∗ and 𝜃_∗ being constants. Hence, on the set 𝑆, the first equation of (2.3) is written as 0=−𝑛sin𝜃_∗, then 𝜃_∗=𝑘𝜋, where 𝑘 is an integer. However, 𝜃_∗∈(−𝜋/2,𝜋/2)because (𝑞,𝑞)̇∈Ω_̃𝑐; consequently 𝜃=0, on the set 𝑆. In a similar way, we can show that 𝑟_∗=0. Then, on the set 𝑆, we have that 𝑞=0 and 𝑞̇=0. Therefore, the largest invariant set 𝑀 contained inside set 𝑆 is given by the single point 𝑥=(𝑞=0,𝑞̇=0). Thus, according to the theorem of LaSalle [19], all the trajectories starting in Ω_̃𝑐 asymptotically converge towards the largest invariant set 𝑀⊂𝑆, which is the equilibrium point 𝑥=0.

We finish this section by presenting the main proposition of this paper.

Proposition 3.4. Consider system (2.3) in closed-loop with (3.31), under conditions of Remarks 3.2 and 3.3. Then the origin of the closed-loop system is locally asymptotically stable with the domain of attraction defined by (3.33).

4. Numerical Simulations

To show the effectiveness of the proposed nonlinear control strategy we have carried out some numerical simulations by means of the Matlab program. The original system p,arameters, with their respective physical restrictions, were set as 𝑚=0.1kg,𝑅=0.015m,𝐽_𝑏=2.25×10⁻⁵kg⋅m²,𝜃=0.5rad,𝑀=0.2kg,𝐿=0.6m,𝐽=0.36kg⋅m²,𝛽=0.2New⋅m/s (4.1) From the above, we have that 𝑏=0.029, 𝑑=0.01477, and 𝑛=0.1448. The physical control parameters were fixed as 𝑘_𝑝=2.5, 𝑘₁=1/𝑛 and 𝑘_𝑑=0.5, while the initial conditions were fixed; as 𝑥₀=(0.55m;0;0.45rad;0). Notice that the proposed set of parameters {𝑑,𝑘_𝑝,𝑘₁,𝑛} are in agreement with the computation of the restricted stability domain, which has been done in the previous section (see Figure 2(a)); besides the initial conditions satisfy the inequality 𝑉(𝑥₀)<𝑐̃=1.08.⁶

Figure 3 shows the corresponding response of system (2.3) in closed-loop with (3.31), under the conditions in the Remark 3.3. From this figure one can see that both the system position coordinates and the system torque asymptotically converge to the origin, assuring that |𝜃|≤𝜃 and |𝑟|≤𝐿.

Figure 3 

Closed-loop response of the BBS to the initial conditions: (a) 𝑥0 = (0.55 m; 0; 0.45 rad; 0).

In order to provide an intuitive idea of how good our nonlinear control strategy (ACL) is in comparison with the control techniques proposed by Yu in [15] and Hauser et al. in [2], here respectively referred as (YCL) and (HCL), we carried out a second experiment using the same setup as before and assuming that 𝛽=0. The obtained characteristic polynomial of our control strategy of the linearized system is given by 𝑝(𝑠)=0.0634+0.255𝑠+0.88𝑠²+1.72𝑠³+𝑠⁴. (4.2) The control parameters of the YNC and the HNC were selected, such that their corresponding characteristic polynomials coincided with the polynomial (4.2). The initial conditions were fixed as (0.3m;0.18m/s;0.35rad;0.18rad/s). The simulation results are shown in Figure 4. As we can see, our control strategy outperforms the closed-loop responses of the YNC and HNC control strategies.

Figure 4 

Closed-loop response of the BBS to the proposed ANC in comparison with YNC and HNC.

Comment 2. A comparative study between our control strategy and other control strategies presented in the literature for solving the stabilization of the BBS, is beyond the scope of this work.

5. Conclusions

In this work we proposed a novel procedure to stabilize the BBS by using the inverse Lyapunov approach in conjunction with the energy shaping technique. This procedure consists of finding the candidate Lyapunov function as if it were an inverse stability problem. To carry it out, we chose a convenient dissipation function of the unknown closed-loop energy function. Then, we proceeded to obtain the needed energy function, which is the addition of the positive potential energy and the positive kinetic energy. Afterwards, we directly derived the stabilizing controller from the already-obtained time derivative of the Lyapunov function. The corresponding asymptotic convergence analysis was done by applying the theorem of LaSalle. To assess the performance and effectiveness of the proposed control strategy, we carried out some numerical simulations. The simulation results allow us to conclude that our strategy behaves quite well in comparison with other well-known control strategies. It is worth mentioning that, to our knowledge, the procedure used here to obtain the needed Lyapunov function has not been used before to control the BBS.

Acknowledgments

This research was supported by the Centro de Investigación en Computación of the Instituto Politecnico Nacional (CIC-IPN) and by the Secretaría de Investigación y Posgrado of the Instituto Politecnico Nacional (SIP-IPN), under Research Grants 20113116 and 20113280. A first version of this work was presented in the AMCA 2011 Conference.

Endnotes

1. Evidently, 𝑆(𝑥)=𝛿𝑟𝜃̇²−𝑛sin𝜃−𝑏𝑟̇0^𝑇𝐹=01^𝑇. (5.1)
2. The set 𝐷 is related with the region of attraction of the closed-loop system.
3. For simplicity, we use 𝐾_𝑐=𝐾(𝑟), 𝑘_𝑖=𝑘_𝑖(𝑟), 𝑘^′_𝑖=𝑑/𝑑𝑟𝑘_𝑖(𝑟), for 𝑖={1,2,3}.
4. Symbol erf stands for the Gauss error function, defined by (𝑥)=2erf√𝜋^𝑥₀exp−𝑠²𝑑𝑠. (5.2)
5. After some simple algebraic manipulations it is easy to show that 𝜁_𝑝=𝑛sin𝜃+𝜕𝑉_𝑝𝜕𝜃;𝜁_𝑑=−𝑑𝑟𝜃̇²+𝑑𝑘₁𝑟𝑟̇𝑟̇+𝜃̇𝑒^−𝑑𝑟2 (5.3)
6. For this particular case, the condition of Remark 3.2 is satisfied, because −𝑏+4𝑘₁−1𝑘_𝑑=𝑏+2exp−𝑑∗𝐿²=13.7>0. (5.4)