Continuous sliding-mode control for underactuated systems: Relative degree one and two

Control Engineering Practice 90 (2019) 342–357

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Continuous sliding-mode control for underactuated systems: Relative degree one and two Luis R. Ovalle a , Héctor Ríos b ,∗, Miguel A. Llama a a

Tecnológico Nacional de México/I.T. La Laguna, División de Estudios de Posgrado e Investigación, Blvd. Revolución y Cuauhtémoc S/N, C.P. 27000, Torreón, Coahuila, Mexico b CONACYT-Tecnológico Nacional de México/I.T. La Laguna, División de Estudios de Posgrado e Investigación, Blvd. Revolución y Cuauhtémoc S/N, C.P. 27000, Torreón, Coahuila, Mexico

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Keywords: Sliding-mode control Underactuated systems Stabilization Continuous control Relative degree

ABSTRACT This paper deals with the design of sliding-mode controllers for the stabilization of some types of underactuated systems. The proposed method takes advantage of the mechanical properties of three different types of underactuated systems instead of using nonlinear transformations. In order to design the controller, some sliding variables with relative degree one and two are introduced. The theoretical differences between these approaches are discussed and some simulation results show the practical differences. Experimental results on a cart–pole system are presented to validate the proposed control strategy.

1. Introduction It is a well-known fact that the concept of relative degree plays a key roll in the design of sliding-mode controllers (Fridman, Moreno, Bandyopadhyay, Kamal, & Chalanga, 2015). Normally, the order of the sliding-mode is obtained depending on the relative degree of the output function of the system with respect to the disturbance, e.g. for an output with relative degree three, a third order sliding-mode is needed (Levant, 2003). Furthermore, it is possible to achieve a continuous control signal by enforcing a sliding-mode of higher order than the relative degree of the system. For instance, the Super-Twisting algorithm (Levant, 1993), a continuous sliding-mode controller, is capable of controlling systems with relative degree one by enforcing a second-order sliding-mode. However, the main drawback of continuous sliding-mode controllers is that the continuous terms could excite unmodeled dynamics generating a different source of chattering (see, e.g. Utkin, 2016). Theoretically speaking, the application of continuous sliding-mode controllers should alleviate the chattering problem since the control signal is continuous (Levant, 2003). However, the advantages of continuous control schemes lessen when the dynamics of the actuator is slow (Pérez-Ventura & Fridman, 2019). Moreover, the price to pay for the continuity is the modification of the class of disturbances these controllers are able to deal with. For instance, while the first-order algorithm is able to cope with bounded perturbations, the Super-Twisting algorithm is insensible to Lipschitz continuous perturbations (Shtessel, Edwards, Fridman, & Levant, 2014). For fully actuated mechanical systems, whose dynamics are modeled as second-order differential equations, the definition of an output

signal with relative degree two is common. However, since the underactuation property has several different natures (Moreno-Valenzuela & Aguilar-Avelar, 2018), e.g. inherent dynamics of the system, actuator faults or optimization of masses and building costs, it is expected that the dynamics of underactuated systems possess different structures. This fact means that, in order to design controllers which are valid for a type of underactuated systems, an appropriate design for a sliding variable is needed. This, in turn, normally means that the finite-time convergence of the states is not achievable. The previous arguments arise the question of whether the use of output functions with relative degree two, with respect to the control input, can be advantageous or not in comparison with usual output functions with relative degree one. The usual procedure for the design of controllers which are valid for a type of underactuated systems relies on a transformation of the dynamics into a normal form (Olfati-Saber, 2002), and then, design a controller for such a normal form. Nonetheless, this approach implies the use of a feedback linearization procedure which is not robust and could cause important problems when dealing with disturbances, unmodeled dynamics and parametric uncertainty. Another problem of such a design is the fact that, since an appropriate output function is needed for the design, the controller is still based on a case-bycase approach. Besides, the use of partial feedback linearization was proposed to decouple the control signal from one of the equations of the underactuated system dynamics (Olfati-Saber, 2000). This implies that the underactuated position signals should have relative degree four with respect to the disturbance. The augmentation of the relative degree could be undesirable since a high relative degree implies the use of a high number of continuous terms.

∗ Corresponding author. E-mail address: [email protected] (H. Ríos).

https://doi.org/10.1016/j.conengprac.2019.07.014 Received 22 February 2019; Received in revised form 16 July 2019; Accepted 19 July 2019 Available online xxxx 0967-0661/© 2019 Elsevier Ltd. All rights reserved.

L.R. Ovalle, H. Ríos and M.A. Llama

Control Engineering Practice 90 (2019) 342–357

In the context of robust control for underactuated systems, it is wellknown that such an area is very active. In Donaire, Romero, Ortega, Siciliano, and Crespo (2017), a robust extension to the interconnection and damping assignment passivity-based controller is proposed to deal with bounded perturbations for systems whose inertia matrices do not depend on the underactuated coordinates. In Rudra, Barai, and Maitra (2014), a block-backstepping controller is employed to stabilize a cart–pole system. On the other hand, a robust controller is proposed by Park (2017) for an uncertain underactuated surface vessel based on a constructive Lyapunov function approach. Sliding-mode control is very common to robustly stabilize underactuated systems. For instance, an adaptive sliding-mode control algorithm is presented in Haghighi and Pang (2018) to solve the robust formation control problem of a swarm of nanosatellites. In Park and Chwa (2009), a coupled sliding-mode controller is proposed for the cart–pole system. In this work the dynamics of the system is not taken into any normal form but the controller is based on a first-order sliding-mode algorithm. In Aguilar, Iriarte, and Fridman (2013), a two relay controller is used for the tracking control of an inertia wheel pendulum. In Vázquez, Fridman, Collado, and Castillo (2015), a Twisting and a Super-Twisting controller are proposed to solve the tracking control problem for a 5 DOF underactuated crane. In Xu and Özgüner (2008), a procedure to control underactuated systems with a strict feedback form is presented. A sliding variable with relative degree one is designed and a first-order sliding-mode controller is used. In Lu, Fang, and Sun (2018) a continuous secondorder sliding-mode controller is presented for a type of underactuated systems, the considered type is the same as the work in Xu and Özgüner (2008). Nevertheless, the sliding variable has relative degree two with respect to a suitable control input, and a controller which is Lipschitz continuous with respect to time is designed. However the controller requires the knowledge of one more derivative of the states. Notice that most of these works perform the design on a caseby-case basis. In this sense, the design of control schemes which are valid for complete types of underactuated systems is a very important and challenging problem. In the context of sliding-mode control, the works presented in Lu et al. (2018) and Xu and Özgüner (2008) are of particular interest since a design of a sliding-mode controller is presented for a complete type of underactuated systems based on a nonlinear change of variables. Motivated by the previous discussion, this paper presents some sliding-mode control designs to robustly stabilize the origin of three different types of underactuated systems without the use of nonlinear transformations. Moreover, different sliding variables are proposed with relative degree one and two. The theoretical differences of the proposed approaches are discussed and their advantages and shortcomings are further investigated by means of simulations. Some experimental results depict the differences of the approaches from the application point of view. This paper is structured as follows: Section 2 introduces the problem statement, where the different types of underactuated systems are introduced. Section 3 presents the design of sliding variables with relative degree one and two with respect to the control input for each type of underactuated system. Section 4 presents the design of the robust controllers. Section 5 shows some simulation results. Section 6 illustrates some experimental results. Finally, Section 7 presents some concluding remarks.

generalized force, 𝑑𝑥 ∈ R represents the disturbance term and 𝒃 = [0, … , 0, 1]𝑇 ∈ R𝑛+1 is an input distribution vector. System (1) can be partitioned as: [ ][ ] [ ] 𝑴 𝒖𝒖 (𝒒 𝒖 , 𝑞𝑎 ) 𝑴 𝒖𝒂 (𝒒 𝒖 , 𝑞𝑎 ) 𝒒̈ 𝒖 𝒈 (𝒒 , 𝑞 ) + 𝐫𝐮 𝒖 𝑎 𝑇 𝑴 𝒖𝒂 (𝒒 𝒖 , 𝑞𝑎 ) 𝑀𝑎𝑎 (𝒒 𝒖 , 𝑞𝑎 ) 𝑞̈𝑎 𝑔𝑟𝑎 (𝒒 𝒖 , 𝑞𝑎 ) [ ][ ] [ ] 𝑪 𝒖𝒖 (𝒒 𝒖 , 𝑞𝑎 , 𝒒̇ 𝒖 , 𝑞̇ 𝑎 ) 𝑪 𝒖𝒂 (𝒒 𝒖 , 𝑞𝑎 , 𝒒̇ 𝒖 , 𝑞̇ 𝑎 ) 𝒒̇ 𝒖 𝟎 + = , (2) 𝑪 𝒂𝒖 (𝒒 𝒖 , 𝑞𝑎 , 𝒒̇ 𝒖 , 𝑞̇ 𝑎 ) 𝐶𝑎𝑎 (𝒒 𝒖 , 𝑞𝑎 , 𝒒̇ 𝒖 , 𝑞̇ 𝑎 ) 𝑞̇ 𝑎 𝜏 + 𝑑𝑥 [ ]𝑇 where 𝒒 𝒖 = 𝑞1 , … , 𝑞𝑛 ∈ R𝑛 represents the underactuated section of the dynamics and 𝑞𝑎 = 𝑞𝑛+1 ∈ R is the actuated part. Let us introduce the following assumption on the mechanical system (2). Assumption 1. For the system (2), the inertia matrix is nonsingular and has a bounded norm for all 𝒒 ∈  ⊆ R𝑛+1 , with  ∶= {𝒒 ∈ R𝑛+1 | ‖𝒒‖ ≤ 𝛾}, for some 𝛾 > 0. Remark 1. Assumption 1 implies that no controllability issues arise and the stabilization is possible. Based on the structural properties of system (2), the following definitions for the Types of underactuated systems are introduced: Definition 1.

System (2) is said to belong to Type I if:

(i) dim(𝑞𝑢 ) = dim(𝑞𝑎 ), i.e. 𝑞𝑢 ∈ R. (ii) The term 𝑀𝑢𝑎 (𝑞𝑢 , 𝑞𝑎 ) is a zero. (iii) At least one of the terms 𝐶𝑢𝑢 (𝑞𝑢 , 𝑞𝑎 , 𝑞̇ 𝑢 , 𝑞̇ 𝑎 ), 𝐶𝑢𝑎 (𝑞𝑢 , 𝑞𝑎 , 𝑞̇ 𝑢 , 𝑞̇ 𝑎 ), 𝑔𝑟𝑢 (𝑞) ∈ R is not equal to zero and depend either on 𝑞𝑎 or 𝑞̇ 𝑎 . (iv) The fact that 𝑞𝑢 = 𝑞̇ 𝑢 = 0 implies that 𝑞𝑎 (𝑡) → 0 as 𝑡 → ∞. Definition 2.

System (2) is said to belong to Type II if:

(i) 𝑀𝑎𝑎 (𝒒 𝒖 ) ∈ R, 𝑴 𝒖𝒂 (𝒒 𝒖 ) ∈ R𝑛×1 , 𝑪 𝒖𝒖 (𝒒 𝒖 , 𝒒̇ 𝒖 ) ∈ R𝑛×𝑛 and 𝒈𝒓𝒖 (𝒒 𝒖 ) ∈ R𝑛 are functions of the underactuated variable 𝒒 𝒖 and its derivative 𝒒̇ 𝒖 . (ii) The relative degree of 𝒒 𝒖 and 𝑞𝑎 with respect to 𝜏 is two. (iii) 𝑪 𝒖𝒂 (𝒒 𝒖 , 𝑞𝑎 , 𝒒̇ 𝒖 , 𝑞̇ 𝑎 ) = 𝟎 for all 𝒒 ∈  and 𝒒̇ ∈ R𝑛+1 . (iv) The only solution to 𝒈𝒓𝒖 (𝒒 𝒖 ) = 𝟎 is given by 𝒒 𝒖 = 𝟎 for all 𝒒 𝑢 ∈ 𝑢 and 𝑢 ⊂ . Definition 3.

System (2) is said to belong to Type III if:

(i) The relative degree of 𝒒 𝒖 and 𝑞𝑎 with respect to 𝜏 is two. (ii) 𝑪 𝒖𝒂 (𝒒 𝒖 , 𝑞𝑎 , 𝒒̇ 𝒖 , 𝑞̇ 𝑎 ) ≠ 𝟎 for all 𝒒 ∈  and 𝒒̇ ∈ R𝑛+1 . (iii) The only solution to 𝒈𝒓𝒖 (𝒒 𝒖 ) = 𝟎 is given by 𝒒 𝒖 = 𝟎 for all 𝒒 𝑢 ∈ 𝑢 and 𝑢 ⊂ . Remark 2. Constraints (iv) of Definition 2 and (iii) of Definition 3 imply that all open-loop equilibria of (19) are given by 𝒒 𝒖 = 𝟎. Remark 3. An example of a Type I system is a 3 DOF helicopter, a cart–pole system for Type II while a 2 DOF flexible joint manipulator belongs to Type III, see Fig. 1. Remark 4. Notice that a system cannot belong to two of the given types. Moreover, if a system does not fall into one of the proposed types, it is always possible to find a diffeomorphism to transform it to Type I as long as dim(𝑞𝑢 ) = dim(𝑞𝑎 ) (Olfati-Saber, 2000).

2. Problem statement Consider the following mechanical system: ̇ 𝒒̇ + 𝒈(𝒒) = 𝒃(𝜏 + 𝑑𝑥 ), 𝑴(𝒒)𝒒̈ + 𝑪(𝒒, 𝒒)

The mechanical system (2) has a state-space representation of the form

(1)

⎡𝒙̇ 𝒖 ⎤ ⎡ ⎤ 𝒙𝒗 ⎢ ⎥ ⎢ ⎥ ̇ 𝒙 𝒇 (𝒙) + 𝒈 (𝒙 , 𝑥 )(𝜏 + 𝑑 ) 𝒖 𝒖 𝑎 𝑥 ⎥ ⎢ 𝒗⎥ = ⎢ 𝒖 , ⎢ 𝑥̇ 𝑎 ⎥ ⎢ ⎥ 𝑥𝑏 ⎢ 𝑥̇ ⎥ ⎢ 𝑓 (𝒙) + 𝑔 (𝒙 , 𝑥 )(𝜏 + 𝑑 ) ⎥ 𝑎 𝒖 𝑎 𝑥 ⎦ ⎣ 𝑏⎦ ⎣ 𝑎

where 𝒒 ∈ R𝑛+1 is a vector of the configuration variables, 𝑴 ∶ R𝑛+1 → R(𝑛+1)×(𝑛+1) is the inertia matrix, 𝑪 ∶ R(𝑛+1)×(𝑛+1) → R(𝑛+1)×(𝑛+1) is a matrix containing the centrifugal and Coriolis forces, 𝒈 ∶ R𝑛+1 → R𝑛+1 represents the forces due to the potential energy, 𝜏 ∈ R is the 343

(3)

L.R. Ovalle, H. Ríos and M.A. Llama

Control Engineering Practice 90 (2019) 342–357

Fig. 1. Examples of Type I, II, and III systems.

with 𝒙𝒖 ∶= 𝒒 𝒖 ∈ R𝑛 , 𝒙𝒗 ∶= 𝒒̇ 𝒖 ∈ R𝑛 , 𝑥𝑎 ∶= 𝑞𝑎 ∈ R, 𝑥𝑏 ∶= 𝑞̇ 𝑎 ∈ R, 𝒇 𝒖 (𝒙) ∈ R𝑛 , 𝑓𝑎 (𝒙) ∈ R, 𝒈𝒖 (𝒙𝒖 , 𝑥𝑎 ) ∈ R𝑛 , 𝑔𝑎 (𝒙𝒖 , 𝑥𝑎 ) ∈ R, 𝒙 ∈  ∶=  × R𝑛+1 , 𝒙 ∶= [𝒙𝒖 𝑇 , 𝑥𝑎 , 𝒙𝒗 𝑇 , 𝑥𝑏 ]𝑇 , and [ ] 𝒇 𝒖 (𝒙) = −𝑴 −1 (𝒙𝒖 , 𝑥𝑎 )[𝑪(𝒙)[𝒙𝒗 , 𝑥𝑏 ]𝑇 + 𝒈(𝒙)], 𝑓𝑎 (𝒙) [ ] 𝒈𝒖 (𝒙𝒖 , 𝑥𝑎 ) = 𝑴 −1 (𝒙𝒖 , 𝑥𝑎 )𝒃. 𝑔𝑎 (𝒙𝒖 , 𝑥𝑎 )

Note that the relation between 𝑧1 and 𝑥𝑢 is linear and that 𝑧2 , 𝑧3 and 𝑧4 are the derivatives of this signal. Thus, this is only a change of variables and not a nonlinear transformation. Due to restriction (𝑖𝑣) in Definition 1, it is known that driving 𝑥𝑢 to zero implies the asymptotic stability of 𝑥𝑎 . Sliding variable with relative degree one Consider the case (𝑎), when 𝑓𝑢 does not explicitly depend on 𝑥𝑏 , i.e. 𝜕𝑓 𝜕𝑓𝑢 = 0 and 𝜕𝑥𝑢 ≠ 0. Then, the following sliding variable is considered 𝜕𝑥

The objective of this paper is to design some sliding-mode controllers to stabilize the origin of systems that belong to the proposed types without the use of nonlinear transformations. All of the designs must employ a continuous control signal and every controller should be robust against Lipschitz continuous disturbances. To this aim, a collection of sliding variables with relative degree one and two are designed.

𝑏

𝑎

(5)

𝑠 = 𝑐1 𝑧1 + 𝑐2 𝑧2 + 𝑐3 𝑧3 + 𝑧4 .

Then, the following theorem describes the stability properties of system (4) on the sliding manifold.

3. Design of the sliding variables

Theorem 1. Assume that 𝑓𝑢 in (4) does not depend explicitly on 𝑥𝑏 , i.e. 𝜕𝑓𝑢 𝜕𝑓 = 0 and 𝜕𝑥𝑢 ≠ 0. Let the parameters 𝑐1 , 𝑐2 and 𝑐3 be designed such that 𝜕𝑥

This section deals with the design of sliding variables with relative degree one and two that allows the stabilization of a given type of underactuated systems.

⎡ 0 𝑨𝒏 = ⎢ 0 ⎢ ⎣−𝑐1

3.1. Design for Type I systems

is Hurwitz. If a sliding-mode is enforced on system (4), with (5) as the sliding variable, then the origin of (4), [𝑥𝑢 , 𝑥𝑎 ]𝑇 = 0, is Asymptotically Stable.

𝑏

(b)

𝜕𝑓𝑢 𝜕𝑥𝑎

𝑑 𝑑𝑡

(

𝜕𝑓 𝜕𝑓𝑢 𝑥 + 𝑢𝑓 𝜕𝑥𝑢 𝑣 𝜕𝑥𝑣 𝑢

) +

𝑑 𝑑𝑡

(

𝜕𝑓𝑢 𝜕𝑥𝑎

) 𝑥𝑏 +

(7)

𝜕𝑓𝑢 𝑓 , 𝜕𝑥𝑎 𝑎

(8)

(9)

with 𝛿𝑥 as a new disturbance term. This closed-loop system has a relative degree equal to one considering 𝑠 as the output and 𝑣 as the input. 𝜕𝑓 For the case (𝑏), when 𝑓𝑢 depends explicitly on 𝑥𝑏 , i.e. 𝜕𝑥𝑢 ≠ 0, the 𝑏 following sliding variable is proposed

Depending on the structure of 𝑓𝑢 , two distinct cases are considered: = 0 and

(6)

reduces the closed-loop system to ) ( 𝜕𝑓𝑢 𝑠̇ = 𝑣 + 𝑔𝑎 𝑑𝑥 = 𝑣 + 𝛿𝑥 , 𝜕𝑥𝑎

(4)

Remark 5. Due to the structure of (4), the constraint (iv) of Definition 1, can be rewritten as: 𝑓𝑢 (0, 𝑥𝑎 , 0, 𝑥𝑏 ) = 0 implies that 𝑥𝑎 → 0 as 𝑡 → ∞.

𝜕𝑓𝑢 𝜕𝑥𝑏 𝜕𝑓𝑢 𝜕𝑥𝑏

0 ⎤ 1 ⎥ ⎥ −𝑐3 ⎦

with 𝜔(𝑥, 𝑡) =

(a)

1 0 −𝑐2

The proof of all the results are given in the Appendix. Hence, the nominal control law ] )−1 [ ( 𝜕𝑓𝑢 𝑔 𝑐1 𝑧2 + 𝑐2 𝑧3 + 𝑐3 𝑧4 + 𝜔(𝑥, 𝑡) − 𝑣 , 𝜏=− 𝜕𝑥𝑎 𝑎

This subsection deals with the application of the main result given by Lu et al. (2018) and Xu and Özgüner (2008) and an extension for systems with relative degree two. Consider the mechanical system (2) and assume that it fulfills the conditions given in Definition 1, i.e. the system belongs to Type I. Due to the restrictions (ii) and (iii) in Definition 1, it is possible to show that the state space representation given by (3) has the following form ⎤ ⎡𝑥𝑢 ⎤ ⎡ 𝑥𝑣 ⎥ ⎢ ⎥ ⎢ 𝑓 (𝒙) 𝑥 𝑑 ⎢ 𝑣⎥ ⎢ 𝑢 ⎥, = ⎥ 𝑥𝑏 𝑑𝑡 ⎢𝑥𝑎 ⎥ ⎢ ⎢𝑥 ⎥ ⎢𝑓 (𝒙) + 𝑔 (𝑥 , 𝑥 )(𝜏 + 𝑑 )⎥ 𝑎 𝑢 𝑎 𝑥 ⎦ ⎣ 𝑏⎦ ⎣ 𝑎

𝑎

(10)

𝑠 = 𝑐1 𝑧1 + 𝑐2 𝑧2 + 𝑧3 .

≠ 0.

Then, the following theorem describes the stability properties of (4) on the sliding manifold.

≠ 0.

Theorem 2. Assume that 𝑓𝑢 in (4) depends explicitly on 𝑥𝑏 , i.e. Let the parameters 𝑐1 and 𝑐2 be designed such that [ ] 0 1 𝑨𝒏 = −𝑐1 −𝑐2

Remark 6. The fulfillment of one of these conditions follows from restriction (iii) in Definition 1. Consider the following change of variables: 𝑧1 = 𝑥𝑢 , 𝑧3 = 𝑓𝑢 (𝑥), 𝑑𝑓𝑢 (𝑥) 𝑧2 = 𝑥𝑣 , 𝑧4 = . 𝑑𝑡

𝜕𝑓𝑢 𝜕𝑥𝑏

≠ 0.

(11)

is Hurwitz. If a sliding-mode is enforced on system (4), with (10) as the sliding variable, then the origin of (4), [𝑥𝑢 , 𝑥𝑎 ]𝑇 = 0 is Asymptotically Stable. 344

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Control Engineering Practice 90 (2019) 342–357

Consider the following nominal control law ] ( )−1 [ 𝜕𝑓𝑢 𝜏=− 𝑔𝑎 𝑐1 𝑧3 + 𝜔(𝑥) ̄ −𝑣 . 𝜕𝑥𝑏

Then, the nominal controller ] ( )−1 [ 𝜕𝑓𝑢 𝑔𝑎 𝜏=− 𝑐1 𝑧2 + 𝑐2 𝑧3 + 𝜔(𝑥) ̄ −𝑣 , 𝜕𝑥𝑏 with

Hence, the closed-loop system is described as

𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜔(𝑥) ̄ = 𝑢 𝑥𝑣 + 𝑢 𝑓𝑢 + 𝑢 𝑥𝑏 + 𝑢 𝑓𝑎 , 𝜕𝑥𝑢 𝜕𝑥𝑣 𝜕𝑥𝑎 𝜕𝑥𝑏 reduces the closed-loop system to ( ) 𝜕𝑓𝑢 𝑠̇ = 𝑣 + 𝑔𝑎 𝑑𝑥 = 𝑣 + 𝛿𝑥 , 𝜕𝑥𝑏

(12)

[ ] ⎡ [ ] ( 𝑠̇ ) ⎤ 𝑠̇ 𝑑 𝑠 ⎥= 𝜕𝑓𝑢 =⎢ , 𝑔 𝑑𝑥 ⎥ ⎢𝑣 + 𝑣 + 𝛿𝑥 𝑑𝑡 𝑠̇ ⎣ ⎦ 𝜕𝑥𝑏 𝑎

(13)

with 𝛿𝑥 as a new disturbance term. 3.1.1. Discussion of the sliding variable design Notice that the structures of the sliding variables, i.e. (5) and (10) for relative degree one; (14) and (17) for relative degree two, provide linear sliding-modes dynamics. Thus, the time response behavior of the system, on the sliding-mode, is modified by means of the parameters 𝑐𝑖 , 𝑖 = 1, 3. Such parameters can be designed based on pole placement or LQR methods in order to provide a certain transient performance.

with 𝛿𝑥 as a new disturbance variable. It is clear that the output of system (13), i.e. s, also has a relative degree equal to one with respect to the input 𝑣. Sliding variable with relative degree two 𝜕𝑓 𝜕𝑓 For the case (𝑎) 𝜕𝑥𝑢 = 0 and 𝜕𝑥𝑢 ≠ 0, the following sliding variable 𝑏 𝑎 is proposed: (14)

𝑠 = 𝑐1 𝑧1 + 𝑐2 𝑧2 + 𝑧3 .

3.2. Design for Type II systems

The relative degree of 𝑠 with respect to 𝜏 is equal to two, this is easily verified by differentiating (14) twice i.e.: 𝑠̇ = 𝑐1 𝑧2 + 𝑐2 𝑧3 + 𝑧4 , 𝑠̈ = 𝑐1 𝑧3 + 𝑐2 𝑧4 + 𝜔(𝑥, 𝑡) +

𝜕𝑓𝑢 𝑓 + 𝜕𝑥𝑎 𝑎

(

Consider the system (2) and assume that it fulfills the conditions given in Definition 3, i.e. this system belongs to Type II. Due to the restriction (i) it is known that this system has the state space representation (3).

) 𝜕𝑓𝑢 𝑔𝑎 (𝜏 + 𝑑𝑥 ), 𝜕𝑥𝑎

with 𝜔 given in (8). Then, the following theorem describes the stability properties of (4) on the sliding manifold.

3.2.1. Sliding variable with relative degree one Consider the following sliding variable ( ) ( ) 𝑠 = 𝜆𝑎 𝜎𝑎 𝑥𝑎 + 𝑥𝑏 + 𝝀𝑇 Ξ𝒙𝒖 + 𝒙𝒗 ,

Theorem 3. Assume that 𝑓𝑢 in (4) does not depend explicitly on 𝑥𝑏 , i.e. 𝜕𝑓𝑢 𝜕𝑓 = 0 and 𝜕𝑥𝑢 ≠ 0. Let the parameters 𝑐1 and 𝑐2 be designed such that 𝜕𝑥𝑏 𝑎 the matrix [ ] 0 1 𝑨𝒏 = (15) −𝑐1 −𝑐2

with 𝜆𝑎 , 𝜎𝑎 > 0 ∈ R, 𝝀 ∈ R𝑛 and 0 < Ξ ∈ R𝑛×𝑛 as parameters to be designed in the sequel. Since the sliding variable depends on the velocities 𝒙𝒗 and 𝑥𝑏 , it follows from (3) that 𝑠 has a relative degree one with respect to the control input. Notice that the stability of the system, once the sliding-mode is enforced, is not trivially verified. For this aim, the following lemma is introduced.

is Hurwitz. If a sliding-mode is enforced on system (4), with (14) as the sliding variable, then the origin of (4), [𝑥𝑢 , 𝑥𝑎 ]𝑇 = 0, is Asymptotically Stable. Consider the following nominal control law ) ( )−1 ( 𝜕𝑓𝑢 𝜕𝑓 𝜏=− 𝑔𝑎 𝑐1 𝑧3 + 𝑐2 𝑧4 + 𝜔(𝑥, 𝑡) + 𝑢 𝑓𝑎 − 𝑣 . 𝜕𝑥𝑎 𝜕𝑥𝑎

Lemma 1. Assume that the parameters of the sliding manifold are chosen such that 𝝀𝑇 𝑴 𝒖𝒖 −1 (𝒙𝒖 )𝑴 𝒖𝒂 (𝒙𝒖 ) ≠ 𝜆𝑎 for all 𝒙 ∈ . Then, on the sliding-mode, 𝑠 = 0, the dynamics of system (3) is given as follows: [ ] 𝒙̇ 𝒗 = − 𝑪 𝟑 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 + 𝒈𝟑 (𝒙𝒖 , 𝑡) + 𝒉𝟐 (𝒙𝒖 , 𝑡) , (19)

(16)

Thus, the closed-loop system is described as [ ] ⎡ [ ] ( 𝑠̇ ) ⎤ 𝑠̇ 𝑑 𝑠 ⎥= 𝜕𝑓𝑢 =⎢ , 𝑔 𝑑𝑥 ⎥ ⎢𝑣 + 𝑣 + 𝛿𝑥 𝑑𝑡 𝑠̇ ⎣ ⎦ 𝜕𝑥𝑎 𝑎 with 𝛿𝑥 as a new disturbance term. On the other hand, for the case (𝑏) sliding variable

with 𝑪 𝟑 (𝒙𝒖 , 𝒙𝒗 ) = 𝑴 𝟐 −1 (𝒙𝒖 )(𝑪 𝒖𝒖 (𝒙𝒖 , 𝒙𝒗 ) + 𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟐 ), 𝜕𝑓𝑢 𝜕𝑥𝑏

𝒈𝟑 (𝒙𝒖 , 𝑡) = 𝑴 𝟐 −1 (𝒙𝒖 )(𝒈𝒓𝒖 (𝒙𝒖 ) + 𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟑 (𝒙𝒖 , 𝑡)), ≠ 0, consider the following

𝒉𝟐 (𝒙𝒖 , 𝑡) = 𝑴 𝟐 −1 (𝒙𝒖 )𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟒 (𝒙𝒖 , 𝑡), 𝑇 𝑴 𝟐 (𝒙𝒖 ) = 𝑴 𝒖𝒖 (𝒙𝒖 ) + 𝑴 𝒖𝒂 𝝍 𝟏 , 𝝍 𝟏 = −𝜆−1 𝑎 𝝀 , 𝑇 −1 𝑇 𝝍 𝟐 = 𝜆−1 𝑎 𝝀 𝑩, 𝝍 𝟑 (𝒙𝒖 , 𝒙𝒗 ) = −𝜆𝑎 𝝀 𝜎𝑎 𝑩𝒂(𝒙𝒖 , 𝑡𝑟 , 𝑡),

(17)

𝑠 = 𝑐1 𝑧1 + 𝑧2 .

𝑇 −𝜎𝑎 (𝑡−𝑡𝑟 ) 𝝍 𝟒 (𝑡) = 𝜎𝑎2 (𝑥𝑎 (𝑡𝑟 ) + 𝜆−1 , 𝑎 𝝀 𝒙𝒖 (𝑡𝑟 ))𝑒

The relative degree of 𝑠 with respect to 𝜏 is also equal to two, consider the following time derivatives of (17):

𝑡

𝒂(𝒙𝒖 , 𝑡𝑟 , 𝑡) = 𝒙𝒖 − 𝜎𝑎

𝑠̇ = 𝑐1 𝑧2 + 𝑧3 , 𝑠̈ = 𝑐1 𝑧3 + 𝜔(𝑥) ̄ +

(18)

𝜕𝑓𝑢 𝑔 (𝜏 + 𝑑𝑥 ), 𝜕𝑥𝑏 𝑎

∫ 𝑡𝑟

𝑒−𝜎𝑎 (𝑡−𝜏) 𝒙𝒖 (𝜏)𝑑𝜏, 𝑩 = 𝜎𝑎 𝑰 𝒏 − Ξ,

and 𝑡𝑟 as the convergence time of a sliding-mode algorithm. The problem now is to prove the stability of the origin of (19). Let us introduce the notation [⋅]𝑖 , which represents the 𝑖th element of a vector. Then, the following theorem is proposed.

with 𝜔̄ given in (12). Thus, the following theorem is introduced 𝜕𝑓

Theorem 4. Assume that 𝑓𝑢 in (4) depends explicitly on 𝑥𝑏 , i.e. 𝜕𝑥𝑢 ≠ 0. 𝑏 Let 𝑐1 in (17) be positive. If a sliding-mode is enforced on system (4), 𝑇 with (17) as the sliding variable, then the origin of (4), [𝑥𝑢 , 𝑥𝑎 ] = 0, is Asymptotically Stable.

Theorem 5.

Assume that the following conditions hold

(i) 𝝀 and 𝜆𝑎 fulfill the conditions of Lemma 1, (ii) Ξ, 𝜎𝑎 > 0 are such that 𝑪 𝟑 (𝒙𝒖 , 𝒙𝒗 ) > 0, 345

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Control Engineering Practice 90 (2019) 342–357

(iii) 𝒈𝒓𝒖 (𝒙𝒖 ), 𝑴 𝒖𝒂 (𝒙𝒖 ) and 𝝍 𝟑 (𝒙𝒖 , 𝑡), defined in Lemma 1, are such that [ ] |[𝒈𝒓𝒖 (𝒙𝒖 )]𝑖 | ≫ | 𝑴 𝒖𝒂 (𝒙𝒖 ) 𝝍 𝟑 (𝒙𝒖 , 𝑡) 𝑖 |,

3.3.1. Sliding variable with relative degree two Consider the following sliding variable

(iv) sign([𝑯(𝒙𝒖 )]𝑖 ) = sign([𝒙𝒖 ]𝑖 ), (v) [𝑯(𝒙𝒖 )]𝑖 is monotonically increasing with respect to [𝒙𝒖 ]𝑖 ,

𝑠 = 𝜆𝑎 𝑥𝑎 + 𝝀𝑇 𝒙𝒖 ,

with 𝑯(𝒙𝒖 ) = 𝑴 𝟐 (𝒙𝒖 )−1 𝒈𝒓𝒖 (𝒙𝒖 ) for all 𝒙 ∈ . Then, the origin of system (19), i.e. [𝒙𝒖 , 𝒙𝒗 ]𝑇 = 𝟎, is Asymptotically Stable. Remark 7. Notice that conditions (i) and (ii) of Theorem 5 hold by design while conditions (iii) and (iv) are structural restrictions on the system. Such conditions are necessary to ensure that the origin of (19) is a stable equilibrium point, which can be checked by means of the Lagrange–Dirichlet theorem (Merkin, 2012) considering a potential 1 energy given by ∫0 𝐻 𝑇 (𝜏𝑥𝑢 )𝑥𝑢 𝑑𝜏.

Lemma 2. Assume that the parameters of the sliding variable (24) are chosen such that 𝝀𝑇 𝑴 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝑴 𝒖𝒖 −1 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ) ≠ 𝜆𝑎 , with 𝝀⋆ = 𝑇 −𝜆−1 𝑎 𝝀 . Then, on the sliding surface, 𝑠 = 0, the dynamics (3) is reduced to: 𝒙̇ 𝒗 = −𝑪 𝟐 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 − 𝒈𝟐 (𝒙𝒖 ),

Once the stability of the sliding-mode dynamics has been proved, the following nominal controller is proposed 𝑇

−1

𝑇

𝜏 = (𝜆𝑎 𝑔𝑎 + 𝝀 𝒈𝒖 ) [𝜆𝑎 (𝑥𝑏 + 𝑓𝑎 ) + 𝝀 (Ξ𝒙𝒗 + 𝒇 𝒖 ) + 𝑣],

(20)

which results in the closed-loop system [ ] 𝑠̇ = 𝑣 + 𝜆𝑎 𝑔𝑎 (𝒙𝒖 , 𝑥𝑎 ) + 𝝀𝑇 𝒈𝒖 (𝒙𝒖 , 𝑥𝑎 ) 𝑑𝑥 = 𝑣 + 𝛿𝑥 .

(21)

3.2.2. Sliding variable with relative degree two Consider the following sliding variable ( ) ( 𝑡

𝑠 = 𝜆𝑎 𝜎𝑎

∫0

𝑡

𝑥𝑎 (𝜏)𝑑𝜏 + 𝑥𝑎 (𝑡)

+𝝀

𝑇

Ξ

∫0

[ 𝑪 𝟐 (𝒙𝒖 , 𝒙𝒗 ) = 𝑴 𝟐 (𝒙𝒖 )−1 𝑪 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 , 𝒙𝒗 , 𝝀⋆ 𝒙𝒗 )𝝀⋆ ] + 𝑪 𝒖𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 , 𝒙𝒗 , 𝝀⋆ 𝒙𝒗 ) , 𝒈𝟐 (𝒙𝒖 ) = 𝑴 𝟐 (𝒙𝒖 )−1 𝒈𝒓𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ), 𝑴 𝟐 (𝒙𝒖 ) = 𝑴 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝝀⋆ + 𝑴 𝒖𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ). In order to prove the stability of the origin of (25), the following theorem is introduced.

)

𝒙𝒖 (𝜏)𝑑𝜏 + 𝒙𝒖 (𝑡) ,

Theorem 6.

(22)

(i) (ii) (iii) (iv)

∫0

∫0

𝝀 and 𝜆𝑎 fulfill the conditions of Lemma 2, 𝝀 and 𝜆𝑎 are such that 𝑪 𝟐 (𝒙𝒖 , 𝒙𝒗 ) > 0, sign([𝑯(𝒙𝒖 )]𝑖 ) = sign([𝒙𝒖 ]𝑖 ), [𝑯(𝒙𝒖 )𝑇 ]𝑖 is monotonically increasing with respect to [𝒙𝒖 ]𝑖 ,

Remark 8. Notice that conditions (𝑖) and (𝑖𝑖) of Theorem 6 hold by design while conditions (𝑖𝑖𝑖) and (𝑖𝑣) are structural restrictions on the system. Such conditions are necessary to ensure that the origin of (25) is a stable equilibrium point, which can be checked by means of the Lagrange–Dirichlet theorem (Merkin, 2012) considering a potential 1 energy given by ∫0 𝐻 𝑇 (𝜏𝑥𝑢 )𝑥𝑢 𝑑𝜏. Since the asymptotic stability of the sliding-mode dynamics has been proved, the following controller is proposed 𝜏 = −[𝜆𝑎 𝑔𝑎 (𝒙𝒖 , 𝑥𝑎 ) + 𝝀𝑇 𝒈𝒖 (𝒙𝒖 , 𝑥𝑎 )]−1 (𝜆𝑎 𝑓𝑎 (𝒙) + 𝝀𝑇 𝒇 𝒖 (𝒙) − 𝑣).

(26)

Then, the following closed-loop system is obtained [ ] [ ] [ ] 𝑠̇ 𝑠̇ 𝑑 𝑠 [ ] = = . 𝑇 𝑣 + 𝜆𝑎 𝑔𝑎 (𝒙𝒖 , 𝑥𝑎 ) + 𝝀 𝒈𝒖 (𝒙𝒖 , 𝑥𝑎 ) 𝑑𝑥 𝑣 + 𝛿𝑥 𝑑𝑡 𝑠̇ Remark 9. To the best of the authors’ knowledge, in the literature, there does not exist a method to control Type III systems by means of a relative degree one sliding variable and without applying nonlinear transformations.

𝑡

𝑥𝑎 (𝜏)𝑑𝜏 = −𝝀𝑇 Ξ

Assume that the following conditions hold

with 𝑯(𝒙𝒖 ) = 𝑴 𝟐 (𝒙𝒖 )−1 𝒈𝒓𝒖 (𝒙𝒖 ), for all 𝒙 ∈ . Then, the origin of system (19), i.e. [𝒙𝒖 , 𝒙𝒗 ]𝑇 = 𝟎, is Asymptotically Stable.

3.2.3. Discussion of the sliding variable design For Type II systems the nominal controller and the sliding-mode dynamics are the same for the sliding variables with relative degree one and two. This fact allows to compare controllers in a more systematic way in contrast to the case for Type I systems, where the sliding-mode dynamics are different. Additionally, if the asymptotic stability of 𝑥𝑎 and 𝑥𝑏 is achieved, from (22), it is possible to see that the relative degree two scheme actually imposes the integral relation 𝑡

(25)

with

and notice that by taking the first time derivative of (22), the sliding variable (18) is recovered; thus, this variable has a relative degree two with respect to the control input 𝜏. Moreover, if the robust controller to be introduced enforces a higher order of the sliding-mode, meaning that at least one more time derivate of 𝑠 is taken to zero; then, by choosing the parameters of (22) according to Lemma 1 and Theorem 5, the stability of the origin of (3) is ensured. Thus, if the nominal control law (20) is applied to system (3), the following closed-loop system is obtained [ ] [ ] [ ] 𝑠̇ 𝑠̇ 𝑑 𝑠 [ ] = = . (23) 𝑇 𝑣 + 𝜆𝑎 𝑔𝑎 (𝒙𝒖 , 𝑥𝑎 ) + 𝝀 𝒈𝒖 (𝒙𝒖 , 𝑥𝑎 ) 𝑑𝑥 𝑣 + 𝛿𝑥 𝑑𝑡 𝑠̇

𝜆 𝑎 𝜎𝑎

(24)

with 𝜆𝑎 ∈ R and 𝝀 ∈ R𝑛 . Since the sliding variable depends only on the position variables, it can be shown that this variable has a relative degree two with respect to the control input. Notice that the stability of the zero dynamics is not easily studied. For this purpose, the following lemma is proposed.

𝒙𝒖 (𝜏)𝑑𝜏.

The proposed method for Type II systems can be described as augmenting the order of the system. Unlike the general procedure, where the added order is introduced as the time derivative of the control input (see, e.g. Lu et al., 2018), in the presented method the order is increased by considering the integral of the position variables as a new state.

3.4. Summary of the results Table 1 presents a summary of properties that the proposed sliding variables possess. Notice that for each type of system, the type of convergence is not compromised by the augmentation of the relative degree of the sliding variable and that the required information, i.e. the number of time derivatives, is lowered. Regarding the number of parameters to be tuned, one can see that for Type I systems such a number depends on the relative degree while for Type II systems one has that this number is the same for both cases. On the other hand, Type III systems seem to require minimum information while maintaining a low number of parameters to be tuned.

3.3. Design for Type III systems Consider the system (2) and assume that it fulfills the conditions given in Definition 3, i.e. this system belongs to Type III. Due to the restriction (i) it is known that this system has the same state space representation than Type II systems. 346

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Control Engineering Practice 90 (2019) 342–357

Table 1 Summary of the properties for each sliding variable.

this means that the task has been reduced to the robust stabilization of such a system by mean of a continuous control law 𝑣. In order to design the robust controller some possibilities include:

Sliding variable

Type of convergence

Information needed

Number of parameters

Type I, case 1 Relative degree 1

Exponential for 𝑥𝑢 Asymptotic for 𝑥𝑎

𝑥𝑢 , 𝑥𝑣 , 𝑥̇ 𝑣 , 𝑥̈ 𝑣

3

(a) Continuous Singular Terminal Sliding-Mode Controller (CST) (Fridman et al., 2015).

Type I, case 1 Relative degree 2

Exponential for 𝑥𝑢 Asymptotic for 𝑥𝑎

𝑥𝑢 , 𝑥𝑣 , 𝑥̇ 𝑣

2

𝜎 = 𝑘𝜎 ⌈𝑠⌋2∕3 + 𝑠,̇ 𝑣 = −𝑘1 ⌈𝜎⌋

1∕2

(32) (33)

+ 𝑣, ̄

Type I, case 2 Relative degree 1

Exponential for 𝑥𝑢 Asymptotic for 𝑥𝑎

𝑥𝑢 , 𝑥𝑣 , 𝑥̇ 𝑣

Type I, case 2 Relative degree 2

Exponential for 𝑥𝑢 Asymptotic for 𝑥𝑎

𝑥𝑢 , 𝑥𝑣

1

Type II Relative degree 1

Asymptotic

𝑥𝑢 , 𝑥𝑣 , 𝑥𝑎 , 𝑥𝑏

4

𝑘𝜎 > 0,

Type II Relative degree 2

Asymptotic

𝑥𝑢 , 𝑥𝑎

4

(b) Continuous Nonsingular Terminal Sliding-Mode Controller (CNST) (Kamal, Moreno, Chalanga, Bandyopadhyay, & Fridman, 2016).

Type III Relative degree 2

Asymptotic

𝑥𝑢 , 𝑥𝑎

4

𝜎 = 𝑠 + 𝑘𝜎 ⌈𝑠⌋ ̇ 3∕2 ,

2

𝑣̄̇ = −𝑘2 ⌈𝜎⌋0 . One possible selection of gains is: 1

𝑘1 = 1.5𝜂 2 ,

𝑣 = −𝑘1 ⌈𝜎⌋

(35)

+ 𝑣, ̄

(36) (37)

According to Kamal et al. (2016), one possible selection of the controller gains is

Once the stability of the sliding-mode dynamics has been guaranteed and a nominal controller has been introduced for every case, the design of the robust controllers will be discussed. Before introducing the controllers, let us introduce the following assumption over the disturbance term 𝛿𝑥 .

𝑘𝜎 = 7.7𝜂 −1∕2 , 𝑘1 = 7.5𝜂 2∕3 , 𝑘2 = 2𝜂. (c) Continuous Twisting Algorithm (CTA) (Torres-González, Sanchez, Fridman, & Moreno, 2017).

There exist a constant 𝜂 > 0 such that

| |𝑑 | | | 𝛿𝑥 (𝒙)| ≤ 𝜂, ∀𝒙 ∈ . | | 𝑑𝑡 | |

1∕3

𝑘2 = 1.1𝜂.

𝑣̄̇ = −𝑘2 ⌈𝜎⌋0 .

4. Robust controller design

Assumption 2.

(34)

(27)

𝑣 = −𝑘1 ⌈𝑠⌋1∕3 − 𝑘2 ⌈𝑠⌋ ̇ 1∕2 + 𝑣̄

(38)

𝑣̄̇ = −(𝑘3 ⌈𝑠⌋0 + 𝑘4 ⌈𝑠⌋ ̇ 0 ).

(39)

In Torres-González et al. (2017), some possible gains can be taken as

Remark 10. Assumption 2 requires the disturbance terms 𝛿𝑥 to be Lipschitz continuous.

𝑘1 = 7𝜂 2∕3 , 𝑘2 = 5𝜂 1∕2 , 𝑘3 = 2.3𝜂, 𝑘4 = 1.1𝜂. (d) Discontinuous Integral Controller (DIC) (Moreno, 2018).

4.1. Relative degree one 𝜎1 = 𝑠 + 𝑘2 ⌈𝑠⌋ ̇ 3∕2 , Notice that up to this point all of the mentioned control schemes for sliding variables with relative degree one have taken the closed-loop system to a perturbed integrator:

𝜎2 = 𝑠 + 𝑘4 ⌈𝑠⌋ ̇ 𝑣 = −𝑘1 ⌈𝜎1 ⌋

(29)

𝑣̄̇ = −𝑘2 ⌈𝑠⌋0 , 𝛾

(41)

+ 𝑣̄

(42) (43)

(30) Theorem 8. Let the system (31) be controlled by one of the continuous controllers (32)–(43); then, 𝑠 = 0 is Finite-Time Stable.

𝛾

where ⌈𝑎⌋ ∶= |𝑎| sign(𝑎) for any 𝑎 ∈ R and any 𝛾 ∈ R≥0 , and some positive constants 𝑘1 and 𝑘2 . In order to proof the stability of the origin of system (28), the following theorem is recalled.

Remark 11. The convergence proofs of the given continuous slidingmodes controllers can be found in the corresponding references.

Theorem 7 (Seeber & Horn, 2017). Let the controller (29) and (30) be applied to system (28), and Assumption 2 hold. If the controller gains are designed as

4.3. Discussion of the robust controller design

1

𝑘1 = 1.5𝜂 2 ,

,

In Moreno (2018) the following gain selection procedure is proposed 𝜂 𝑘2 > 0, 𝑘4 ∈ R, 𝑘3 = , 𝑘1 > 0, 𝑙 for some 0 < 𝑙 < 1.

In order to robustly stabilize the perturbed integrator (28), by means of a continuous control signal, the Super-Twisting (ST) algorithm (Levant, 1993) is used. This algorithm has the form 𝑣 = −𝑘1 ⌈𝑠⌋1∕2 + 𝑣, ̄

(40)

𝑣̄̇ = −𝑘3 ⌈𝜎2 ⌋0 .

(28)

𝑠̇ = 𝑣 + 𝛿𝑥 .

1∕3

3∕2

All of the controllers presented in this section are robust against Lipschitz continuous (LC) perturbations, provide a continuous control effort and ensure that the trajectories of the system reach the set 𝑠 = 0 in a finite time. The main differences between the controllers are the required information for their implementation, design complexity and the obtained order of the sliding-mode. Notice that for the ST controller only two gains need to be tuned and that the control law depends only on the sliding variable by enforcing a second-order sliding-mode, while the controllers for relative degree two require the sliding variables and its derivative, have at least

𝑘2 = 1.1𝜂,

then, 𝑠 = 0 is Finite-Time Stable. 4.2. Relative degree two The mentioned control schemes for sliding variables with relative degree two have rendered the closed-loop system to [ ] [ ] 𝑠̇ 𝑑 𝑠 = , (31) 𝑣 + 𝛿𝑥 𝑑𝑡 𝑠̇ 347

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three gains to be tuned and enforce a third-order sliding-mode, which implies a better precision with respect to the output (Levant, 1993). The difference in required information actually seems to imply that the ST controller has an advantage over the schemes for relative degree two. Nevertheless, as can be seen in Table 1, the relative degree one schemes require more information for computing the sliding variable. Thus, from a theoretical point of view, the differences between the different proposed schemes do not seem to be significant. Note that the structure of the CST and CNST are very similar to the ST algorithm, although such controllers require the calculation of an auxiliary variable 𝜎 in order to be computed. The CTA avoids the calculation of this signal, but the discontinuous part has a bigger upper bound. This implies that the correction terms will be more aggressive and, if chattering is induced, the oscillations would be more considerable. The DIC can be seen as an ST-based controller. The fact that 𝑘4 can be selected arbitrarily means that the computation could be made less complex in certain situations, e.g. when 𝑘4 = 0, and the discontinuous part of the algorithm is bounded by only one term. The main drawback of such an algorithm is that, when 𝜂 is unknown, the tuning of such a controller is significantly harder. Moreover, since the procedure is not systematic, the application of such a controller becomes more difficult. Table 2 shows the main theoretical properties of the control schemes.

Fig. 2. Schematic representation of the 3 DOF helicopter.

The dynamics of such a system is given by: 𝜖̈ = 𝜙1 + 𝜙2 𝑢𝑠 , 𝜃̈ = 𝜙4 sin(𝜌), 𝜌̈ = 𝜙3 (𝜏 + 𝑑𝑥 ), where 𝜖 represents the elevation angle measured over the 𝑥 axis, 𝜌 is the pitch angle measured over the 𝑧 axis, 𝜃 is the travel angle measured over the 𝑦 axis, 𝑢𝑠 is the vertical force applied by the propellers, 𝜏 represent the torque applied over the 𝑧 axis and 𝜙1 , 𝜙2 , 𝜙3 and 𝜙4 are positive constants that depend on the inertial parameters of the system. This system is composed of two main subsystems: the elevation dynamics which is decoupled and fully actuated and as such will not be considered, and the 𝜌 and 𝜃 dynamics; this dynamics can be rewritten in the form (1) with: [ ] 1 1 ̇ = 0, 𝒈(𝒒) = [sin(𝑞𝑎 ) 0]𝑇 , 𝑴(𝒒) = diag , , 𝑪(𝒒, 𝒒) 𝜙4 𝜙3

5. Simulation results In this section a number of simulations are carried out in order to analyze the behavior of the relative degree one and two approaches with some particular control algorithms, i.e. ST and CST. In order to do so, the CST sliding-mode controller is considered for the relative degree two schemes. This controller has been chosen due to its similarities with the ST controller. Notice that if 𝑘𝜎 = 0, then an ST algorithm with respect to 𝑠̇ is recovered. Recall that there does not exist a sliding variable with relative degree one for Type III systems. Then, for this type, the objective is to analyze the effect of the parameter 𝑘𝜎 on the system response. The simulation results show a comparison for Type I and III systems. The comparison for a Type II system will be carried out in the next section. All of the simulations are carried out considering the Euler numerical method with a sampling time equal to 0.001[s]. The simulations present the responses to different values of 𝑘𝜎 modifying the convergence time of s. In order to provide some quantitative measure for the performance of the controllers, the RMS value for the position tracking error is considered. This RMS value is computed according to the expression: √ 𝑡 1 [𝒒 (𝑢)𝑇 𝑞𝑎 (𝑢)][𝒒 𝒖 (𝑢)𝑇 𝑞𝑎 (𝑢)]𝑇 𝑑𝑢, 𝑥𝑅𝑀𝑆 (𝑡) = ∫ 𝛥𝑇 𝑡−𝛥𝑇 𝒖 √ 𝜏𝑅𝑀𝑆 (𝑡) =

with 𝑞𝑎 = 𝜌 and 𝑞𝑢 = 𝜃. This system clearly fulfills the conditions for Assumption 1 for all 𝑞𝑎 ∈ (−𝜋, 𝜋) and for Definition 1 as well. Then such a system belongs to Type I. For the simulations, the following parameters and initial conditions were considered: 𝜙3 = 0.2431, 𝜙4 = −0.4975, 𝑞𝑎 (0) = 1 [rad], 𝑞𝑢 (0) = 0.5 [rad], 𝑞̇ 𝑎 (0) = 𝑞̇ 𝑢 (0) = 0 [rad/s]. For the relative degree one case, the robust controller (29) and (30), along with the nominal controller (7) and the sliding variable (5), is considered. On the other hand, for the relative degree two controller, the CST (32)–(34), along with the nominal controller (16) and the sliding variable (14), is considered. To test the robustness of the scheme, a perturbation signal, 𝑑𝑥 = sin(𝑡) + 0.1[V], is considered. In order to provide a fair comparison, all of the simulations use the same parameters for 𝑘1 and 𝑘2 , given by 𝑘1 = 1.06 and 𝑘2 = 0.55, which fulfill the stability conditions for the controllers and the parameter 𝑘𝜎 of (32) takes the values of 𝑘𝜎 = {5, 1, 0.1}. The sliding variables parameters were chosen specifically to have all of the poles of the characteristic polynomial in 𝑠 = −1. Fig. 3 shows the evolution for the position variables 𝑞𝑢 and 𝑞𝑎 from top to bottom. Notice that for a lower value of 𝑘𝜎 , the transient response for 𝑞𝑢 is slower. This figure shows that the precision added by the CST is evident once the sliding-mode is reached. However, if 𝑘𝜎 is low, a slow reaching-time is performed. For 𝑞𝑎 , a sufficiently large value of 𝑘𝜎 may render the response faster; however, some overshoot is present. Thus, the response of the position values show a trade-off between speed and accuracy of such a response. Fig. 4 shows the response of the sliding variable 𝑠 and the control input. The added precision for 𝑠 of the third order algorithm is evident. Moreover, a higher value for 𝑘𝜎 implies that, numerically, the trajectories of the system will be closer to the set 𝑠 = 0. However, a bigger value for 𝑘𝜎 implies a higher oscillation level of the control signal. Fig. 5 shows the RMS values for the position variables and control input. Notice that all of the RMS values for the position variables

𝑡

1 𝜏 2 (𝑢)𝑑𝑢, 𝛥𝑇 ∫𝑡−𝛥𝑇

which represents a measure of performance over the last 𝛥𝑇 seconds, i.e. the calculation is performed in a time window of width 𝛥𝑇 and any prior information is not considered. Throughout the simulations the term RD-1 makes reference to the relative degree one with ST controller and the term RD-2 means the relative degree two with CST controller. 5.1. Simulation for a Type I system Consider a three degrees of freedom helicopter (Pérez, Capello, Punta, Perea, & Fridman, 2018) (see Fig. 2). Such a system consists on a spherical joint to allow angular movement in three axis while anchoring the structure. 348

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Table 2 Summary of the theoretical properties of the robust control schemes. Property

ST

CST

CNST

CTA

DIC

Continuous control law Type of disturbance Type of convergence Needed information Number of parameters Order of the sliding-mode

yes LC Finite-time 𝑠 2 2

yes LC Finite-time 𝑠, 𝑠̇ 3 3

yes LC Finite-time 𝑠, 𝑠̇ 3 3

yes LC Finite-time 𝑠, 𝑠̇ 4 3

yes LC Finite-time 𝑠, 𝑠̇ 4 3

Fig. 3. Position variables for the Type I system considered.

Fig. 4. Sliding variable and control effort for the Type I system considered.

reach a similar value in steady state. This implies the robustness of the control schemes. However, as 𝑘𝜎 grows bigger, the rate of convergence is slower, implying an overall slower response for the system. For the control input, it is evident that this value will have very similar values in steady-state. However, a small difference on these results can be seen for the ST controller. This is due to the higher initial control action of the ST controller. Although the RMS value is smaller for the CST, Fig. 4 shows a higher level of oscillations for the control input. This means that a higher chattering effect is induced by numerical errors. Thus, the ST controller shows a faster transient response, which can be explained by the more active control activity. This result shows a tradeoff between a faster response and a more active control input for the ST

controller, while the 𝑘𝜎 parameter in the CST allows to better shape the system response to achieve a compromise between these characteristics. Table 3 shows the maximum, minimum and average value of 𝑥𝑅𝑀𝑆 and 𝑢𝑅𝑀𝑆 . The minimum values were computed for 𝑡 ≥ 𝛥𝑇 to eliminate the initial conditions of the integration. In this table, the previously mentioned trade-off, between speed of the response and control activity, can be seen. While the ST controller has a lower average for the RMS value of the position signals, the RMS value of the control input is higher. This behavior can be almost matched for the CST controller with 𝑘𝜎 = 5. Moreover, the simulation study shows that the value for 𝑘𝜎 can be lowered to achieve a lower control activity at the price of a slower response. 349

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Fig. 5. RMS values for the Type I system considered.

𝑞𝑢 (0) = 0.3491 [rad], 𝑞𝑎 (0) = 0.2618 [rad], 𝑞̇ 𝑢 (0) = 𝑞̇ 𝑎 (0) = 0 [rad/s]. For this system, the CST (32)–(34), along with the nominal controller (26) and the sliding variable (24) is considered. To test the robustness of the scheme, a perturbation signal, 𝑑𝑥 = sin(𝑡) + 1[Nm] is considered. In order to provide a fair comparison, all of the simulations use the same parameters for the control gains given by 𝑘1 = 9.75 and 𝑘2 = 14.3, which fulfill the stability conditions for the controller, and the parameter 𝑘𝜎 of (32) takes the values of 𝑘𝜎 = {5, 1, 0.1, 0.01}. The sliding variable parameters are designed according to the procedure described in Appendix A.9. These parameters are given by

Fig. 6. Schematic representation of the 2 DOF flexible joint manipulator.

Table 3 Quantitative simulation results for the Type I system.

RD-2, RD-2, RD-2, RD-1,

𝑘𝜎 = 5 𝑘𝜎 = 1 𝑘𝜎 = 0.1 𝑆𝑇

RD-2, RD-2, RD-2, RD-1,

𝑘𝜎 = 5 𝑘𝜎 = 1 𝑘𝜎 = 0.1 𝑆𝑇

max(𝑥𝑅𝑀𝑆 )

min(𝑥𝑅𝑀𝑆 )

Average of 𝑥𝑅𝑀𝑆

0.715 0.637 0.635 0.633

0.015 0.015 0.012 0.015

0.098 0.101 0.23 0.086

max(𝑢𝑅𝑀𝑆 )

min(𝑢𝑅𝑀𝑆 )

Average of 𝑢𝑅𝑀𝑆

4.462 3.647 3.625 4.898

0.665 0.568 0.563 0.706

1.195 1.067 1.064 1.271

𝜆𝑎 = 7.3607, 𝜆 = 3.8139, which fulfill the conditions of Lemma 2 and Theorem 6. Fig. 7 shows the evolution for the position variables 𝑞𝑢 and 𝑞𝑎 from top to bottom. Notice that, for 𝑞𝑎 , the effect of 𝑘𝜎 seems to be small, impacting only the precision of the steady-state response as 𝑘𝜎 grows bigger. However, for 𝑞𝑢 the impact of such a value is extremely important. For higher values of 𝑘𝜎 a faster response for 𝑞𝑢 is obtained, achieving a more oscillatory response. Fig. 8 shows the response of the sliding variable 𝑠 and the control input. As expected, the convergence time is highly impacted by the value of 𝑘𝜎 . However, it is in the control signal that a better conclusion can be drawn. Notice that the oscillations of such a signal are extremely high for the value of 𝑘𝜎 = 5. This is a sign of the appearance of chattering. Notice that a compromise between convergence time and alleviation of oscillations must be made. However, it is less straight forward between convergence time and level of oscillations, since a slower reaching time implies a higher initial overshoot from 𝑞𝑎 . Fig. 9 shows the RMS values for the position variables and control input. The results shown in this figure are very straight forward. It is obvious, from Fig. 7 that the RMS values are expected to decrease as 𝑘𝜎 increases and, since the control input adapts to the perturbation faster as this parameter grows, the control signal should reach a value corresponding to the disturbance faster. However, as previously stated, a compromise is still present between a faster response and a lower lever of oscillations. Table 4 shows the maximum, minimum and average value of 𝑥𝑅𝑀𝑆 and 𝑢𝑅𝑀𝑆 . The minimum values were computed for 𝑡 ≥ 𝛥𝑇 to eliminate the initial conditions of the integration. In this table a clear advantage of a higher value for 𝑘𝜎 is shown in terms of both accuracy and control activity. However, as can be seen in Fig. 8, an excessive value for such a parameter can cause oscillations in the control input, implying the need for caution when tuning the controller.

5.2. Simulation for a Type III system Consider a 2 DOF robot with a flexible joint (Fantoni & Lozano, 2002) (see Fig. 6). Such a system consists in a link that is driven by motor and a second link which is not actuated and is connected to the first link by means of a spring. This system can be seen as an approximation of a single flexible link robot (Fantoni & Lozano, 2002). Its dynamics is given by: [ ][ ] [ ] 𝜃2 𝜃2 + 𝜃3 cos(𝑞𝑢 ) 𝑞̈𝑢 𝑘𝑞𝑢 + 𝜃2 + 𝜃3 cos(𝑞𝑢 ) 𝜃1 + 𝜃2 + 2𝜃3 cos(𝑞𝑢 ) 𝑞̈𝑎 0 [ ][ ] [ ] 𝑓𝑣𝑢 𝜃3 sin(𝑞𝑢 )𝑞̇𝑎 𝑞̇ 𝑢 0 + = , −𝜃3 sin(𝑞𝑢 )𝑞̇ 𝑢 − 𝜃3 sin(𝑞𝑢 )𝑞̇ 𝑎 −𝜃3 sin(𝑞𝑢 )𝑞̇ 𝑢 + 𝑓𝑣𝑎 𝑞̇ 𝑎 𝜏 + 𝑑𝑥 where 𝑞𝑎 represents the angular position of the actuated link, 𝑞𝑢 is the position of the second link, 𝜏 represents the torque applied to the actuated link and 𝜃1 , 𝜃2 , 𝜃3 , 𝑓𝑣𝑢 , 𝑓𝑣𝑎 and 𝑘 are parameters which depend on the lengths and inertias of the links and the stiffness of the spring. This system clearly fulfills the conditions for Assumption 1 for all 𝑞𝑢 and for Definition 3 as well. Thus, this system belongs to Type III. For the simulations, the following parameters and initial conditions were considered: 𝜃1 = 0.0799, 𝜃2 = 0.0244, 𝜃3 = 0.0205, 𝑘 = 1, 𝑓𝑣𝑎 = 0.001, 𝑓𝑣𝑢 = 0.005 350

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Fig. 7. Position variables for the Type III system considered.

Fig. 8. Sliding variable and control effort for the Type III system considered.

Fig. 9. RMS values for the Type III system considered.

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Table 4 Quantitative simulation results for the Type III system.

RD-2, RD-2, RD-2, RD-2,

𝑘𝜎 𝑘𝜎 𝑘𝜎 𝑘𝜎

=5 =1 = 0.1 = 0.01

RD-2, RD-2, RD-2, RD-2,

𝑘𝜎 𝑘𝜎 𝑘𝜎 𝑘𝜎

=5 =1 = 0.1 = 0.1

Table 5 Quantitative experimental results for the Type II system.

max(𝑥𝑅𝑀𝑆 )

min(𝑥𝑅𝑀𝑆 )

Average of 𝑥𝑅𝑀𝑆

0.244 1.435 3.407 4.223

0.005 0.004 0.004 2.579

0.033 0.173 0.754 3.579

max(𝑢𝑅𝑀𝑆 )

min(𝑢𝑅𝑀𝑆 )

Average of 𝑢𝑅𝑀𝑆

0.008 0.031 0.034 0.035

0.004 0.004 0.004 0.005

0.006 0.008 0.01 0.01

RD-2, RD-2, RD-2, RD-1,

𝑘𝜎 = 4 𝑘𝜎 = 1 𝑘𝜎 = 0.1 𝑆𝑇

RD-2, RD-2, RD-2, RD-1,

𝑘𝜎 = 4 𝑘𝜎 = 1 𝑘𝜎 = 0.1 𝑆𝑇

max(𝑥𝑅𝑀𝑆 )

min(𝑥𝑅𝑀𝑆 )

Average of 𝑥𝑅𝑀𝑆

0.156 0.186 0.178 0.195

0.023 0.046 0.023 0.019

0.039 0.063 0.056 0.051

max(𝑢𝑅𝑀𝑆 )

min(𝑢𝑅𝑀𝑆 )

Average of 𝑢𝑅𝑀𝑆

26.025 24.435 25.372 26.204

15.759 16.624 15.0125 14.355

17.353 18.156 19.039 17.82

shown in this section are carried out in an output-feedback scheme by means of a homogeneous differentiator detailed in Cruz-Zavala and Moreno (2016). The stability of such a scheme for the ST controller has been studied in Ovalle et al. (2019). On the other hand, for the CST, a proof can be made based on finite-time input-to-state stability properties of the closed-loop system, as detailed in Ríos, Falcón, González, and Dzul (2019). The rail of the plant is just 1[m] from side to side. It is too short to implement an aggressive swing-up strategy. Thus, a discontinuous swing-up controller (Aström & Furuta, 2000) is used to lift the pendulum without crashing into the limits of the rail. In order to keep the results consistent, the swing-up controller is not shown, nor is it considered for the calculations for the RMS values. This means that the responses will have different initial conditions. Nonetheless the characteristics of each control scheme are notorious regardless of such differences. The considered disturbance signal is given by 𝑑𝑥 (𝑡) = 5.25 sin(3𝜋𝑡) + 1[N] and the variation of 𝑘𝜎 is given by 𝑘𝜎 = {0, 0.1, 1, 4}. Fig. 11 shows the position variables for each experiment. The initial overshoot of the CST is increased with respect to the ST. Nevertheless, the oscillations are significantly damped. This means that a more aggressive strategy is employed and that a faster behavior is observed. Fig. 12 shows the response of the sliding variable 𝑠 and the control input. In such a figure, a better accuracy with respect to the output can be seen for the ST algorithm. This fact may appear counterintuitive at first. However, since the CST controller appears to be more aggressive than the ST algorithm, a higher amplitude of some oscillations is expected. Moreover, the control input shows a more oscillatory response for the CST as 𝑘𝜎 grows bigger. Fig. 13 shows the RMS values for the position variables and control input. Notice that although the ST algorithm seems to rival the performance of the CST controller in the steady state, the latter has a far superior transient response, provided a big enough value of 𝑘𝜎 is considered. The differences in the RMS values for the control input actually show the fact that the proposed second order controller achieves a faster cancellation of the disturbance by reaching the steady state quicker. Notice that, for 𝑘𝜎 = 1, the CST controller appears to degrade in performance. This is mainly because the cart is unable to reach the zero position due to the high dry friction coefficient, however, as can be seen in Fig. 11, this difference is less than 3 [cm], which is not very significant. Table 5 shows the maximum, minimum and average value of 𝑥𝑅𝑀𝑆 and 𝑢𝑅𝑀𝑆 . The minimum values were computed for 𝑡 ≥ 𝛥𝑇 to eliminate the initial conditions of the integration. The results show that a bigger value for 𝑘𝜎 yields a better overall performance. However, as can be seen in Fig. 12, such an improvement comes at the cost of a higher level of oscillations, and that the choice of 𝑘𝜎 is critical, in an experimental setting, for a good performance. If such a value is too low, the CST controller cannot provide a performance that matches the ST controller. However, as this value grows, the parasitic dynamics may cause an undesirable level of oscillations.

Fig. 10. Cart–pole system to be controlled.

6. Experimental results In this section, some experimental results are shown for a Type II system. To this aim, consider a cart–pole system (Ovalle, Ríos, & Llama, 2019) (see Fig. 10). Such a system consists on a freely moving pendulum mounted on top of a cart that can move in one axis. The system dynamics is given by: ( )] [ ] [ ( ) ] [ [ ] 𝛼 𝛽 cos 𝑞𝑢 𝑞̈𝑢 𝑓𝑣𝜃 𝑞̇ 𝑢 − 𝜂 sin 𝑞𝑢 0 ( ) ( ) 2 = + , (44) 𝛽 cos 𝑞𝑢 𝛾 𝑞̈𝑎 𝑓𝑣𝑥 𝑞̇ 𝑎 − 𝛽 sin 𝑞𝑢 𝑞̇ 𝑢 𝜏 + 𝑑𝑥 where 𝑞𝑎 represents the cart displacement, 𝑞𝑢 is the angular position of the pendulum, 𝜏 represents the linear force applied to the cart and 𝛼, 𝛽, 𝛾, 𝜂, 𝑓𝑣𝜃 and 𝑓𝑣𝑥 represent the parameters of such dynamics. This system clearly fulfills the conditions for Assumption 1 for all 𝑞𝑢 ∈ (−𝜋, 𝜋) and for Definition 2 as well and so, this system belongs to Type II. The following parameters are considered for the experimental results: 𝛼 = 0.052, 𝛽 = 0.1115, 𝛾 = 2.608, 𝜂 = 1.094, 𝑓𝑣𝜃 = 0.03, 𝑓𝑣𝑥 = 6.33. For the relative degree one case, the robust controller (29) and (30), along with the nominal controller (20) and the sliding variable (18), is considered. For the relative degree two controller, the CST (32)–(34), along with the nominal controller (20) and the sliding variable (22), is considered. The procedure to find the appropriate sliding variable parameters is given in Appendix A.10. Such parameters are given by: 𝜆 = 1, 𝜆𝑎 = 0.7363, 𝜎𝑎 = 2.6877, 𝛯 = 5.3753. Note that, for systems with high friction coefficients, the application of continuous sliding-mode algorithms is a challenging task. This is due to the effects of dry friction. Such effects require a very high set of gains in order to be compensated. In this context, as stated in Llama, De La Torre, Jurado, and Garcia-Hernandez (2015) and Ovalle et al. (2019), the presented application becomes very meaningful due to the high friction of the system. This platform is only equipped with position sensors, meaning that the velocities should be numerically estimated. To this aim, the results 352

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Fig. 11. Experimental results for the position variables.

Fig. 12. Experimental results for the sliding variable and control effort.

Fig. 13. Experimental results for the RMS values.

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7. Conclusion

A.3. Proof of Theorem 3

In this paper, some sliding-mode controllers were presented to robustly stabilize the origin of three different types of underactuated systems without the use of nonlinear transformations. Moreover, different sliding variables were proposed with relative degree one and two. The theoretical differences of the proposed approaches were discussed and their advantages and shortcomings were further investigated by means of simulations. Some experimental results depicted the differences of the approaches from the application point of view. The presented analysis has shown that there do not exist significant theoretical differences of the controllers for systems with relative degree one and two. Therefore, the differences are based on practical considerations such as actuator dynamics and bounds of the disturbances. Moreover, due to the trade-off between the speed of the response and the needed control activity shown by the simulation and experimental results, the advantages and shortcomings in terms of transient response and control activity of each technique should also be considered when implementing a regulator.

Since a sliding-mode is enforced, then it follows from (14) that the dynamics of (4) collapses to 𝑧3 = −𝑐1 𝑧1 − 𝑐2 𝑧2 . This equation represents a linear dynamics which can be rewritten as 𝑧̇ = 𝐴𝑛 𝑧 with 𝑧 = [𝑧1 , 𝑧2 , 𝑧3 ]𝑇 and 𝐴𝑛 given in (15). Thus, if 𝐴𝑛 is Hurwitz, from the definition of 𝑧1 and 𝑧2 it follows that 𝑥𝑢 = 0 is exponentially stable. As stated by the constraint (iv) of Definition 1, the imposed restriction on the structure of system (4) implies that 𝑥𝑎 → 0 whenever 𝑥𝑢 = 𝑥̇ 𝑢 = 0; therefore, one may conclude the asymptotic stability of 𝑥𝑎 = 0. This concludes the proof. A.4. Proof of Theorem 4 Assume that a sliding-mode is enforced, then it follows from (17) that the dynamics of (4) collapses to 𝑧2 = −𝑐1 𝑧1 This equation represents a linear dynamics which is stable for any 𝑐1 > 0. Thus, from the definition of 𝑧1 it follows that 𝑥𝑢 = 0 is exponentially stable. As stated by the constraint (iv) of Definition 1, the imposed restriction on the structure of system (4) implies that 𝑥𝑎 → 0 whenever 𝑥𝑢 = 𝑥̇ 𝑢 = 0; therefore, one may conclude the asymptotic stability of 𝑥𝑎 = 0. This concludes the proof.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

A.5. Proof of Lemma 1

Acknowledgments

Let us assume that a sliding-mode takes place in the sliding manifold after a time 𝑡𝑟 , i.e. 𝑠(𝑡) = 0, ∀ 𝑡 ≥ 𝑡𝑟 . This implies that ( ) 𝑇 𝑥𝑏 = −𝜎𝑎 𝑥𝑎 − 𝜆−1 Ξ𝒙𝒖 + 𝒙𝒗 , (45) 𝑎 𝝀

The authors gratefully acknowledge the financial support from Tecnológico Nacional de México research projects. L. Ovalle and H. Ríos also acknowledge the financial support from CONACYT, Mexico 591548 and 270504, respectively.

and thus ( ) 𝑇 𝑥̇ 𝑏 = −𝜎𝑎 𝑥𝑏 − 𝜆−1 Ξ𝒙𝒗 + 𝒙̇ 𝒗 . 𝑎 𝝀

Appendix

(46)

Substituting (45) in (46), one gets A.1. Proof of Theorem 1

𝑇 𝑇 𝑥̇ 𝑏 = 𝜎𝑎2 𝑥𝑎 + 𝜆−1 𝑎 (𝝀 𝜎𝑎 Ξ𝒙𝒖 + 𝝀 (𝜎𝑎 𝑰 𝒏 − Ξ𝒙𝒗 ) − 𝒙̇ 𝒗 ).

Since a sliding-mode is enforced, then it follows from (5) that the dynamics of (4) collapses to

The solution to (45) is given by [( ] ) −𝜎 (𝑡−𝑡 ) 𝑇 𝑎 𝑟 − 𝜆−1 𝝀𝑇 𝒙 (𝑡) 𝑥𝑎 (𝑡) = 𝑥𝑎 (𝑡𝑟 ) + 𝜆−1 𝒖 𝑎 𝝀 𝒙𝒖 (𝑡𝑟 ) 𝑒 𝑎

(47)

𝑡

𝑧4 = −𝑐1 𝑧1 − 𝑐2 𝑧2 − 𝑐3 𝑧3 .

𝑇 +𝜆−1 𝑎 𝝀 (𝜎𝑎 𝑰 𝒏 − Ξ)

This equation represents a linear dynamics which can be rewritten as 𝑧̇ = 𝐴𝑛 𝑧 with 𝑧 = [𝑧1 , 𝑧2 , 𝑧3 ]𝑇 and 𝐴𝑛 given in (6). Thus, if 𝐴𝑛 is Hurwitz, from the definition of 𝑧1 , 𝑧2 and 𝑧3 it follows that 𝑥𝑢 = 0 is exponentially stable. As stated by the constraint (iv) of Definition 1, the imposed restriction on the structure of system (4) implies that 𝑥𝑎 → 0 whenever 𝑥𝑢 = 𝑥̇ 𝑢 = 0; therefore, one may conclude the asymptotic stability of 𝑥𝑎 = 0. This concludes the proof.

∫𝑡𝑟

𝒙𝒖 (𝜏)𝑒−𝜎𝑎 (𝑡−𝜏) 𝑑𝜏.

Then, substituting 𝑥𝑎 in (47), one obtains: ] [ ( ) 𝑡 𝑇 −𝜎𝑎 (𝑡−𝜏) ̇ 𝑥̇ 𝑏 = −𝜆−1 𝝀 𝒙 + 𝜎 𝑩 𝒙 − 𝜎 𝑒 𝒙 (𝜏)𝑑𝜏 𝒗 𝑎 𝒖 𝑎 𝒖 𝑎 ∫ 𝑡𝑟 [ ] −𝜎 (𝑡−𝑡 ) 2 −1 𝑇 𝑇 +𝜎𝑎 𝑥𝑎 (𝑡𝑟 ) + 𝜆𝑎 𝝀 𝒙𝒖 (𝑡𝑟 ) 𝑒 𝑎 𝑟 + 𝜆−1 𝑎 𝝀 𝑩𝒙𝒗 , where 𝑩 = 𝜎𝑎 𝑰 𝒏 − Ξ. Define

A.2. Proof of Theorem 2

𝑇 −1 𝑇 𝝍 𝟏 = −𝜆−1 𝑎 𝜆 , 𝝍 𝟐 = 𝜆𝑎 𝝀 𝑩, 𝑇 𝝍 𝟑 (𝒙𝒖 , 𝒙𝒗 ) = −𝜆−1 𝑎 𝝀 𝜎𝑎 𝑩𝒂(𝒙𝒖 , 𝑡𝑟 , 𝑡),

Since a sliding-mode is enforced, then it follows from (10) that the dynamics of (4) collapses to

𝑇 −𝜎𝑎 (𝑡−𝑡𝑟 ) 𝝍 𝟒 (𝑡) = 𝜎𝑎2 (𝑥𝑎 (𝑡𝑟 ) + 𝜆−1 𝑎 𝝀 𝑥𝑢 (𝑡𝑟 ))𝑒 𝑡

𝑧3 = −𝑐1 𝑧1 − 𝑐2 𝑧2 .

𝒂(𝒙𝒖 , 𝑡𝑟 , 𝑡) = 𝒙𝒖 − 𝜎𝑎

This equation represents a linear dynamics which can be rewritten as 𝑧̇ = 𝐴𝑛 𝑧 with 𝑧 = [𝑧1 , 𝑧2 , 𝑧3 ]𝑇 and 𝐴𝑛 given in (11). Thus, if 𝐴𝑛 is Hurwitz, from the definition of 𝑧1 and 𝑧2 it follows As stated by the constraint (iv) of Definition 1, the imposed restriction on the structure of system (4) implies that 𝑥𝑎 → 0 whenever 𝑥𝑢 = 𝑥̇ 𝑢 = 0; therefore, one may conclude the asymptotic stability of 𝑥𝑎 = 0. This concludes the proof.

∫𝑡𝑟

𝑒−𝜎𝑎 (𝑡−𝜏) 𝒙𝒖 (𝜏)𝑑𝜏.

Then, it follows that 𝑥̇ 𝑏 = 𝝍 𝟏 𝒙̇ 𝒗 + 𝝍 𝟐 𝒙𝒗 + 𝝍 𝟑 (𝒙𝒖 , 𝑡) + 𝝍 𝟒 (𝑡),

(48)

substituting (48) in the general equation for the system dynamics, (2), the equation for 𝑞̈𝑢 yields 𝑴 𝟐 (𝒙𝒖 )𝒙̇ 𝒗 + 𝑪 𝟐 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 + 𝒈𝟐 (𝒙𝒖 , 𝑡) + 𝒉𝟏 (𝒙𝒖 , 𝑡) = 𝟎, 354

(49)

L.R. Ovalle, H. Ríos and M.A. Llama

Control Engineering Practice 90 (2019) 342–357

A.7. Proof of Lemma 2

with 𝑴 𝟐 (𝒙𝒖 ) = 𝑴 𝒖𝒖 (𝒙𝒖 ) + 𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟏 , 𝑪 𝟐 (𝒙𝒖 , 𝒙𝒗 ) = 𝑪 𝒖𝒖 (𝒙𝒖 , 𝒙𝒗 ) + 𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟐 , 𝒈𝟐 (𝒙𝒖 , 𝑡) = 𝒈𝒓𝒖 (𝒙𝒖 )+𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟑 (𝒙𝒖 , 𝑡) and 𝒉𝟏 (𝒙𝒖 , 𝑡)𝑴 𝒖𝒂 𝝍 𝟒 (𝑡). Notice that, because 𝑴 𝒖𝒂 𝝍 𝟏 is a rank one matrix product, if its inverse exists, it can be computed by means of the Sherman–Morrison formula (Sherman & Morrison, 1950), i.e. 𝑴 𝟐 −1 (𝒙𝒖 ) = 𝑴 𝒖𝒖 −1 (𝒙𝒖 ) −

𝑴 𝒖𝒖 −1 (𝒙𝒖 )𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟏 𝑴 𝒖𝒖 −1 (𝒙𝒖 ) 1 + 𝝍 𝟏 𝑴 𝒖𝒖 −1 (𝒙𝒖 )𝑴 𝒖𝒂 (𝒙𝒖 )

Let us assume that a third order sliding-mode takes place in the sliding manifold 𝑠 = 0, i.e. 𝑠(𝑡) = 𝑠(𝑡) ̇ = 𝑠(𝑡) ̈ = 0, for all 𝑡 ≥ 𝑡𝑟 > 0. This implies that the following relations hold 𝑥𝑎 = 𝝀⋆ 𝒙𝒖 , 𝑥𝑏 = 𝝀⋆ 𝒙𝒗 , 𝑥̇ 𝑏 = 𝝀⋆ 𝒙̇ 𝒗 .

.

Then, the first equation of (2) can be rewritten as

Due to boundedness properties of mechanical systems, 𝑀𝑢𝑢 (𝑥𝑢 ) is positive definite, at least locally, which implies that 𝑀2 (𝑥𝑢 ) is not singular if and only if 𝝀𝑇 𝑴 𝒖𝒖 −1 𝑴 𝒖𝒂 ≠ 𝜆𝑎 . Because of this, one may write (49) as: [ ] 𝒙̇ 𝒗 = − 𝑪 𝟑 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 + 𝒈𝟑 (𝒙𝒖 , 𝑡) + 𝒉𝟐 (𝒙𝒖 , 𝑡) ,

𝑴 𝒖𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝒙̇ 𝒗 + 𝑴 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝝀⋆ 𝒙̇ 𝒗 + 𝑪 𝒖𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 , 𝒙𝒗 , 𝝀⋆ 𝒙𝒗 )𝒙𝒗 + 𝑪 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 , 𝒙𝒗 , 𝝀⋆ 𝒙𝒗 )𝝀⋆ 𝒙𝒗 + 𝒈𝒓𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ) = 𝟎. By the definition of 𝑴 𝟐 (𝑥𝑢 ), it follows that 𝑴 𝟐 (𝒙𝒖 )𝒙̇ 𝒗 + 𝑪 𝒖𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 , 𝒙𝒗 , 𝝀⋆ 𝒙𝒗 )𝒙𝒗 + 𝒈𝒓𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ) + 𝐶𝑢𝑎 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 , 𝒙𝒗 , 𝝀⋆ 𝒙𝒗 )𝝀⋆ 𝒙𝒗 = 𝟎,

which concludes the proof.

which can be rewritten as (25). Since 𝑀2 = 𝑴 𝒖𝒖 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ) + 𝑴 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝝀⋆ , its inverse can be computed by means of the Sherman– Morrison formula (Sherman & Morrison, 1950), i.e.

A.6. Proof of Theorem 5 [ ] Notice that due to the restriction |[𝒈𝒖 ]𝑖 | ≫ | 𝑴 𝒖𝒂 (𝒙𝒖 )𝝍 𝟑 (𝒙𝒖 , 𝑡) 𝑖 |, (19) can be rewritten as [ ] 𝒙̇ 𝒗 = − 𝑪 𝟑 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 + 𝑴 𝟐 −1 (𝒙𝒖 )𝒈𝒓𝒖 (𝒙𝒖 ) + 𝒉𝟐 (𝒙𝒖 , 𝑡) ,

𝑴 𝟐 −1 (𝒙𝒖 ) = −

𝑴 𝒖𝒖 −1 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝑴 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝝀⋆ 𝑴 𝒖𝒖 −1 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )

+ 𝑴 𝒖𝒖

which resembles a vector nonautonomous Liénard equation (Tunç, 2013), with the exception of the time dependent term ℎ2 (𝑥𝑢 , 𝑡). Thus, inspired by Tunç (2013), the following Lyapunov candidate function is proposed

1 + 𝝀⋆ 𝑴 𝒖𝒖 −1 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝑴 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ) −1

(𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ),

which is nonsingular if the relation 𝝀𝑇 𝑴 𝒖𝒂 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 )𝑴 𝒖𝒖 −1 (𝒙𝒖 , 𝝀⋆ 𝒙𝒖 ) ≠ 𝜆𝑎 holds. This concludes the proof. A.8. Proof of Theorem 6

1

1 𝑯(𝜏𝒙𝒖 )𝑇 𝒙𝒖 𝑑𝜏. 𝑉 (𝑥𝑢 , 𝑥𝑣 ) = 𝒙𝒗 𝑇 𝒙𝒗 + ∫0 2

(50)

The derivative of 𝑉 is given by Notice that (25) is a vector Lié equation. Then, the following Lyapunov candidate function is proposed

Since sign([𝑯(𝜏𝒙𝒖 )]𝑖 ) = sign([𝑯(𝒙𝒖 )]𝑖 ) = sign([𝒙𝒖 ]𝑖 ) for all 𝜏 > 0, and { } 𝑯(𝟎) < 𝑯(𝒙𝒖 ), for all 𝒙𝒖 ∈ 𝑢 = 𝒙𝒖 ∈ R𝑛 ||[𝒙𝒖 𝑇 , 𝑥𝑎 ]𝑇 ∈  , it follows that the integrand of the second term is positive definite for all 𝑥𝑢 ∈ 𝑢 . Notice that the integral vanishes when 𝒙𝒖 = 𝟎 and is proportional to 𝒙𝒖 and 𝜏. Therefore (50) is positive semi-definite with respect to 𝒙𝒖 ∈ 𝑢 , implying the positive definiteness of (50) for all [𝒙𝒖 𝑇 , 𝒙𝒗 𝑇 ]𝑇 ∈ 𝑢 × R𝑛 . The derivative of 𝑉 is given by

1

𝑉 (𝑥𝑢 , 𝑥𝑣 ) =

(51)

≤ ‖𝒙𝒗 ‖‖𝒉𝟐 ‖, ≤ (1 + ‖𝒙𝒗 ‖2 )‖𝒉𝟐 ‖ ≤ (1 + 2𝑉 )‖𝑴 𝟐 −1 𝑴 𝒖𝒂 (𝒙𝒖 )‖‖𝝍 𝟒 , ‖ ≤ 𝜇𝜈𝑒−𝜎𝑎 𝑡 (1 + 2𝑉 ),

(53)

Since sign([𝑯(𝜏𝒙𝒖 )]𝑖 ) = sign([𝑯(𝒙 { 𝒖 )]𝑖 ) = sign([𝒙𝒖 ]𝑖 ) for all } 𝜏 > 0, and 𝑯(𝟎) < 𝑯(𝒙𝒖 ), for all 𝒙𝒖 ∈ 𝑢 = 𝒙𝒖 ∈ R𝑛 ||[𝒙𝒖 𝑇 , 𝑥𝑎 ]𝑇 ∈  , it follows that the integrand of the second term is positive definite for all 𝑥𝑢 ∈ 𝑢 . Notice that the integral vanishes when 𝒙𝒖 = 𝟎 and is proportional to 𝒙𝒖 and 𝜏. Therefore (50) is positive semi-definite with respect to 𝒙𝒖 ∈ 𝑢 , implying the positive definiteness of (50) for all [𝒙𝒖 𝑇 , 𝒙𝒗 𝑇 ]𝑇 ∈ 𝑢 × R𝑛 . The derivative of 𝑉 is given by

𝑉̇ = [𝑴 𝟐 −1 𝒈𝒓𝒖 (𝒙𝒖 )]𝑇 𝒙𝒗 − 𝒙𝒗 𝑇 (𝑪 𝟑 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 + 𝑴 𝟐 −1 𝒈𝒓𝒖 (𝒙𝒖 ) + 𝒉𝟐 (𝒙𝒖 , 𝑡)) = −𝒙𝒗 𝑇 𝑪 𝟑 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 − 𝒙𝒗 𝑇 𝒉𝟐 (𝒙𝒖 , 𝑡),

1 𝑇 𝒙 𝒙 + 𝑯(𝜏𝒙𝒖 )𝑇 𝒙𝒖 𝑑𝜏. 2 𝒗 𝒗 ∫0

𝑉̇ = [𝑴 𝟐 −1 𝒈𝒓𝒖 (𝒙𝒖 )]𝑇 𝒙𝒗 − 𝒙𝒗 𝑇 (𝑪 𝟐 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗 + 𝑴 𝟐 −1 𝒈𝒓𝒖 (𝒙𝒖 ))

(52)

= −𝒙𝒗 𝑇 𝑪 𝟐 (𝒙𝒖 , 𝒙𝒗 )𝒙𝒗

where 𝜇 = max∀𝒙𝒖 ∈𝑢 ‖𝑴 𝟐 −1 (𝒙𝒖 )𝑴 𝒖𝒂 (𝒙𝒖 )‖, 𝜆∗ ≥ 𝝀𝑇 𝒙𝒖 (0) and 𝜈 = ∗ 𝑚𝜎𝑎2 [𝑥𝑎 (0) + 𝜆−1 𝑎 𝜆 ], and the equivalence between the norms ‖𝒂‖ ≤ 𝑚‖𝒂‖∞ for all 𝒂 ∈ R𝑚 is used. Integrating both sides of (52), from the time the sliding-mode is reached, 𝑡𝑟 , up to 𝑡, it follows that: ( ) 𝑡 𝑒−𝜎𝑎 𝑠 |𝑡 −𝜎𝑎 𝑠 𝑉 (𝑡) ≤𝑉 (𝑡𝑟 ) + 𝜇𝜈 2𝑒 𝑉 (𝑠)𝑑𝑠 − | , ∫𝑡 𝜎𝑎 |𝑡𝑟

Since 𝑪 𝟐 is positive definite, the derivative of the Lyapunov function is negative semidefinite. Notice that the set for which the derivative of the Lyapunov function vanishes is given by 𝒙𝒗 = 𝟎. This implies, by means of the invariance principle (Khalil, 1996), that 𝒙𝒗 → 𝟎 as 𝑡 → ∞. Moreover, (19) converges asymptotically to 𝒙̇ 𝒗 = 𝑴 𝟐 −1 𝒈𝒓𝒖 (𝒙𝒖 ) as 𝑡 → ∞. Therefore, since all equilibria for the system are contained in 𝑢 , the origin of (19) is asymptotically stable.

𝑟

𝑡

𝑒−𝜎𝑎 𝑡𝑟 + 2𝜇𝜈𝑒−𝜎𝑎 𝑠 𝑉 (𝑠)𝑑𝑠. 𝑉 (𝑡) ≤𝑉 (𝑡𝑟 ) + ∫𝑡𝑟 𝜎𝑎

A.9. Design of the sliding variable parameters for the flexible joint manipulator

Then, applying the Gronwall–Bellman inequality (Khalil, 1996), one has that ( ) −2𝜇𝜈 𝑒−𝜎̄ 𝑎 𝑡 𝑒−𝜎̄ 𝑎 𝑡𝑟 𝑉 (𝑡) ≤ 𝑉 (𝑡𝑟 ) + , 𝑒 𝜎̄ 𝑎 𝜎̄ 𝑎

According to Lemma 2 𝑀2 (𝑥𝑢 ) can be computed as 𝑀2 (𝑥𝑢 ) = 𝜃2 − (𝜃2 + 𝜃3 cos(𝑥𝑢 ))

which implies boundness of the trajectories of the system (49). Then, since the trajectories are bounded, it follows that (51) is bounded, which means that 𝑉 is bounded and uniformly continuous and, by Barbalat’s lemma (Slotine & Li, 1991), it follows that 𝑉̇ → 0 as 𝑡 → ∞. Then, since 𝒉𝟐 is exponentially decreasing with respect to the time for bounded positions, it follows from (51), that 𝒙𝒗 → 0 as 𝑡 → ∞. Similarly, (19) converges asymptotically to 𝒙̇ 𝒗 = 𝑴 𝟐 −1 𝒈𝒓𝒖 (𝒙𝒖 ) as 𝑡 → ∞. Therefore, since all equilibria for the system are contained in 𝑢 , the origin of (19) is asymptotically stable.

𝜆 . 𝜆𝑎

𝜃 +𝜃 ̄ Thus, if 𝜆𝑎 = 𝑘̄ 2𝜃 3 , for some 𝑘̄ > 0, it follows that, for any 𝜆 < 𝑘, 2 𝑀2 (𝑥𝑢 ) is positive. According to Theorem 6 the variable 𝐶3 (𝑥𝑢 , 𝑥𝑣 ) takes the form

𝐶3 (𝑥𝑢 , 𝑥𝑣 ) = 𝑀2−1 (𝑓𝑣𝑢 + 𝜃3 sin(𝑥𝑢 )

𝜆2 𝑥𝑣 ). 𝜆2𝑎

Notice that the positiveness of 𝐶3 cannot be guaranteed for an unbounded value of 𝑥𝑣 ; therefore, an upper bound for 𝑥𝑣 , |𝑥𝑣 | < 𝑥𝑣𝑚 ∈ 355

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̄ it follows (0, ∞), is introduced. Moreover, since 𝑀2 (𝑥𝑢 ) > 0 for 𝜆 < 𝑘, 𝜆2 that 𝐶3 (𝑥𝑢 , 𝑥𝑣 ) > 0 ⇒ 𝑓𝑣𝑢 > 𝜃3 2 𝑥𝑣𝑚 . This implies, from the definition 𝜆𝑎 √ 𝑓𝑣𝑢 𝜃2 +𝜃3 of 𝜆𝑎 , that 𝜆 < 𝑘̄ . Thus, choosing 𝜃3 𝑥𝑣𝑚

( √ 𝜆 = 𝑘̄ min 1,

References Aguilar, L., Iriarte, R., & Fridman, L. (2013). Second order sliding mode tracking controller for inertia wheel pendulum. Journal of the Franklin Institute, 350(1), 92–106. Aström, K., & Furuta, K. (2000). Swinging up a pendulum by energy control. Automatica, 36(2), 287–295. Cruz-Zavala, E., & Moreno, J. (2016). Lyapunov functions for continuous and discontinuous differentiators. IFAC-PapersOnLine, 49(18), 660–665. Donaire, A., Romero, J., Ortega, R., Siciliano, B., & Crespo, M. (2017). Robust ida-pbc for underactuated mechanical systems subject to matched disturbances. International Journal of Robust and Nonlinear Control, 27(6), 1000–1016. Fantoni, I., & Lozano, R. (2002). Non-linear control for underactuated mechanical systems. Springer Science & Business Media. Fridman, L., Moreno, J. A., Bandyopadhyay, B., Kamal, S., & Chalanga, A. (2015). Continuous nested algorithms: The fifth generation of sliding mode controllers. In Recent advances in sliding modes: from control to intelligent mechatronics (pp. 5–35). Springer. Haghighi, R., & Pang, C. K. (2018). Robust concurrent attitude-position control of a swarm of underactuated nanosatellites. IEEE Transactions on Control Systems Technology, 26(1), 77–88. Kamal, S., Moreno, J. A., Chalanga, A., Bandyopadhyay, B., & Fridman, L. M. (2016). Continuous terminal sliding-mode controller. Automatica, 69, 308–314. Khalil, H. (1996). Nonlinear systms. Prentice-Hall. Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6), 1247–1263. Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, 76(9–10), 924–941. Llama, M., De La Torre, W., Jurado, F., & Garcia-Hernandez, R. (2015). Robust takagisugeno fuzzy dynamic regulator for trajectory tracking of a pendulum-cart system. Mathematical Problems in Engineering, 2015. Lu, B., Fang, Y., & Sun, N. (2018). Continuous sliding mode control strategy for a class of nonlinear underactuated systems. IEEE Transactions on Automatic Control, 63(10), 3471–3478. Merkin, D. (2012). Introduction to the theory of stability, vol. 24. Springer Science & Business Media. Moreno, J. A. (2018). Discontinuous integral control for systems with relative degree two. In New perspectives and applications of modern control theory (pp. 187–218). Springer. Moreno-Valenzuela, J., & Aguilar-Avelar, C. (2018). Motion control of underactuated mechanical systems. Springer. Olfati-Saber, R. (2000). CaScade normal forms for underactuated mechanical systems. In Proceedings of the 39th IEEE conference on decision and control (pp. 2162–2167). Olfati-Saber, R. (2002). Normal forms for underactuated mechanical systems with symmetry. IEEE Transactions on Automatic Control, 47(2), 305–308. Ovalle, L., Ríos, H., & Llama, M. (2019). Robust output-feedback control for the cartpole system: A coupled super-twisting sliding-mode approach. IET Control Theory & Applications, 13(2), 269–268. Park, B. S. (2017). A simple output-feedback control for trajectory tracking of underactuated surface vessels. Ocean Engineering, 143, 133–139. Park, M., & Chwa, D. (2009). Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method. IEEE Transactions on Industrial Electronics, 56(9), 3541–3555. Pérez, U., Capello, E., Punta, E., Perea, J., & Fridman, L. (2018). Fault detection and isolation for a 3-dof helicopter with sliding mode strategies. In 2018 15th international workshop on variable structure systems (VSS) (pp. 279–284). Pérez-Ventura, U., & Fridman, L. (2019). When is it reasonable to implement the discontinuous sliding-mode controllers instead of the continuous ones? frequency domain criteria. International Journal of Robust and Nonlinear Control, 29(3), 810–828. Ríos, H., Falcón, R., González, O. A., & Dzul, A. (2019). Continuous sliding-mode control strategies for quadrotor robust tracking: Real-time application. IEEE Transactions on Industrial Electronics, 66(2), 1264–1272. Rudra, S., Barai, R. K., & Maitra, M. (2014). Nonlinear state feedback controller design for underactuated mechanical system: A modified block backstepping approach. ISA Transactions, 53(2), 317–326. Seeber, R., & Horn, M. (2017). Stability proof for a well-established super-twisting parameter setting. Automatica, 84, 241–243. Sherman, J., & Morrison, W. J. (1950). Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. The Annals of Mathematical Statistics, 21(1), 124–127. Shtessel, Y., Edwards, C., Fridman, L., & Levant, A. (2014). Sliding mode control and observation. Springer. Slotine, J., & Li, W. (1991). Applied nonlinear control. NJ: Prentice hall Englewood Cliffs. Torres-González, V., Sanchez, T., Fridman, L. M., & Moreno, J. A. (2017). Design of continuous twisting algorithm. Automatica, 80, 119–126.

𝜃2

𝑓𝑣𝑢 𝜃2 + 𝜃3 𝜃3 𝑥𝑣𝑚 𝜃2

) ,

the positivity of 𝐶3 (𝑥𝑢 , 𝑥𝑣 ) is ensured. The variable 𝐻(𝑥𝑢 ) takes the form 𝐻(𝑥𝑢 ) =

𝑘𝑥𝑢 𝜃2 − (𝜃2 + 𝜃3 cos(𝑥𝑢 )) 𝜆𝜆

.

𝑎

Since the denominator of such a function is positive for the proposed values of 𝜆 and 𝜆𝑎 and then, it follows that sign(𝐻(𝑥𝑢 )) = sign(𝑥𝑢 ) and since the denominator decreases as |𝑥𝑢 | increases, it follows that 𝐻2 (𝑥𝑢 ) is strictly increasing for all |𝑥𝑢 | < 𝜋2 . Finally, if the values 𝑘̄ = 4 and 𝑥𝑣𝑚 = 0.9085 are chosen, the sliding variable parameters 𝜆𝑎 = 7.3607 and 𝜆 = 3.8139 fulfill the conditions of Lemma 2 and Theorem 6.

A.10. Design of the sliding variable parameters for the cart–pole system Le us consider 𝜆 = 1 for simplicity. From (44), it is possible to see that 𝑀𝑢𝑎 (𝑥𝑢 ) is zero whenever 𝑥𝑢 = ±𝜋∕2. Thus let us define some 0 < 𝜀 ≪ 𝜋∕2 and consider 𝜃 = 𝜋∕2 − 𝜀. Then, the set ̄ = {𝑥𝑢 ∈ R| |𝑥𝑢 | < 𝜃} is introduced. According to Lemma 1, 𝑀2 (𝑥𝑢 ) = 𝛼 − 𝜆−1 𝑎 𝛽 cos(𝑥𝑢 ). Thus, if 𝜆𝑎 = 𝛼𝛽 cos(𝜃), it follows that 𝑀2 (𝑥𝑢 ) is not singular for all ̄ |𝑥𝑢 | ∈ . To test the conditions of Theorem 5, consider the variable 𝐶3 (𝑥𝑢 , 𝑥𝑣 ) =

𝑓𝑣𝜃 + 𝜆−1 𝑎 𝛽 cos(𝑥𝑢 )(𝜎𝑎 − 𝛯) 𝛼 − 𝜆−1 𝑎 𝛽 cos(𝑥𝑢 )

. 𝑓

This variable is positive if the relation 𝛯 > 𝛼𝑣𝜃 + 𝜎𝑎 holds. Moreover, 𝑔𝑟𝑢 = −𝜂 sin(𝑥𝑢 ). Notice that the structure of the cart– pole system is such that 𝜂 = 𝛽𝑔𝑟 , with 𝑔𝑟 as the acceleration of gravity. From the definition of 𝜓3 , it is possible to see that the term 𝑎(𝑥𝑢 , 𝑡𝑟 , 𝑡) = 𝑡 𝑥𝑢 − 𝜎𝑎 ∫𝑡 𝑒−𝜎𝑎 (𝑡−𝜏) 𝑥𝑢 (𝜏)𝑑𝜏, represents the difference of 𝑥𝑢 and a low 𝑟 pass filtered version of the same signal with unit gain. Thus, it follows that |𝑎(𝑥𝑢 , 𝑡𝑟 , 𝑡)| ≤ |𝑥𝑢 |. Then, |𝑀𝑢 𝑎(𝑥𝑢 )𝜓3 | ≤ 𝜆𝑎 𝜎𝑎 𝐵|𝑥𝑢 |𝛽 cos(𝑥𝑢 ). On the other hand, |𝑔𝑟𝑢 (𝑥𝑢 )| = 𝛽𝑔𝑟 sin(|𝑥𝑢 |). One must ensure that 𝛽𝑔𝑟 sin(|𝑥𝑢 |) ≫ 𝜆−1 𝑎 𝜎𝑎 |𝐵| |𝑥𝑢 |𝛽 cos(𝑥𝑢 ). Since 𝑔𝑟 ≈ 10, it suffices to prove that sin(|𝑥𝑢 |) > 𝜆−1 𝑎 𝜎𝑎 |𝐵| |𝑥𝑢 | cos(𝑥𝑢 ), which is feasible as long as 𝜎𝑎 (𝛯 − 𝜎𝑎 ) < 𝜆𝑎 . Lastly, consider the term 𝐻(𝑥𝑢 ), which is given by 𝐻(𝑥𝑢 ) =

−𝛽𝑔𝑟 sin(𝑥𝑢 ) . 𝛼 − 𝛽𝜆𝑎 cos(𝑥𝑢 )

By the definition of 𝜆𝑎 and 𝜃, the denominator of 𝐻(𝑥𝑢 ) is negative ̄ Therefore, it follows that sign(𝐻(𝑥𝑢 )) = sign(sin(𝑥𝑢 )) = for all 𝑥𝑢 ∈ . ̄ Moreover, sign(𝑥𝑢 ) for all 𝑥𝑢 ∈ . 𝑑𝐻(𝑥𝑢 ) 𝛽𝑔 cos(𝜃) 1 − cos(𝑥𝑢 ) cos(𝑥𝑢 ) ̄ = 𝑟 > 0 ∀𝑥𝑢 ∈ . 𝑑𝑥𝑢 𝛼 (cos(𝑥𝑢 ) − cos(𝜃))2 ̄ This means that 𝐻(𝑥𝑢 ) is strictly increasing with respect to 𝑥𝑢 ∈ . Thus, considering a value of 𝜃 = 1.221, it is possible to show that the following choice of parameters fulfill the conditions of Lemma 1 and Theorem 5 𝜆 = 1, 𝜆𝑎 = 0.7363, 𝜎𝑎 = 2.6877, 𝛯 = 5.3753. 356

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