A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems

ISA Transactions xxx (xxxx) xxx

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Research article

A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems ∗

Mohamed Boukattaya , Hamdi Gassara, Tarak Damak Laboratory of Sciences and Techniques of Automatic control & computer engineering (Lab-STA), National School of Engineering of Sfax, University of Sfax, Postal Box 1173, 3038 Sfax, Tunisia

highlights • A novel time-varying global sliding-mode control is presented to eliminate the reaching phase and to ensure the global robustness of the system. • The tracking errors are proven to converge to zero within a pre-determined time, which can be specified by the designer according to mission requirement.

• An adaptive tuning law is proposed to eliminate the requirement of the prior knowledge of the upper bound of the lumped uncertainty. • Many practical situations are taken into account such as unmodeled dynamics and external disturbance, singularity avoidance, and input saturation.

article

info

Article history: Received 19 August 2018 Received in revised form 3 June 2019 Accepted 2 July 2019 Available online xxxx Keywords: Time-varying sliding-mode control Desired convergence time Reaching phase System uncertainties Robust control Adaptive control

a b s t r a c t In this paper, a global time-varying sliding-mode control scheme with prespecified convergence time is proposed for the tracking problem of a class of uncertain nonlinear systems under parameters uncertainties and external disturbances. Firstly, a novel time-varying sliding manifold with appropriate coefficients is presented. These coefficients are tuned to eliminate the reaching phase and to drive the system states to the equilibrium in a specified time. Hence, the system states are constrained to the sliding surface from the beginning of the motion which enables the global robustness, the reduction of the initial control effort, and the meet of the convergence time requirement. Moreover, in order to address the more practical case that the upper bound of the system uncertainties and disturbances is unavailable, an adaptive time-varying sliding mode control algorithm is derived, by which the tracking error vanish as time tends to infinity. The stability of the system has been proved by the Lyapunov stability theorem, and simulation studies are conducted to show the effectiveness of the suggested control schemes. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Trajectory tracking under disturbances and uncertainties is a challenging and important problem for nonlinear systems, especially robotic systems [1–3]. For this purpose, many advanced control algorithms have been widely applied to command such perturbed systems. Among these control approaches, slidingmode control is an important control procedure, due to its rapid response, acceptable transient performance, and insensitivity to external disturbances and variation of the parameters [4–20]. Thanks to these advantages, the sliding-mode technique has been widely used in many applications, including robotics [1– 5], electrical and mechanical systems [20,21], aeronautical, and ∗ Corresponding author. E-mail addresses: [email protected] (M. Boukattaya), [email protected] (H. Gassara), [email protected] (T. Damak).

aerospace engineering [22,23]. In conventional linear slidingmode (LSM) controller, only asymptotic stability is ensured which implies the asymptotic convergence of the system states to the origin. Obviously, for practical applications that require an extremely high precision, the convergence rate may be insufficient [4–6]. To realize the finite-time convergence for uncertain dynamical systems, Terminal Sliding-Mode (TSM) controller was developed [7–9]. In addition to rapid convergence, the TSM algorithm gives higher accuracy, better disturbance compensation, as well as its good robustness to uncertainties [9–13]. Although the TSM control has solved the finite time convergence, the singularity problem may occur when a negative fractional power exists in the control signals. Hence, Nonsingular Terminal Sliding-mode (NTSM) [14–19] and Nonsingular Fast Terminal Sliding-mode (NFTSM) [24] control algorithms were developed to overcome these drawbacks. It should be noted that the phase trajectory of the systems controlled by sliding-mode technique is composed of

https://doi.org/10.1016/j.isatra.2019.07.003 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

2

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two parts namely reaching phase and sliding phase. The robustness of the controller is assured only during the sliding phase [20–22]. Thus, when initial state is located far from the sliding surface, the control performances are seriously degraded because of the long duration of the reaching phase. One way is to speed up the reaching phase by high switching control gains [21]. However, this method will induce excessive control input at the beginning of the motion and may excite the undesired chattering problem. Another approach is to replace the time-invariant sliding surface by the time-varying sliding surface, which is known as the TimeVarying Sliding-Mode (TVSM) technique [23,25,26]. By using the TVSM approach, the reaching phase can be totally eliminated and the system states is guaranteed to be located in the sliding surface from the beginning of the trajectory, and consequently the global robustness of the system. Despite the efficiency of the above-mentioned works, they only realize the convergence in bounded finite-time and this time cannot be imposed in advance by the user. However, for real engineering applications that require high-precision and fast convergence, it is compulsory to drive the system states to the origin in a desired time whose value can be fixed according to the mission requirements. To the authors’ knowledge, there are only a few references dealing with the time specified sliding-mode control. In [27], a higher order sliding-mode control scheme for uncertain nonlinear systems is proposed. The control law is composed of ideal control part to stabilize the system states to the origin in the desired finite-time and integral sliding-mode control part to reject the uncertainties and disturbances. However, this algorithm is only designed for Single Input Single Output (SISO) systems. In [28], TVSM control algorithm with piecewise defined function is integrated into the sliding surface to eliminate the reaching phase and to ensure the finite-time convergence of system states to the origin. For the proposed controller, the convergence time is an explicit parameter which can be designed to meet the time requirement of the mission. In [29], nonsingular TVSM control scheme is proposed for rigid robot manipulators with parametric uncertainties and external disturbances. The reaching phase is removed to ensure the global robustness of the system. In addition, the convergence time is not only finite, but can be also specified explicitly by using a parameter selection method. A robust fault-tolerant controller for surface vessel is developed in [30] by integrating an adaptive neural network function into a time-varying sliding-mode control. By exploiting the proposed method, the tracking errors are proved to converge to the origin in bounded finite time, whose value can be assigned in advance according to the mission obligation. Recently, a time specified nonsingular TSM control scheme is proposed in [31] to resolve the problem of trajectory tracking for robotic airships. A pre-specified nonlinear function is proposed to handle the problems of singularity and time-depended of TSM. However, the reaching phase of this control law is not totally eliminated and hence the robustness in not ensured all the time. Although the effectiveness of the above mentioned terminal sliding-mode methods with desired finite time convergence, they suffer from two main problems. The first is that they use a highly complicated algorithm that involves the computation of several parameters to obtain the desired finite time convergence. Second, they need prior knowledge of the upper bounds of the system uncertainties, which is difficult to have in practical situations. Also, the actuator saturation was not taken into account which may be harmful to the actuators when a large variation in the control effort occurs. Inspired by the efficiency of the advanced guidance technique for homing missiles with impact time and angle constraint [32–35], a novel global adaptive time-varying sliding-mode control scheme with prespecified convergence time is proposed in this paper for the tracking problem of dynamical systems under unmodeled dynamics and time-varying external disturbances.

Compared to other approaches, the contributions of this work are: (1) A novel global time-varying sliding-mode control technique is proposed. The convergence time of the tracking error to zero can be set in advance in this research, while the methods in [23–30] only guarantee bounded convergence time; (2) In contrast to the existing approaches based on TSM [7–9], NTSM [14–19], or NFTSM [24] techniques, the proposed control law can eliminate the reaching phase so that the robustness is established during the whole response of the system, and consequently, faster convergence rate and smaller input command amplitude can be achieved; (3) To overcome the need of the prior knowledge of the upper bound of the system uncertainties, an adaptive approach is developed to estimate these parameters and then used in the controller so that the effects of disturbances and uncertainties can be eliminated and the asymptotical convergence can be guaranteed; (4) The control saturation due to the physical limit of the actuator is taken into account in this work which is convenient for practical implementation; (5) The proposed controllers were firstly designed for uncertain second-order nonlinear system and then adapted to control multi-link robotic manipulator. In the same way, the proposed approaches can be easily adapted to the control other practical systems such as underwater vehicle [18,19], vibratory gyroscope [20], piezoelectric actuators [21], rigid spacecraft [22,23], surface vessel [36], etc. The remainder of this paper is organized as follows: In Section 2, the design procedures of the proposed global TVSMC and the stability analysis are presented. First, it was derived for a generalized second-order nonlinear system and then adapted to multi-link robot manipulators. Numerical simulation results are described in Section 3. The research is concluded in Section 4. 2. Design of a time-varying sliding-mode control 2.1. Time-varying sliding-mode control of the generalized secondorder nonlinear system Consider the following generalized second-order nonlinear system described by the following canonical state-space form x˙ 1 (t ) = x2 (t ) x˙ 2 (t ) = f (x, t ) + b (x, t ) τ (t) + τd (t) y (t ) = x1 (t )

[

(1)

]T

where x = x1 x2 is the vector of the state variables of the system, f (x, t ) and b (x, t ) are nonlinear functions, y (t ) is the output signal, τd (t ) denotes the external disturbances and τ (t ) is the control input. In practice, it may be impossible to determine f (x, t ) and b (x, t ) exactly, hence f (x, t ) = f0 (x, t ) + ∆f (x, t ) b (x, t ) = b0 (x, t ) + ∆b (x, t )

(2)

where f0 (x, t ) and b0 (x, t ) are the known terms, ∆f (x, t ) and ∆b (x, t ) are the uncertain terms induced by unmodeled dynamics and structural variation of the system. According to (2), the dynamic equation (1) can be rewritten as x˙ 1 (t ) = x2 (t ) x˙ 2 (t ) = f0 (x, t ) + b0 (x, t ) τ (t) + d(x, t) y (t ) = x1 (t )

(3)

where d (x, t ) = τd (t ) + ∆f (x, t ) + ∆b (x, t ) τ (t) is called the compound disturbances. Here, it is assumed that the desired output yd (t) is a twice continuously differentiable function in term of t. Let us define the tracking error and its derivative as e1 (t) = y(t) − yd (t) e2 (t) = y˙ (t) − y˙ d (t)

(4)

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

M. Boukattaya, H. Gassara and T. Damak / ISA Transactions xxx (xxxx) xxx

Then, the system model (3) can be defined in the error-state space form as follows e˙ 1 (t ) = e2 (t ) e˙ 2 (t ) = f0 (x, t ) − y¨ d (t) + b0 (x, t ) τ (t ) + d (x, t )

(5)

In this work, the control objective is to design a global timevarying sliding mode control scheme for the second-order nonlinear system (3) such that: (i) The tracking error and its derivative will both converge to zero at the desired finite-time tf ; (ii) The desired finite-time convergence is ensured without any singularity and the robustness of the system is guaranteed from the beginning of the motion. This global TVSMC method will be used later to solve the trajectory tracking problem of a robotic manipulator. For this purpose, we consider the following lemma and assumptions: Lemma 1. Suppose the vector x =

∑n

i=1

x1 ,

[

|xi | is the absolute norm and ∥x∥ =

··· √∑ n

i=1

, xn

]T

and |x| =

x2i is the Euclidean

norm, the following inequality holds

∥x∥ ≤ |x|

2.1.1. Robust time-varying sliding-mode control (RTVSMC) In general, the time-varying sliding-mode approach is composed of two phases. The first phase is to choose an appropriate time-varying sliding surface to drive the states to the origin within a prespecified finite-time. The second phase is to design a time-varying sliding mode input torque that guarantees desired finite-time convergence, good robustness, high precision, and smooth input command. To begin with, a new time-varying sliding surface is designed as follows n e1 (t) − z (t ) (8) s(t) = e2 (t) + tf − t where tf is a prespecified finite-time convergence, and z (t ) is a time-varying piecewise function defined as z (t ) =

⎧ ( ) ⎨ A tf − t p p

for 0 ≤ t ≤ tf

tf

⎩

(9)

for t > tf

0

The parameters n, p and A are constants that will be determined later.

(6) Theorem 1. Assuming that the error-states are always constrained to the surface s(t) = 0 with p > n − 1 > 0. Then, e1 (t) and e2 (t) will both converge to zero at t = tf .

Proof. See [37]. Assumption 1. The matrix b0 (x, t) is invertible ∀(x, t). Assumption 2. The state variables of the system x1 and x2 are available for measurement.

Proof. By setting s(t) = 0, the following second-order differential equation is obtained as e˙ 1 (t) +

Assumption 3 ([24,38]). The norm of the compound disturbances d(x, t) is assumed to be bounded such that

∥d (x, t )∥ ≤ δ δ = a0 + a1 ∥x∥ + a2 ∥x∥2

3

n

e1 (t) −

tf − t

A (

p tf

tf − t

Assumption 4 ([22]). The velocity tracking error is assumed to be zero at t = 0, i.e. e2 (0) = 0. Remark 1. Assumption 3 indicates that the compound disturbance which is a function of the system states should be bounded. It is based on the consideration that in a most physical system, the velocity and acceleration are always limited by the motors. Hence, the control input may vary with time, but cannot be infinite. Though conservative, it can still cover many realistic applications, as cited in [39,40]. Therefore, Assumption 3 is reasonable and practical. We also note that Assumption 3 is designed based on the fact that we have considered the generalized form of the second-order nonlinear dynamical system. It is worth to notice that this hypothesis is not restrictive since it can still remain for many dynamical systems in literature, such as robotic systems [1–3], unmanned underwater vehicle [18,19], rigid spacecraft [22,23], unmanned aerial vehicle [31], autonomous surface vessel [36], etc. We will discuss later how to formulate an appropriate control algorithm for a multi-link robotic manipulators. Remark 2. According to [38], the∑ upper bound of the lumped unr i certainties can be written as δ = i=0 ai ∥x∥ where r is the order of the lumped uncertainty. However, for periodic disturbances, it is well represented by a constant value and hence we can choose r = 0. Obviously, the choice of r = 2 covers more area in the range space of the lumped uncertainty. So, it is quite realistic to consider δ = a0 + a1 ∥x∥ + a2 ∥x∥2 .

=0

(10)

The analytic solution for e1 (t) is obtained after solving the previous equation, it follows

(7)

where δ is the upper bound of the compound disturbances in which a0 , a1 and a2 are positive constants.

)p

(

e1 (t) = C tf − t

)n

−

A

(

p

tf (p − n + 1)

tf − t

)p+1

(11)

where C is a constant which can be set by the initial condition, thus C =

e1 (0) tfn

+

A p−n+1

tf1−n

(12)

Differentiating (11), the analytic solution of e2 (t) can be obtained as

(

e2 (t) = −Cn tf − t

)n−1

+

A(p + 1) p tf (p

− n + 1)

(

tf − t

)p

(13)

where p and n should be a positive integer, with n > 1 and p > n − 1, it can be concluded from Eqs. (11) and (13) that both e1 (t) and e2 (t) converge to zero as t goes to tf . From Theorem 1, once the error-states are constrained on the basic sliding surface from the start, the error-states can be driven to) zero at the desired finite-time tf . The term z (t ) = ( p A in the sliding surface (8) is called the forcing term in p tf − t tf

which the coefficient A is determined so that the sliding surface is constrained at zero from the beginning of the motion, i.e., at the initial time t = 0, s(t) = 0, it follows n n A = e2 (0) + e1 (0) = e1 (0) (14) tf tf After developing a suitable time-varying sliding surface, the next step is to determine an input signal to guarantee that the errorstates converge to the origin at a predetermined time tf even in the presence of external disturbances and parametric uncertainty. In general, the time-varying sliding approach is composed of an

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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equivalent control part to stabilize the nominal system when the sliding surface is established from the beginning of the process and a switching control part to keep the error-states staying on the sliding surface in the presence of disturbances and uncertainties. Thus, the equivalent control law can be obtained by setting

s˙(t) = e˙ 2 (t) + n

e2 (t) tf − t

+n

− z˙ (t ) = 0

(tf − t)2

(15)

Considering the nominal system without any disturbances and uncertainties, one can get f0 (x, t ) − y¨ d (t ) + b0 (x, t ) τ (t ) + n

e2 (t) tf − t

+n

e1 (t) (tf − t)2

− z˙ (t ) = 0 (16)

Then, the expression of the equivalent control law is obtained as 1 τeq (t ) = −b− (x, t ) (0 ) e2 (t) e1 (t) × f0 (x, t ) − y¨ d (t) + n +n − z˙ (t ) tf − t (tf − t)2

(17) Furthermore, the switching control law is designed as 1 τsw (t ) = −b− 0 (x, t ) (Ks + (δ + η) sign(s))

(18)

where K is a positive-definite diagonal gain, η > 0 is a positive scalar and δ = a0 + a1 ∥x∥ + a2 ∥x∥2 > 0 is the upper bound of the compound disturbances norm defined in Assumption 3. Thus, the overall input command may be designed as follows

1 = −b− 0 (x, t )

+n

e1 (t) (tf − t)2

f0 (x, t ) − y¨ d (t) + n

e2 (t) (19)

tf − t

)

− z˙ (t ) + Ks + (δ + η) sign(s)

From the above analysis, the result can be summarized in the following theorem: Theorem 2. Consider the uncertain second-order nonlinear system (3) where the compound disturbances satisfy the constraint (7). If the time-varying sliding surface is chosen as (8) and (9) in which the coefficient A is selected as (14) and the control input is designed as (19), then: (i) The tracking errors e1 (t) and e2 (t) will both converge to zero at the prespecified time tf . [(ii) The] global sliding-mode is accomplished, i.e. s ≡ 0 for t ∈ 0 tf . Proof. Consider the following Lyapunov function candidate: V =

1 2

s2

(20)

Differentiating V with respect to time and using (15) yields

(

V˙ = ss˙ = s e˙ 2 (t) + n

e2 (t) tf − t

+n

V˙ ≤ ∥d (x, t )∥ |s| − Ks2 − (δ + η) |s| = (∥d (x, t )∥ − δ) |s| − Ks2 − η |s|

(24)

Considering Assumption 3, we have V˙ ≤ −Ks2 − η |s|

e1 (t)

τ (t) = τeq (t) +(τsw (t)

Having in mind Lemma 1, it follows that

e1 (t) (tf − t)2

) − z˙ (t )

(25)

It is obvious that since K and η are positive gains, V˙ is nonpositive. Consequently, V is non-increasing and hence V (t) ≤ V (0). Also, one can find from the value of the coefficient A in Eq. (14) that s(0) = 0 which implies that V (0) = 21 s2 (0) = 0 and thus V (t) ≤ V (0) = 0. On the other hand, V (t) ≥ 0 is achieved according to Eq. (20). Consequently, it is concluded that V ≡ 0 [ ] gives s ≡ 0 for t ∈ 0 tf . Thus, the global sliding-mode is obtained. In addition, according to Theorem 1, the convergence of the tracking errors e1 (t) and e2 (t) is guaranteed when t tends to tf . Remark 3. The control input (19) is designed so that the states of the system converge to the origin, exactly at time t = tf . The control input will maintain this equilibrium state, even after the terminal time, i.e. t > tf , provided that the term z (t ) is eliminated at the terminal time. However, when z (t ) = 0 occurs according to the last interval of (9), the second sliding surface s0 (t) = e2 (t)+ t n−t e1 (t) is obtained and the corresponding analytic f

(

)n

(

)n−1

solutions are (e1 (t) . It is ) = C (tf )− t and e2 (t) = −Cn tf − t clear that e1 tf = e2 tf = 0 and consequently s0 (t ) would be zero at t = tf . Hence, it can be concluded that the transition between s and s0 is smooth. Similar to the proof of Theorem 2, when setting V0 = 21 s20 , one can have V˙ 0 = sT0 s˙0 ≤ 0 under the control law (19). Inspired by the analysis in [41, Remark 3], sT0 s˙0 = 0 is established only at s0 = 0, which means that the magnitude of the second sliding mode is monotonic, and once s0 = 0 is reached, the second sliding mode and its derivative will stay at s0 = s˙0 = 0 for t > tf . Furthermore, according ] [ to Theorem 2, we can get s = 0 for t ∈ 0 tf . Then, global robustness is accomplished. In addition, when the second-sliding mode reaches s0 = 0, we have e2 (t) = t−−nt e1 (t), which means the f variation of e1 is also monotonic, and once e1 = 0 is established, the tracking error and its derivative will stay at e1 = e2 = 0 for t > tf . 2.1.2. Adaptive time-varying sliding-mode control (ATVSMC) It is worth noting that the upper bound of the compound disturbances is used in the design of the RTVSMC and it is assumed to be known in advance. However, this upper bound is difficult to obtain in practical situations due to the complex structure of disturbances and uncertainties. To solve this problem, ATVSMC scheme is proposed for the estimation of unknown upper bounds of the compound disturbances. Hence, an adaptive switching input torque is obtained by modifying Eq. (19) as follows

( ( ) ) 1 ˆ τasw (t) = −b− 0 (x, t ) Ks + δ + η sign(s) ( ( ) ) 1 2 ˆ ˆ ˆ = − b− 0 (x, t ) Ks + a0 + a1 ∥x∥ + a2 ∥x∥ + η sign(s) (26)

(21)

The substitution of the error-dynamics (5) and the input torque (19) into (21), gives

where aˆ 0 , aˆ 1 and aˆ 2 be the estimates of a0 , a1 and a2 , respectively. As a result, the total input command can be designed as follows

V˙ = s (d (x, t ) − Ks − (δ + η) sign(s)) = d (x, t ) s − Ks2 −(δ + η) |s|

τ (t) = τeq (t) + τasw((t)

(22)

−˙z (t )

It is obvious that V˙ ≤ ∥d (x, t )∥ ∥s∥ − Ks2 − (δ + η) |s|

1 ¨ = −b− 0 (x, t ) f0 (x, t ) − yd (t) + n

(23)

e2 (t) tf − t

+n

e1 (t) (tf − t)2

(27)

)

+Ks + aˆ 0 + aˆ 1 ∥x∥ + aˆ 2 ∥x∥2 + η sign(s)

(

)

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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By defining the adaptation error as ˜ a0 = aˆ 0 − a0 ,˜ a1 = aˆ 1 − a1 , and ˜ a2 = aˆ 2 − a2 , the parameters aˆ 0 , aˆ 1 , and aˆ 2 are tuned using the following update laws a˙ˆ 0 = µ0 |s|

(28)

a˙ˆ 1 = µ1 |s| ∥x∥

(29)

a˙ˆ 2 = µ2 |s| ∥x∥2

(30)

where µ0 , µ1 and µ2 are positive constants. The results of the proposed ATVSMC can be summarized in the following theorem: Theorem 3. Consider the uncertain second-order nonlinear system (3) where the upper bound of the compound disturbances in (7) is assumed to be unknown. If the time-varying sliding surface is chosen as (8) and (9) in which the coefficient A is selected as (14) and the adaptive control input is designed as (27) under the update laws (28)–(30), then, the tracking errors e1 (t) and e2 (t) will both converge to zero as t → ∞. Proof. Consider the following Lyapunov function candidate: V =

1 2

s2 +

2 ∑ )2 1 ( aˆ i − ai 2µi

(31)

i=0

Differentiating V with respect to time yields V˙ = ss˙ +

2 ∑ ) 1 ( aˆ i − ai a˙ˆ i µi

(32)

i=0

Considering (15), (32) can be written as follows

) ∑ 2 ) 1 ( e1 (t) e2 (t) ˙ + +n − z t aˆ i − ai a˙ˆ i V˙ = s e˙ 2 (t) + n () tf − t (tf − t)2 µi (

i=0

5

Thus, we can deduce that V is non-increasing function and hence V (t) ≤ V (0). Consequently, V is bounded, which imply that the signals s,˜ a0 , ˜ a1 , ˜ a2 are bounded. Hence, the signals e, e˙ , τ , a˙ˆ 0 , a˙ˆ 1 , a˙ˆ 2 are bounded. Based on (16) we can deduce that s˙ is bounded, which ensure that V¨ is bounded based on (38). Thus, V˙ is uniformly continuous. Based on Barbalat’s lemma [1], we can draw a conclusion that V˙ → 0 when t → ∞, which implies that the tracking errors e1 (t) and e2 (t) will both go to zero as t → ∞. So, the asymptotical convergence is verified. Remark 4. Unlike the RTVSMC algorithm in Theorem 2, which needs the knowledge of the upper bound norm for the compound disturbances, the proposed ATVSMC algorithm in Theorem 3 does not require such information. This is because the adaptation law (28)–(30) which are used to estimate these unknown parameters. The estimated quantities are then used in the input torque (27). Therefore, the proposed control schemes enhance the robustness against external disturbances and parameter variations. Remark 5. With the designed adaptive estimation algorithm, only boundedness of the estimated parameters can be concluded along with the asymptotic tracking control. It is shown in many works that the estimated parameters asymptotically converge to their real values if persistent excitation condition is satisfied [1,6]. It has also proved that persistent excitation condition is essential for good estimation, but is not required for good output prediction. Hence, many adaptive control methods do not focus on parameter estimation and both robustness and performance can be guaranteed without any persistent excitation condition (see, e.g., [42,43]). Remark 6. It is obvious from the analytical expressions of the tracking errors e1 (t) and e2 (t) in Eqs. (11) and (13) that the convergence speed may be set by adjusting the parameter n. However, a larger value of n will result in a faster convergence rate.

(33) The substitution of the error-dynamics (5) and the input torque (27) into (33), gives V˙ = s d (x, t ) − Ks − aˆ 0 + aˆ 1 ∥x∥ + aˆ 2 ∥x∥2 + η sign(s) ) ) ) 1 ( 1 ( 1 ( + aˆ 0 − a0 aˆ˙ 0 + aˆ 1 − a1 a˙ˆ 1 + aˆ 2 − a2 a˙ˆ 2

(

(

µ0

)

µ1

)

µ2

) = d (x, t ) s − Ks − aˆ 0 + aˆ 1 ∥x∥ + aˆ 2 ∥x∥ + η |s| ) ) ) 1 ( 1 ( 1 ( aˆ 0 − a0 a˙ˆ 0 + aˆ 1 − a1 a˙ˆ 1 + aˆ 2 − a2 a˙ˆ 2 + µ0 µ1 µ2 2

2

(

(34) Considering the update laws (28)–(30), it follows

)

) ) ) ( ( ( + aˆ 0 − a0 |s| + aˆ 1 − a1 |s| ∥x∥ + aˆ 2 − a2 |s| ∥x∥2 ) ( = d (x, t ) s − Ks2 − η |s| − a0 + a1 ∥x∥ + a2 ∥x∥2 |s|

(35)

V˙ ≤ ∥d (x, t )∥ ∥s∥ − Ks2 − η |s| − a0 + a1 ∥x∥ + a2 ∥x∥2 |s| (36)

(

for t > tf

Thus, as long as n > 2 and p > 1, the singularities are avoided in the control laws.

)

(37)

In practice, many electromechanical structures are secondorder nonlinear systems governed by a set of state-space equation in the form of (3) with the satisfaction of Assumptions 1–3. Thus, the above proposed RTVSMC and ATVSMC can be extended to control robot manipulators. To do this, consider the well-known dynamic equation of multi-link robotic manipulators

(38)

M(q)q¨ + C (q, q˙ )q˙ + G(q) = τ (t) + τd (t)

Considering Lemma 1, we can obtain 2 2 V˙ ≤ (∥d (x, t )∥ |s| − ( Ks − η |s| a0 + a1 ∥2 x))∥ + a2 ∥x2∥ |s| = ∥d (x, t )∥ − a0 + a1 ∥x∥ + a2 ∥x∥ |s| − Ks − η |s|

(

for 0 ≤ t ≤ tf

2.2. Case study: time-varying sliding-mode control applied on a robotic manipulator

It is obvious that

)

According to Assumption 3, we can get V˙ ≤ −Ks2 − η |s|

( ⎧ ( )n−2 ⎪ ⎪ −1 ⎪ − b x , t f0 (x, t ) − y¨ d (t) + Cn (1 − n) tf − t ( ) ⎪ 0 ⎪ ⎪ ⎪ ⎪ ) ( )p−1 ⎪ ⎪ ⎪ ( ) t −t ⎪ ⎪ + 1 + Ap2 p f + Ks + + η) sign(s) (δ ⎪ ⎨ t (p − n + 1) ( f τ (t ) = ⎪ ( )n−2 ⎪ −1 ⎪ ⎪ −b0 (x, t ) f0 (x, t ) − y¨ d (t) + Cn (1 − n) tf − t ⎪ ⎪ ⎪ ⎪ ) ⎪ ( )p−1 ⎪ ⎪ tf − t ⎪ ⎪ ⎪ + p + Ks + (δ + η) sign(s) ⎩ tf (p − n + 1)

(39)

V˙ = d (x, t ) s − Ks2 − aˆ 0 + aˆ 1 ∥x∥ + aˆ 2 ∥x∥2 + η |s|

(

Remark 7. The presence of the term tf − t in the denominator of the control laws (19) may lead to a singularity when t = tf . By substituting the analytic solution (11) and (13) into (19), we have

(40)

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

6

M. Boukattaya, H. Gassara and T. Damak / ISA Transactions xxx (xxxx) xxx

where q ∈ ℜn is the vector of joint position, M(q) ∈ ℜn×n is the inertia matrix, C (q, q˙ ) ∈ ℜn×n is the centrifugal and Coriolis matrix, G(q) ∈ ℜn is the gravitational forces vector, τ (t) ∈ ℜn is the input torque vector and τd (t) ∈ ℜn is external disturbances vector. The dynamic parameters can be expressed in term of a known nominal part and unknown or uncertain part, as follows M(q) = M0 (q) + ∆M(q) C (q, q˙ ) = C0 (q, q˙ ) + ∆C (q, q˙ ) G(q) = G0 (q) + ∆G(q)

(41)

where M0 (q), C0 (q, q˙ ), G0 (q) are the nominal parts and ∆M(q), ∆C (q, q˙ ), ∆G(q) are the uncertain parts induced by unmodeled dynamics and payload variation. Thus, the dynamic model (39) can be written in the following form M0 (q)q¨ + C0 (q, q˙ )q˙ + G0 (q) = τ (t) + ρ (q, q˙ )

(42)

where ρ (q, q˙ ) = τd (t) − ∆M(q)q¨ − ∆C (q, q˙ )q˙ − ∆G(q) is the compound disturbances. The following assumptions are made about the root dynamics

∥M(q)∥ ≤ α0

(43)

∥C (q, q˙ )∥ ≤ β0 + β1 ∥q∥ + β2 ∥˙q∥2

(44)

∥G(q)∥ ≤ γ0 + γ1 ∥q∥

(45)

∥τ (q, q˙ )∥ ≤ c0 + c1 ∥q∥ + c2 ∥˙q∥2

(46)

∥ρ (q, q˙ )∥ ≤ d0 + d1 ∥q∥ + d2 ∥˙q∥2

(47)

Remark 8. The assumptions (43)–(45) are obtained according to the well-known model properties of the robotic manipulator. Thus, there have been widely used in many control methods [3– 10]. Remark 9. It is seen from Eq. (46) that the control input is bounded by a positive function of the position and velocity measurements. In practice, the joints of the robot are equipped with actuators, generally DC motors for generating the input torque and adequate sensor for the measurement of the position and velocity. Thus, assumption (46) is reasonable and practical. Remark 10. It is proved in [44] that the system uncertainty is input-related which means that the upper bound of the system uncertainties is not only related to the properties of the system, but also to the input command. Obviously, when the input command includes position and velocity feedback, then the bounded property in (47) is obtained. Consider the position tracking error e1 = q − qd and the velocity tracking error e2 = q˙ − q˙ d in which qd is the reference position. Hence, the equation of the error dynamic under uncertainties and disturbances can be expressed as

{

e˙ 1 = e2 e˙ 2 = M0−1 (q) [−C0 (q, q˙ )q˙ − G0 (q) − M0 (q)q¨ d + τ + ρ ] e˙ 1 = e2 e˙ 2 = F (q, q˙ ) − q¨ d + B(q)τ + D(q, q˙ )

(49)

where F (q, q˙ ) = −M0−1 (q) [C0 (q, q˙ )q˙ + G0 (q)], B(q) = M0−1 (q) and D(q, q˙ ) = M0 (q)ρ (q, q˙ ) −1

= M0−1 (q) [τd − ∆M(q)q¨ − ∆C (q, q˙ )q˙ − ∆G(q)] From the above definition, and the boundedness property (47), the following inequality is obtained

∥D(q,q˙ )∥ ≤ ∆ ( ) ∆ = M0−1 (q) d0 + d1 ∥q∥ + d2 ∥˙q∥2

Theorem 4. Consider the robot manipulator system (42) with the satisfaction of the constraint (50) and suppose that the parameters d0 , d1, and d2 are perfectly known. If the time-varying sliding manifold is chosen as n s(t) = e2 (t) + e1 (t) − z (t ) t f −t ⎧ ( ) ⎨ A t − t p for 0 ≤ t ≤ t (51) f f p z (t ) = tf ⎩ 0 for t > tf where tf is the desired convergence time, n > 1, p > n − 1 and A = tn e1 (0), and the applied input torque is designed as f

τ (t) = τeq (t) + τsw (t) τeq (t) = M0 (q)q¨ d + C0 (q, q˙ )q˙ + G0 (q) ) ( e2 (t) e1 (t) ˙ − M0 (q) n +n − z t ( ) tf − t (tf − t)2 τsw (t) = −M0 (q) (Ks + (∆ + η) sign(s))  ( ) ∆ = M0−1 (q) d0 + d1 ∥q∥ + d2 ∥˙q∥2

(52)

(53) (54) (55)

where η is a positive scalar and K is a positive definite diagonal constant matrix. Then, (i) The tracking errors e1 (t) and e2 (t) will both converge to zero at the prespecified time tf . ] [ (ii) The global sliding-mode is achieved, i.e. s ≡ 0 for t ∈ 0 tf . Proof. The analysis is similar to the one developed in Theorem 1. For this reason, it will be ignored in this paragraph. Theorem 5. Consider the robot manipulator system (42) with the satisfaction of the constraint (50) and suppose that the parameters d0 , d1, and d2 are completely unknown. If the time-varying sliding manifold is chosen as n s(t) = e2 (t) + e1 (t) − z (t ) tf − t ⎧ ⎨ A (t − t )p for 0 ≤ t ≤ t (56) f f p z (t ) = tf ⎩ 0 for t > tf where tf is the desired convergence time, n > 1, p > n − 1 and A = tn e1 (0), and the applied adaptive input torque is f

(48)

Eq. (48) can still be expressed in the form

{

where d0 , d1 and d2 are positive constants. It is trivial that (49) has the same form as (5). Thus, the above proposed RTVSMC and ATVSMC schemes can be adapted to multilink robotic manipulators. The essential results of the proposed approaches can be outlined in the following theorems.

(50)

τ (t) = τeq (t) + τasw (t) τeq (t) = M0 (q)q¨ d + C0 (q, q˙ )q˙ + G0 (q) ( ) e2 (t) e1 (t) − M0 (q) n +n − z˙ (t ) tf − t (tf − t)2

( ( ) ) ˆ + η sign(s) τasw (t) = −M0 (q) Ks + ∆ )  ( ˆ = M0−1 (q) dˆ 0 + dˆ 1 ∥q∥ + dˆ 2 ∥˙q∥2 ∆

(57)

(58)

(59) (60)

where dˆ 0 , dˆ 1 and dˆ 2 are the estimates of d0 , d1 and d2 , respectively, which are updated by the following laws

˙

dˆ 0 = λ0 M0−1 (q) |s|





(61)

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

M. Boukattaya, H. Gassara and T. Damak / ISA Transactions xxx (xxxx) xxx

˙



˙



dˆ 1 = λ1 M0−1 (q) ∥q∥ |s|



7

(62)

dˆ 2 = λ2 M0−1 (q) ∥˙q∥2 |s|



(63)

where K is a positive definite diagonal matrix, η > 0 is a positive scalar, and λi ; i = 0, 1, 2 are positive scalars. Then, the tracking errors e1 (t) and e2 (t) will both converge to zero as t → ∞. Proof. Consider the following Lyapunov function candidate: V =

1 2

s2 +

2 )2 ∑ 1 ( dˆ i − di 2λi

(64)

i=0

Differentiating V with respect to time yields V˙ = ss˙ +

2 ) ∑ 1 ( ˙ dˆ i − di dˆ i λi

(65)

Fig. 1. The architecture of two-link robotic manipulator.

i=0

With respect to Eq. (15), it follows

) ∑ 2 ) 1 ( ˙ ˙V = s e˙ 2 (t) + n ˙ dˆ i − di dˆ i +n − z (t ) + 2 tf − t (tf − t) λi (

e2 (t)

e1 (t)

i=0

(66) Considering the error dynamic (49) and the adaptive input torque (57)–(60), one obtains 2 ) ∑ 1 ( ˙ dˆ i − di dˆ i λi i=0 2 ) ( ) ∑ 1 ( ˙ 2 ˆ + η |s| + = D(q, q˙ )s − Ks − ∆ dˆ i − di dˆ i λi

(

(

)

)

ˆ + η sign(s) + V˙ = s D(q, q˙ ) − Ks − ∆

(67) Substituting the update laws (61)–(63), we have V˙ = D(q, q˙ )s − Ks2

) ] [ ( − M0−1 (q) dˆ 0 + dˆ 1 ∥q∥ + dˆ 2 ∥˙q∥2 + η |s| ( ) [( )   + dˆ 0 − d0 M0−1 (q) |s| + dˆ 1 − d1 M0−1 (q) ∥q∥ |s| ] ( )  + dˆ 2 − d2 M0−1 (q) ∥˙q∥2 |s| (68) Simplifying (68) yields V˙ = D(q, q˙ )s − Ks2 − η |s| (69)

= D(q, q˙ )s − Ks2 − η |s| − ∆ |s| It is trivial to verify that V˙ ≤ ∥D(q, q˙ )∥ ∥s∥ − Ks2 − η |s| − ∆ |s|

(70)

max

3. Simulation results To demonstrate the benefits of the proposed algorithm, a simulation study was carried out using a planar 2 DOF robot manipulators (see Fig. 1). The dynamic equation of the system can be described by

[

][ ] [ q¨ 1 −c(q2 )q˙ 1 + q¨ 2 0 [ ] [ ] g (q , q ) τ +g 1 1 2 = 1 g2 (q1 , q2 ) τ2

m11 (q2 ) m21 (q2 )

m12 (q2 ) m22

−2.c(q2 )q˙ 1 c(q2 )q˙ 2

][ ] q˙ 1 q˙ 2 (74)

where m12 (q2 ) = m21 (q2 ) = m2 L22 + m2 L1 L2 cos(q2 )

(71)

m22 = m2 L22 + J2 c(q2 ) = m2 L1 L2 sin(q2 )

According to the boundedness property (50), we can get V˙ ≤ −Ks2 − η |s|

In most of the literature [46,47], this problem was resolved by using the sign saturation function as follows: Γ = sign (τmax ) if τ ≥ τmax ; otherwise, Γ = τ where τmax is the maximum permitted torque. However, this function has sharp corners as τ = τmax which is undesirable for practical application. Recently, in [48,49] Rahimi et al. have proposed a smooth hyperbolic tangent function Γ = τmax tanh (τ /τmax ) to have all functions being differentiable. Inspired by this idea, we propose here a new algorithm for actuator saturation avoidance using the following smooth saturation τ . function Γ = τmax |τ |+τ

m11 (q2 ) = (m1 + m2 ) L21 + m2 L22 + 2m2 L1 L2 cos(q2 ) + J1

Considering Lemma 1, we can obtain V˙ ≤ ∥D(q, q˙ )∥ |s| − Ks2 − η |s| − ∆ |s| = (∥D(q, q˙ )∥ − ∆) |s| − Ks2 − η |s|

⎧ ⎨sign (s) if ∥s∥ > ε fsat (s) = sig ρ (s) (73) ⎩ if ∥s∥ ≤ ε ερ where ε > 0 is a sufficiently small constant and 0 < ρ < 1. Remark 12. Due to physical limitations on the driven actuator, special attention should be paid to actuator saturation to maintain the performance of systems when the limitation occurs.

i=0

 ( ) − M0−1 (q) d0 + d1 ∥q∥ + d2 ∥˙q∥2 |s|

Remark 11. The controllers (19), (27), (54) and (59) contain the discontinuous function sign (s) which causes undesirable chattering. To solve this problem, it is replaced by the function fsat (s) [36,45] defined as following

(72)

Similar to the previous analysis, the tracking errors e1 (t) and e2 (t) will both converge to zero as t → ∞.

g1 (q1 , q2 ) = (m1 + m2 ) L1 cos(q2 ) + m2 L2 cos(q1 + q2 ) g2 (q1 , q2 ) = m2 L2 cos(q1 + q2 ) where mi is the mass of link i, Li is the length of link i, Ii is the inertia of link i (i = 1, 2).

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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Fig. 2. Positions tracking for joints 1 and 2.

The nominal parameters of the manipulators are chosen as L01

= 1 m, I10

L02

= 0.5 m,

= 0.2 kg m , 2

I20

m01

= 4 kg,

m02

= 2 kg,

= 0.2 kg m

2

It is well known that a large variation in the control effort is not desired in most of the control applications, since it may be harmful to the actuators. Hence, the control signals are assumed to be saturated within |τ (t)| ≤ τmax = 50 N m [49]. The reference trajectory is chosen as

[ qd =

q1d q2d

]

⎡

7

⎢ =⎣

5

1.25 −

e−t +

1.25 + e

−t

7

e−4t

⎤

⎥ 20 ⎦ 1 −4t + e 4

(75)

The initial configuration and velocity are: q(0) = [1, 1.5]T and q˙ (0) = [0, −2]T . We assume that the dynamic parameters of the robot are uncertain with a variance of 20%, which follows m1 = m01 + 0.2.m01 , m2 = m02 + 0.2.m02 , I1 = I10 + 0.2.I10 , I2 = I20 + 0.2.I20

(76)

We consider also the following time-varying external disturbances

τd (t) =

[ ] [ ] τ1d (t) 2 sin (t ) + 0.5 sin (200π t ) = τ2d (t) cos (2t ) + 0.5 sin (200π t )

(77)

Measurement noise in the output signal is neglected in this simulation. 3.1. Performance evaluation of the RTVSMC design In this section, numerical simulations are presented to show the effectiveness of the proposed RTVSMC scheme in Theorem 4, in which the upper bound of the system uncertainty is assumed to be known in advance. Thus, these bounds are selected as d0 = 9.5, d1 = 2.2, d2 = 2.8. The parameters of the controller are chosen as n = 5, p = 5, η = 0.5, K = 120, and the desired convergence time is set to tf = 1 s. The coefficient A in (14) is obtained as [ ]T n n A = e1 (0) = (q (0) − qd (0)) = 4 −5 tf tf . Tracking performances when uncertainties and external disturbances occur are illustrated in Figs. 2–6. From these results, it is obvious that good tracking performance is achieved with the

satisfaction of the convergence time requirement. However, it can be observed that the convergence to the desired trajectories is achieved with small tracking errors for both position and velocity, and this exactly at the prescribed convergence time. Fig. 5 shows that the sliding mode manifold responses converge to zero from the start of the trajectory which verifies the elimination of the reaching phase and thus, the insensitivity of the system to external disturbances and parameter variations during the whole tracking process. Furthermore, the input signals generated by the proposed controller in Fig. 6 are smooth without exceeding its maximum admissible value. To further illustrate the efficacy of the control law to meet different desired time demands, various simulations are performed by different convergence time tf , the values of tf are selected as 1, 2 and 3, respectively and the associated results are shown in Figs. 7 and 8. These results clearly demonstrate that for each designated convergence time, the convergence of the tracking errors to the neighbors of zero has always achieved exactly at the imposed time. The tracking errors are small enough, indication that the high precision and the time constraint are well satisfied. For further analysis, we show in Figs. 9 and 10 the performance of the controller with a different value of n. As can be seen, the value of n has an impact on the convergence rate. However, a bigger value of n would lead to a quicker convergence rate. Thus, one can conclude that the designed RTVSMC algorithm gives a good tracking performance, and it can work robustly and smoothly in uncertainties and disturbances all the time. This can be achieved by a simple adjustment of the parameters. However, the desired time of convergence is obtained by setting tf , the convergence rate is improved by increasing n, and the global robustness is guaranteed by adjusting the coefficient A. 3.2. Performance evaluation of the ATVSMC design In this subsection, numerical simulations are presented to show the effectiveness of the proposed ATVSMC scheme, in which the system uncertainty upper bound is assumed to be completely unknown. The parameters of the controller are chosen as n = 5, p = 5, η = 0.5, K = 120, λ0 = 0.01, λ1 = 0.01, λ2 = 0.01, [ ]T A = 4 −5 , the initial conditions are selected as dˆ 0 (0) = 0, dˆ 1 (0) = 0, dˆ 2 (0) = 0, and the desired convergence time is set to tf = 1 s. The performances under disturbances and uncertainties are shown in Figs. 11–16. It can be observed that the reference trajectories were tracked faithfully with a small tracking error. Moreover, the control input remains smooth in the admissible range limit, and without any

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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9

Fig. 3. Velocities tracking for joints 1 and 2.

Fig. 4. Position and velocity tracking errors.

Fig. 5. Sliding manifold for joints 1 and 2.

chattering which clearly demonstrates the effectiveness of the

3.3. Comparative study evaluation of the ATVSMC approach

proposed ATVSMC. The time response of the sliding manifold in Fig. 14 converges to zero after a short period, which is due to the time taken by the uncertain parameters to stabilize at constant values (Fig. 16).

In this subsection, a comparative study will be conducted to demonstrate the superiority of the proposed ATVSMC. Two recent sliding-mode type controller are suggested. The first one is the Adaptive Nonsingular Fast Terminal Sliding-Mode Control

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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Fig. 6. Control torques for joints 1 and 2.

Fig. 7. Positions tracking for joints 1 and 2 with a different value of tf (n = 5).

Fig. 8. Velocities tracking for joints 1 and 2 with a different value of tf (n = 5).

(ANFTSMC) of M. Boukattaya et al. (2018) [24] and the second

torque of [24] are given by

is the Adaptive Integral Terminal Sliding-Mode Control (AITSMC) of A. Riani et al. (2018) [50]. The sliding manifold and the input

s(t) = e1 + k |e1 |α sign(e1 ) + k2 |e2 |β sign(e2 )

(78)

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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11

Fig. 9. Positions tracking for joints 1 and 2 with a different value of n (t f = 1).

Fig. 10. Velocities tracking for joints 1 and 2 with a different value of n (t f = 1).

Fig. 11. Positions tracking for joints 1 and 2.

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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Fig. 12. Velocities tracking for joints 1 and 2.

Fig. 13. Position and velocity tracking errors.

Fig. 14. Sliding manifold for joints 1 and 2.

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

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13

Fig. 15. Control torque for joints 1 and 2.

where 1 < β < 2 with α > β , k1 and k2 are positive diagonal matrices.

τ (t) = τ1 (t) + τ2 (t) τ1 (t) = M0 (q)q¨ d + C0 (q, q˙ )q˙ + G0 (q) ( ) M0 (q) |e2 |2−β 1 + α.k1 |e1 |α−1 sign(e2 ) − β.k2 [ ( ) ] τ2 (t) = −M0 (q) k.s + bˆ 0 + bˆ 1 |q| + bˆ 2 |˙q|2 + η sign(s)

(79)

where bˆ 0 , bˆ 1 and bˆ 2 are the estimates of b0 , b1 and, b2 respectively, which are updated by the following adaptive laws

˙

bˆ 0 = λ0 |s| . |e2 |β−1

(80)

˙ bˆ 1 = λ1 |s| . |e2 |β−1 |q| ˙ bˆ 2 = λ2 |s| . |e2 |β−1 |˙q|2

(81) (82)

The values of the parameters are: α = 2, β = 5/3, η = 0.5, k1 = 1, k2 = 1, k = 250, λ0 = λ1 = λ2 = 0.01, bˆ 0 (0) = bˆ 1 (0) = bˆ 2 (0) = 0. The sliding surface and the input torque of [50] are given by s(t) = e2 +

∫ t(

β/α

k1 e2

β/(2α−β)

+ k2 e1

)

Table 1 Quantitative analysis of the controllers. Controller

dt

where α and β are positive odd integers such that α > β > 0, k1 and k2 are positive diagonal matrices.

τ (t) = τ1 (t) + τ2 (t) τ1 (t) = M0 (q)q¨ d + C0 (q, q˙ )q˙ + G0 (q) ( ) β/α β/(2α−β) + M0 (q) k1 e2 + k2 e1 ] [ s τ2 (t) = −M0 (q) k.s + Ψ T Aˆ ∥ s∥

IAE

(83)

0

(84)

where Aˆ is the estimate of A and is updated by the following adaptive law

Proposed ATVSMC ANFTSMC [24] AITSMC [50]

ISV

Joint 1

Joint 2

Joint 1

Joint 2

0.338 0.425 0.748

0.396 0.493 0.616

0.104 0.177 0.343

0.157 0.248 0.481

where µ > 0 is a positive constant. The values of the parameters are: α = 7, β = 9, η = 0.5, ˆ k1 = 3, k2 = 1, k = 100, Γ = 100, A(0) = 0. The following performance functions measures are adopted to ensure a fair comparison tf

∫ IAE =

|e1 (t )| dt

(87)

τ 2 (t ) dt

(88)

0

˙

Aˆ = Γ Ψ χ (s)

(85)

where     [ ]T  β/α   β/(2α−β)  Ψ = 1 1 + ∥q∥ 1 + ∥˙q∥ 1 + ∥˙q∥2 1 + e2  1 + e2  1 + ∥s ∥ ,

Γ is a positive diagonal matrix, and χ (s) is a scalar function defined as

χ (s) =

Fig. 16. Estimation of the unknown parameters.

{

∥s∥ + µ 0

if ∥s∥ ̸ = 0 if ∥s∥ = 0

(86)

And

∫ ISV =

tf

0

where tf is the final running time (see Table 1). To have a fair comparison, simulations are elaborated with the same system parameters, initial condition, and other common parameters. Fig. 17 shows the response of the position tracking trajectories of the system obtained by the proposed ATVSMC,

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

14

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Fig. 17. Comparison of the positions tracking for joints 1 and 2.

Fig. 18. Comparison of the sliding manifolds for joints 1 and 2.

ANFTSMC and AITSMC methods. Clearly, the proposed method exhibits a faster response than the other methods. In addition, by comparing the quantitative measures, one can see that the proposed control method provides lesser IAE values than those with the ANFTSMC and AITSMC methods, illustrating the superiority of the designed algorithm in terms of high precision. The time responses of the three sliding mode surfaces are superposed in Fig. 18. It is clearly observed the existence of the reaching phase when using the ANFTSMC and the AITSMC methods, while the reaching phase is reduced by the proposed ATVSMC. Therefore, the ATVSMC algorithm can realize the terminal sliding mode, which makes the algorithm insensitive to disturbances and uncertainties during the entire response of the system. The control inputs for these control methods are displayed in Fig. 19. It is obvious that the torques of the ANFTSMC and AITSMC methods are both switching dramatically, especially at the beginning. This undesirable phenomenon has been significantly reduced for the proposed ATVSMC where the resulting input signal has appropriate amplitude compared to the other approaches. In addition, the actuator saturation was taken into account for the proposed controller and one can see that the generated control input does not exceed its maximum admissible value (50 N m). Consequently, its corresponding ISV is smaller than those with ANFTSMC and AITSMC manifesting the superiority of the proposed controller in terms of minimal IAE and less control effort amplitude.

4. Conclusion In this research, a novel time-varying sliding mode control laws are proposed for the trajectory tracking in the presence of uncertainties and disturbances. These algorithms are derived using time-varying sliding-mode concept to drive the system states to the equilibrium in prescribed time of convergence. Moreover, and in order to guarantee the global convergence, the reaching phase was eliminated by adding a dummy variable to the original sliding surface. Hence, the strong robustness against disturbances and uncertainties is ensured from the very beginning of the motion and the control signals are smooth without undesirable chattering or singularity. In addition, an adaptive approach was proposed for avoiding prior knowledge of the upper bound of the system uncertainty. The main advantage of this adaptive method is that the upper bounds of the system uncertainties are not needed to be known in advance and the convergence of the tracking error to zero asymptotically is ensured, in spite the presence of uncertainties and disturbances. The control saturation due to the physical limit of the actuator is also taken into account in this work. Extensive simulations have been carried out to demonstrate to verify the effectiveness of the proposed methods. It is shown that the proposed adaptive time-varying sliding mode control schemes, offer several advantages such as the convergence to the origin at the desired time, high precision, fast convergence rate, and global robustness to parameter

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.

M. Boukattaya, H. Gassara and T. Damak / ISA Transactions xxx (xxxx) xxx

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Fig. 19. Comparison of the input torques for joints 1 and 2.

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Mohamed Boukattaya received his diploma in Electromechanical Engineering from the National School of Engineers of Sfax, Tunisia, in 2002. From the same school, he received his M.S. degree in Automatic and Industrial Informatics and the Ph.D. in Electrical Engineering, in 2006 and 2011 respectively. He is currently an assistant professor at the Preparatory Institute of Engineers of Sfax. His research interests include robot modeling and control, especially mobile manipulators, mobile platforms, and redundant systems. E-mail: [email protected] Hamdi Gassara received the Ph.D. in automatic control from the University of Picardie Jules Verne (UPJV), in 2011. Prior to Ph.D., he received his Master degree from UPJV in 2008. His teaching experience started when he was Ph.D. student in UPJV France from 2008 to 2011. He is currently an Assistant Professor in Electrical Department at National School of Engineering of Sfax, Tunisia. His research focuses on analysis and control for fuzzy models with time delay, fault tolerant control, diagnosis, saturation, and polynomial fuzzy models. E-mail: [email protected] Tarak Damak received his diploma in Electrical Engineering from the National School of Engineers of Sfax, Tunisia, in 1989 and his D.E.A degree in Automatic Control from the Institut National des Sciences Appliquées de Toulouse, France, in 1990. He received his Ph.D. from the Université Paul Sabatier de Toulouse, France, in 1994. In 2006. He then obtained the University Habilitation from the National School of Engineers of Sfax. He is currently a professor in the Department of Mechanical Engineering of the National School of Engineers of Sfax, Tunisia. His current research interests are in the fields of distributed parameter systems, sliding mode control and observers, adaptive nonlinear control. E-mail: [email protected]

Please cite this article as: M. Boukattaya, H. Gassara and T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.003.