Adaptive finite-time command-filtered backstepping sliding mode control for stabilization of a disturbed rotary-inverted-pendulum with experimental validation

Abstract

In this paper, the finite-time stabilization of the disturbed and uncertain rotary-inverted-pendulum system is studied based on the adaptive backstepping sliding mode control procedure. For this purpose, first of all, the dynamical equation of the rotary-inverted-pendulum system is obtained in the state-space form in the existence of external disturbances and model uncertainties with unknown bound. Afterward, a novel command filter is defined to enhance the control strategy by consideration of a virtual control input. Therefore, the differential signal is replaced by the output of the command filter to reduce the complicated computing in the control process. Hence, the finite-time convergence of the sliding surface to the origin is attested by using the backstepping sliding mode control scheme according to the Lyapunov theory. Besides, the unknown upper bound of the exterior perturbation and uncertainty is approximated providing the adaptive control technique. Finally, simulations and experimental results are done to demonstrate the impression and proficiency of the suggested method.

Keywords

Backstepping control sliding mode control finite-time convergence robust stability inverted-pendulum

1. Introduction

The inverted-pendulum (IP) system was first introduced as cart-inverted pendulum which has limitation related to its length (Huang et al., 2010, 2019a, 2019b; Roose et al., 2017; Vassiliou et al., 2017). For this reason, the IP system was extended by introducing the Rotary-IP (RIP) system which has two main components including the arm which can rotate in the horizontal plane and the pendulum connected to the arm which can rotate in the vertical plane (Dwivedi et al., 2017; Du et al., 2019; Fukushima et al., 2014; Nath and Dewan, 2017; Watson et al., 2019), whereas the RIP system is an underactuated system with interesting features including nonlinear characteristics and availability, so it is considered as an experimental case study (Kim et al., 2019; Mehedi et al., 2020; Park and Chwa, 2009; Pujol-Vazquez et al., 2018; Roy et al., 2021; Wang et al., 2014). All in all, stability of the position and angular velocity of the arm and up-right balancing of the pendulum are the fundamental control objectives concerned to the RIP system (Hamza et al., 2019; Huang et al., 2012; Li et al., 2017; Wang et al., 2020b; Yiğit, 2017). Due to this reason, some control methods have been applied in the target of stability and swing-up control of the RIP system (Hazem et al., 2020; Patil et al., 2018; Sun et al., 2018; Wasiwitono et al., 2021; Rahmani et al., 2021). In Govind and Kumar (2020), an LQR-based PID state-feedback controller for the stability and balancing control of the RIP system was designed. In addition, the experimental results based on the recommended method were provided to acknowledge the efficiency of this method. Nevertheless, finite-time stability control of the RIP system is not considered in this research. An optimal control technique based on the LQR scheme for the stability of the RIP system was introduced in Wang et al. (2020a) though the influence of the model uncertainty and external disturbance is not investigated in this article. In Sarkar et al. (2020), the stability and swing-up control for the RIP system was planned based on the trial and error LQR technique, and some experimental results for this suggested technique were implemented to confirm the proficiency. However, this method does not offer fast convergence and does not study the effects of the model uncertainty and exterior perturbation. In Al-Araji (2019), the finite-time swing-up control of the RIP system was designed and implemented based on the adaptive sliding mode control (ASMC) technique. Also, an intelligence culture-bees algorithm was used for the adjustment of control parameters even though this article does not present the experimental results to validate the proposed method. In Dhouibi et al. (2019), a second-order ASMC was proposed for the nonlinear system with unknown bounded uncertainties. Also, the IP system is considered as a case study to illustrate the effectiveness of the proposed method. However, the recommended method is not able to overcome the model uncertainties. In Dao and Nguyen (2021), an adaptive control technique based on the sliding mode observer was presented for the finite-time convergence of the wheeled IP system though the experimental results are not presented in the studied research. In Goswami et al. (2017), for the stability control of the wheeled IP system, an adaptive backstepping sliding mode control (SMC) technique was designed. Also, the control strategy was presented in two subsystems, so the control laws for each subsystem have been obtained based on the backstepping SMC combined Lyapunov theory. But the method of this paper suffers from the complicated computation, which is created because of employment of the backstepping technique. In Yang and Zheng (2018), the swing-up and stability control procedures of the RIP system are considered as two separate subsystems. An adaptive neural network method using linear matrix inequality (LMI) is designed for each subsystem. However, the proficiency of the suggested method is not investigated using experimental results. In Nguyen et al. (2020), a fuzzy algorithm based on the super-twisting SMC technique is proposed for the stability and swing-up control of the underactuated RIP system. Also, the efficiency of this method is acknowledged by experimental results. Nevertheless, the impression of external disturbance and model uncertainty is not considered in this work. In Guo et al. (2020), position control of the electro-hydraulic system is presented based on the adaptive backstepping procedure combined with the block-strict-feedback model. Moreover, an adaptive control scheme is adopted for high-robustness and performance of the system against hydraulic parametric uncertainties and external load. Also, this article presented a desirable performance using prescribed performance control. In Guo et al. (2018b), an adaptive neural network technique combined with the backstepping procedure and prescribed performance control is recommended for the control of the robot manipulator driven by the electro-hydraulic system. In Guo et al. (2018a), the finite-time tracking control of the electro-hydraulic system under both hydraulic parametric uncertainties and the external load is investigated using the backstepping scheme and fractional-type Lyapunov function.

From the review of the above researches which have investigated the stability control of the pendulum system, it can be found that no work has considered the problem of stabilization control of the uncertain RIP system with external disturbances by means of backstepping sliding mode control based on the command-filtered technique and it is still open in literature. In this paper, the command filter combined with the virtual control input is adopted to eliminate high-order derivatives of the virtual control inputs. Furthermore, for the removal of differential signals of the command filter, the compensation error system is employed. After that, the sliding mode variables based on the tracking errors and compensation error system are presented. Afterward, finite-time convergence of sliding surfaces is demonstrated by applying the backstepping strategy and Lyapunov stability theory. Furthermore, for high-robustness of the closed-loop system against external disturbances and parameter uncertainties, the unknown upper bounds of the perturbations are estimated via the adaptive control technique. Therefore, the substantial innovations of this study can be listed as follows:

Recommendation of ASMC based on the finite-time backstepping procedure for stabilization of the perturbed and uncertain RIP system;

presentation of the command filter system mixed with the compensation error system based on the virtual control input for the simplification of the control strategy;

definition of the sliding mode variable based on the tracking errors and the compensation error system; and

finite-time reachability of the sliding surface to the origin based on the Lyapunov concept and backstepping method.

The remainder of this paper can be reported as follows: In Section 2, the nonlinear dynamic model of the RIP system is obtained. In Section 3, the state-space form of the RIP system as well as the fundamental lemmas and assumptions are given. The control strategy with respect to the finite-time backstepping ASMC is presented in Section 4. Simulation and experimental results on the real RIP system are reported in Section 5. Conclusions of this paper are explained in Section 6.

2. Dynamic model of the RIP system

The structure of the RIP system is illustrated in Figure 1. As one can observe that $q_{1} = θ_{1}$ is the angular deviation of the arm as well as $q_{2} = θ_{2}$ is the angular deviation of the pendulum.

Presume dynamic equation of the rotary-inverted-pendulum is given as

M (q) \ddot{q} + V_{m} (q, \dot{q}) \dot{q} + G_{m} (q) = τ

(1)

where

q = {[q_{1}, q_{2}]}^{T}

and

M

is the mass matrix which is given as

M = [\begin{matrix} J_{1} + m_{2} l_{2} + m_{2} c_{2}^{2} s_{q_{2}}^{2} & m_{2} l_{1} c_{2} c_{q_{2}} \\ m_{2} l_{1} c_{2} c_{q_{2}} & J_{2} + m_{2} c_{2}^{2} \end{matrix}]

(2)

while

m_{1}

l_{1}

, and

c_{1}

signify the mass, length, and distance to the center of the arm as well as

m_{2}

l_{2}

, and

c_{2}

specify the mass, length, and distance to the center of the pendulum, respectively. In addition,

J_{1}

and

J_{2}

are the moment of inertia with respect to the arm and pendulum, respectively. Also, we define

\sin (q_{2}) = s_{q_{2}}

\cos (q_{2}) = c_{q_{2}}

\sin (2 q_{2}) = s_{2 q_{2}}

, and

\cos (2 q_{2}) = c_{2 q_{2}}

.The term

V_{m}

is the Coriolis and centripetal matrix which is represented by

V_{m} = \frac{1}{2} m_{2} [\begin{matrix} l_{1}^{2} {\dot{q}}_{2} s_{2 q_{2}}^{2} & - 2 l_{1} c_{2} {\dot{q}}_{2} s_{q_{2}} + c_{2}^{2} {\dot{q}}_{1} s_{2 q_{2}} \\ - c_{2} {\dot{q}}_{1} c_{2 q_{2}} & 0 \end{matrix}]

(3)

Also,

G_{m}

is the gravitational matrix which is given as

G_{m} = [\begin{matrix} 0 \\ - m_{2} c_{2} g s_{q_{2}} \end{matrix}]

(4)

where

g

represents the gravitational acceleration. The expression of

τ

implicates the torque matrix which can be represented by

τ = [\begin{matrix} τ_{1} \\ 0 \end{matrix}]

(5)

such that

τ_{1}

is the applied torque. Equation (1) can be rewritten by multiplying the inversion of matrix

M

into both sides of the equation as follows

\ddot{q} + M^{- 1} (q) V_{m} (q, \dot{q}) \dot{q} + M^{- 1} (q) G_{m} (q) = M^{- 1} (q) τ

(6)

where

M^{- 1}

is the inversion of matrix

M

. Substituting equations (4) and (5) into equation (6), one achieves

[\frac{{\ddot{q}}_{1}}{{\ddot{q}}_{2}}] = - [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}] [\begin{matrix} v_{m 11} & v_{m 12} \\ v_{m 21} & v_{m 22} \end{matrix}] [\begin{matrix} {\dot{q}}_{1} \\ {\dot{q}}_{2} \end{matrix}] - [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}] [\begin{matrix} 0 \\ - m_{2} c_{2} g s_{q_{2}} \end{matrix}] + [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}] [\begin{matrix} τ_{1} \\ 0 \end{matrix}]

(7)

such that

M_{11}

M_{12}

M_{21}

, and

M_{22}

denote the components of

M^{- 1}

. Besides

v_{m 11}

v_{m 12}

v_{m 21}

, and

v_{m 22}

are the elements of the matrix

V

. Now, by doing some mathematical operations, it can get

[\begin{matrix} {\ddot{q}}_{1} \\ {\ddot{q}}_{2} \end{matrix}] = - [\begin{matrix} M_{11} v_{m 11} + M_{12} v_{m 21} & M_{11} v_{m 12} + M_{12} v_{m 22} \\ M_{21} v_{m 11} + M_{22} v_{m 21} & M_{21} v_{m 12} + M_{22} v_{m 22} \end{matrix}] [\begin{matrix} {\dot{q}}_{1} \\ {\dot{q}}_{2} \end{matrix}] + [\begin{matrix} M_{12} m_{2} c_{2} g s_{q_{2}} \\ M_{22} m_{2} c_{2} g s_{q_{2}} \end{matrix}] + [\begin{matrix} M_{11} τ_{1} \\ M_{21} τ_{1} \end{matrix}]

(8)

Now, consider

Η_{1} = - M_{11} v_{m 11} - M_{12} v_{m 21}

Η_{2} = - M_{11} v_{m 12} - M_{12} v_{m 22}

Η_{3} = M_{12} m_{2} c_{2} g s_{q_{2}}

Η_{4} = - M_{21} v_{m 11} - M_{22} v_{m 21}

Η_{5} = - M_{21} v_{m 12} - M_{22} v_{m 22}

, and

Η_{6} = M_{22} m_{2} c_{2} g s_{q_{2}}

, so the system’s dynamical equation is represented as

{\ddot{q}}_{1} = Η_{1} {\dot{q}}_{1} + Η_{2} {\dot{q}}_{2} + Η_{3} + M_{11} τ_{1}

(9)

{\ddot{q}}_{2} = Η_{4} {\dot{q}}_{1} + Η_{5} {\dot{q}}_{2} + Η_{6} + M_{21} τ_{1}

(10)

3. Problem formulation and preliminaries

Based on this part, state-space formation of the disturbed and uncertain RIP system is defined first. Then, some assumptions and lemmas related to the control strategy are presented.

Assume that the state-space vector $Z = {[z_{1}, z_{2}, z_{3}, z_{4}]}^{T} = {[q_{1}, {\dot{q}}_{1}, q_{2}, {\dot{q}}_{2}]}^{T}$ and exterior perturbation and uncertain vector $Θ = {[Θ_{1}, Θ_{2}, Θ_{3}, Θ_{4}]}^{T}$ . Then, state-space formulation of the disturbed and uncertain RIP system (9) and (10) can be stated as

{\dot{z}}_{1} (t) = z_{2} (t) + Θ_{1}

(11)

{\dot{z}}_{2} (t) = Η_{1} z_{2} (t) + Η_{2} z_{4} (t) + Η_{3} + M_{11} τ_{1} (t) + Θ_{2}

(12)

{\dot{z}}_{3} (t) = z_{4} (t) + Θ_{3}

(13)

{\dot{z}}_{4} (t) = Η_{4} z_{2} (t) + Η_{5} z_{4} (t) + Η_{6} + M_{21} τ_{1} (t) + Θ_{4}

(14)

Assumption 1

For the bounded uncertainty and external disturbances $Θ_{i}, i = 1, \dots, 4$ , the following inequalities are satisfied

| L_{1} Θ_{1} + L_{2} Θ_{3} | \leq θ_{1}

(15)

| L_{3} Θ_{2} + L_{4} Θ_{4} | \leq θ_{2}

(16)

where

θ_{1}

and

θ_{2}

are the unknown positive constants. Also,

L_{i}'s, i = 1, \dots, 4

are some positive constants.

Lemma 1

(Qian and Lin, 2001): If $Υ_{1}, Υ_{2}$ are real values, so for any positive real numbers c₁,c₂ and real-valued function $x$ , the subsequent inequality is acknowledged

{| Υ_{1} |}^{c_{1}} {| Υ_{2} |}^{c_{2}} \leq \frac{c_{1}}{c_{1} + c_{2}} x {| Υ_{1} |}^{c_{1} + c_{2}} + \frac{c_{2}}{c_{1} + c_{2}} x^{- \frac{c_{1}}{c_{2}}} {| Υ_{2} |}^{c_{1} + c_{2}}

(17)

Lemma 2

(Huang et al., 2005): If $x_{k} \in R (k = 1, .., n)$ and $0 < p \leq 1$ , thus, the subsequent inequality is accepted

{(\sum_{k = 1}^{n} | x_{k} |)}^{p} \leq \sum_{k = 1}^{n} {| x_{k} |}^{p} \leq n^{1 - p} {(\sum_{k = 1}^{n} | x_{k} |)}^{p}

(18)

Lemma 3

(Yu et al., 2018): For any system with the form of $\dot{x} = φ (x, u)$ , it is verified that the trajectory of the system is stable with finite-time convergence rate, if the following condition holds

\dot{V} (x) \leq - ϱ_{1} V (x) - ϱ_{2} V^{γ} + ϵ

(19)

where

ϱ_{1} > 0

ϱ_{2} > 0

0 < γ < 1

, and

0 < ϵ < \infty

are real numbers and

V (x)

is the continuous function. Then,

V (x)

will reach into the region

V (x) \leq \min {\frac{ϵ}{(1 - ξ) ϱ_{1}}, {(\frac{ϵ}{(1 - ξ) ϱ_{2}})}^{γ^{- 1}}}

(20)

when

t \to T_{r}

and

0 < ξ < 1

. So, the settling time (20) is obtained as

T_{r} \leq \max {t_{1}, t_{2}}, t_{1} = t_{0} + \frac{ϵ}{ξ ϱ_{1} (1 - ρ)} l n \frac{ϱ_{2} + ξ ϱ_{1} V^{1 - γ} (t_{0})}{ϱ_{2}}, t_{2} = t_{0} + \frac{ϵ}{ϱ_{1} (1 - ρ)} l n \frac{ξ ϱ_{2} + ϱ_{1} V^{1 - γ} (t_{0})}{ξ ϱ_{2}}

(21)

4. Design of adaptive command-filtered backstepping SMC

According to this section, the problem of stabilization for the disturbed and uncertain RIP system is examined. Due to this reason, a method based on the adaptive backstepping SMC scheme is designed. First, the command filter is offered as

x = - a_{1} {| y_{1} - u |}^{\frac{1}{2}} sign (y_{1} - u) + y_{2}

(22)

{\dot{y}}_{1} = x

(23)

{\dot{y}}_{2} = - a_{2} sign (y_{1} - u)

(24)

where

a_{1}, a_{2}

are the positive constants and

u

is the virtual control input. Now, the outputs of command filter are defined as

x_{2}^{c} = y_{1}

(25)

{\dot{x}}_{2}^{c} = x

(26)

Assume that tracking error is defined as

e_{1} (t) = L_{1} z_{1} (t) + L_{2} z_{3} (t)

(27)

e_{2} (t) = L_{3} z_{2} (t) + L_{4} z_{4} (t) - x_{2}^{c}

(28)

Because the differential signals which are created by command filter make the computation more difficult, the error compensating system can be defined as

{\dot{E}}_{1} = - η_{1} E_{1} + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) (x_{2}^{c} - u) + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) E_{2}

(29)

{\dot{E}}_{2} = - η_{2} E_{2} - (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) E_{1}

(30)

where

E_{1}

and

E_{2}

are the state variables of the compensation system, and

η_{1}

and

η_{2}

signify the positive constants.

Lemma 4

(Dong et al., 2011): The responses of the compensation error system (29) and (30) are bounded and satisfy the following inequality

\sum_{i = 0}^{n} E_{i} \leq \frac{k_{p 0}}{2 k_{0}} (1 - e^{- 2 k_{0} (t - T_{1})})

(31)

where

| x_{i}^{c} - u | \leq k_{p 0}

and

k_{0} = \frac{1}{2} η_{i}

Here, the sliding surface is proposed as

σ_{1} = e_{1} - E_{1}

(32)

σ_{2} = e_{2} - E_{2}

(33)

Taking the derivative of equatioins (32) and (33) with respect to time, it can get

{\dot{σ}}_{1} = {\dot{e}}_{1} - \dot{E}_{1}

(34)

{\dot{σ}}_{2} = {\dot{e}}_{2} - {\dot{E}}_{2}

(35)

Now, equations (27) and (28) are substituted into (34) and (35); thus, it results as

{\dot{σ}}_{1} = L_{1} {\dot{z}}_{1} (t) + L_{2} {\dot{z}}_{3} (t) - \dot{E}_{1}

(36)

{\dot{σ}}_{2} = L_{3} {\dot{z}}_{2} (t) + L_{4} {\dot{z}}_{4} (t) - {\dot{x}}_{2}^{c} - {\dot{E}}_{2}

(37)

From equations (11-14) and substituting them into equations (36) and (37), we have

{\dot{σ}}_{1} = L_{1} z_{2} (t) + L_{2} z_{4} (t) + (L_{1} Θ_{1} + L_{2} Θ_{3}) - \dot{E}_{1}

(38)

{\dot{σ}}_{2} = (L_{3} Η_{1} + L_{4} Η_{4}) z_{2} (t) + (L_{3} Η_{2} + L_{4} Η_{5}) z_{4} (t) + (L_{3} Η_{3} + L_{4} Η_{6}) + (L_{3} M_{11} + L_{4} M_{21}) τ_{1} (t) + (L_{3} Θ_{2} + L_{4} Θ_{4}) - {\dot{x}}_{2}^{c} - {\dot{E}}_{2}

(39)

Substituting the terms of

z_{2} (t)

and

z_{4} (t)

from equation (28) into equation (38), we can get

{\dot{σ}}_{1} = L_{1} (\frac{1}{L_{3}} e_{2} - \frac{L_{4}}{L_{3}} z_{4} (t) + \frac{1}{L_{3}} x_{2}^{c}) + L_{2} (\frac{1}{L_{4}} e_{2} - \frac{L_{3}}{L_{4}} z_{2} (t) + \frac{1}{L_{4}} x_{2}^{c}) + (L_{1} Θ_{1} + L_{2} Θ_{3}) - \dot{E}_{1}

(40)

Doing some simplifications and adding and subtracting the expression

(\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) u

, it yields

\begin{array}{l} {\dot{σ}}_{1} = (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) e_{2} + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) (x_{2}^{c} - u) + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) \\ u - \frac{L_{2} L_{3}}{L_{4}} z_{2} (t) - \frac{L_{1} L_{4}}{L_{3}} z_{4} (t) + (L_{1} Θ_{1} + L_{2} Θ_{3}) - \dot{E}_{1} \end{array}

(41)

Therefore, the finite-time adaptive backstepping SMC controller

τ_{1} (t)

and virtual control input

u

are considered as

u = \frac{1}{(\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}})} [- η_{1} e_{1} - κ_{1} σ_{1}^{α} - {\hat{θ}}_{1} sign (σ_{1}) + \frac{L_{2} L_{3}}{L_{4}} z_{2} (t) + \frac{L_{1} L_{4}}{L_{3}} z_{4} (t)]

(42)

τ_{1} (t) = - \frac{1}{L_{3} M_{11} + L_{4} M_{21}} [(L_{3} Η_{1} + L_{4} Η_{4}) z_{2} (t) + (L_{3} Η_{2} + L_{4} Η_{5}) z_{4} (t) + (L_{3} Η_{3} + L_{4} Η_{6}) - {\dot{x}}_{2}^{c} + η_{2} e_{2} + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) e_{1} + κ_{2} σ_{2}^{α} + {\hat{θ}}_{2} sign (σ_{2})]

(43)

where

κ_{1}

and

κ_{2}

are the positive constants and

α = \frac{α_{1}}{α_{2}}

, while

α_{1}

and

α_{2}

signify two odd integers with the condition

0 < α_{1} < α_{2} < 1

. Besides,

{\hat{θ}}_{1}

and

{\hat{θ}}_{2}

denote the estimation of the upper bounds of model uncertainties and exterior perturbations. Thus, the approximation errors are given as

{\tilde{θ}}_{1} = θ_{1} - {\hat{θ}}_{1}

(44)

{\tilde{θ}}_{2} = θ_{2} - {\hat{θ}}_{2}

(45)

Also, adaptation laws are recommended as

{\dot{\hat{θ}}}_{1} = μ_{1}^{- 1} (| σ_{1} | - ν_{1} {\hat{θ}}_{1})

(46)

{\dot{\hat{θ}}}_{2} = μ_{2}^{- 1} (| σ_{2} | - ν_{2} {\hat{θ}}_{2})

(47)

while

μ_{1}

μ_{2}

ν_{1}

, and

ν_{2}

denote the positive constants.

Now, it can be pointed that the objective of this study is the finite-time stability control of the disturbed and uncertain RIP system applying the adaptive backstepping SMC technique.

Theorem 1

If the disturbed and uncertain rotary-inverted-pendulum system is considered as (11–14) and adaptive laws are designed as equations (46) and (47), it can be proved that the sliding surfaces (32) and (33) converge to the region near the origin in the finite-time based on the adaptive backstepping SMC method. Therefore, the stabilization of rotary-inverted-pendulum is satisfied.

Proof

Presume that the candidate Lyapunov functional is constructed as

V_{1} = 0.5 σ_{1}^{2} + 0.5 μ_{1} {\tilde{θ}}_{1}^{2}

(48)

By taking the derivative of (48) with respect to time, it can achieve

{\dot{V}}_{1} = σ_{1} {\dot{σ}}_{1} + μ_{1} {\tilde{θ}}_{1} {\dot{\tilde{θ}}}_{1}

(49)

From (44), we have

{\dot{\tilde{θ}}}_{1} (t) = - {\dot{\tilde{θ}}}_{1} (t)

. Additionally, equation (41) and adaptive control law (46) are substituted into (49), so the following result is obtained

\begin{array}{l} {\dot{V}}_{1} = σ_{1} [(\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) e_{2} + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) \\ (x_{2}^{c} - u) + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) u - \frac{L_{2} L_{3}}{L_{4}} z_{2} (t) \\ - \frac{L_{1} L_{4}}{L_{3}} z_{4} (t) + (L_{1} Θ_{1} + L_{2} Θ_{3}) - {\dot{E}}_{1}] - {\tilde{θ}}_{1} (| σ_{1} | - ν_{1} {\hat{θ}}_{1}) \end{array}

(50)

where substituting the virtual control input

u

(42) into

{\dot{E}}_{1}

(29) leads to

{\dot{V}}_{1} = σ_{1} [(\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) e_{2} - η_{1} e_{1} - κ_{1} σ_{1}^{α} - {\hat{θ}}_{1} sign (σ_{1}) + (L_{1} Θ_{1} + L_{2} Θ_{3}) + η_{1} E_{1} - - (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) E_{2}] - {\tilde{θ}}_{1} (| σ_{1} | - ν_{1} {\hat{θ}}_{1} (t))

(51)

From equations (32) and (33) and Assumption 1, we get

{\dot{V}}_{1} \leq (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) σ_{1} σ_{2} - η_{1} σ_{1}^{2} - κ_{1} σ_{1}^{α + 1} - {\hat{θ}}_{1} | σ_{1} | + θ_{1} | σ_{1} | - {\tilde{θ}}_{1} (| σ_{1} | - ν_{1} {\hat{θ}}_{1} (t))

(52)

where considering equation (44), it yields

{\dot{V}}_{1} \leq (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) σ_{1} σ_{2} - η_{1} σ_{1}^{2} - κ_{1} σ_{1}^{α + 1} + ν_{1} {\tilde{θ}}_{1} {\hat{θ}}_{1}

(53)

Now, the new candidate Lyapunov function is formed as

V_{2} = V_{1} + 0.5 σ_{2}^{2} + 0.5 μ_{2} {\tilde{θ}}_{2}^{2}

(54)

So, the time-derivative of equation (54) is gained as

{\dot{V}}_{2} = {\dot{V}}_{1} + σ_{2} {\dot{σ}}_{2} + μ_{2} {\tilde{θ}}_{2} {\dot{\tilde{θ}}}_{2}

(55)

From (44), we find

{\dot{\hat{θ}}}_{2} (t) = - {\dot{\hat{θ}}}_{2} (t)

; in addition, equation (39) and adaptation controller (47) are substituted into equation (55), so the subsequent equation is resulted as

{\dot{V}}_{2} = {\dot{V}}_{1} + σ_{2} [(L_{3} Η_{1} + L_{4} Η_{4}) z_{2} (t) + (L_{3} Η_{2} + L_{4} Η_{5}) z_{4} (t) + (L_{3} Η_{3} + L_{4} Η_{6}) + (L_{3} M_{11} + L_{4} M_{21}) τ_{1} (t) + (L_{3} Θ_{2} + L_{4} Θ_{4}) - {\dot{x}}_{2}^{c} - {\dot{E}}_{2}] - {\tilde{θ}}_{2} (| σ_{2} | - ν_{2} {\hat{θ}}_{2})

(56)

By substituting equations (30) and (43) into equation (56), we have

{\dot{V}}_{2} = {\dot{V}}_{1} + σ_{2} [- η_{2} e_{2} - (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) e_{1} - κ_{2} σ_{2}^{α} - {\hat{θ}}_{2} sign (σ_{2}) + (L_{3} Θ_{2} + L_{4} Θ_{4}) + η_{2} E_{2} + (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) E_{1}] - {\tilde{θ}}_{2} (| σ_{2} | - ν_{2} {\hat{θ}}_{2})

(57)

Considering equations (32), (33), (55) and Assumption 1, we get

\begin{array}{l} {\dot{V}}_{2} \leq (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) σ_{1} σ_{2} - η_{1} σ_{1}^{2} - κ_{1} σ_{1}^{α + 1} \\ + ν_{1} {\tilde{θ}}_{1} {\hat{θ}}_{1} - (\frac{L_{1}}{L_{3}} + \frac{L_{2}}{L_{4}}) σ_{1} σ_{2} \\ - η_{2} σ_{2}^{2} - κ_{2} σ_{2}^{α + 1} + ν_{2} {\tilde{θ}}_{2} {\hat{θ}}_{2} \end{array}

(58)

After some simplifications, it can achieve

{\dot{V}}_{n} \leq \sum_{i = 1}^{n} - η_{i} σ_{i}^{2} - κ_{i} σ_{i}^{α + 1} + ν_{i} {\tilde{θ}}_{i} {\hat{θ}}_{i} n = 2

(59)

where

ν_{i} {\tilde{θ}}_{i} {\hat{θ}}_{i} = ν_{i} {\tilde{θ}}_{i} (θ_{i} - {\tilde{θ}}_{i}) = ν_{i} {\tilde{θ}}_{i} θ_{i} - ν_{i} {\tilde{θ}}_{i}^{2}

. Here, with respect to Young’s inequality as

(α β \leq \frac{1}{2} α^{2} + \frac{1}{2} β^{2})

, we can get

ν_{i} {\tilde{θ}}_{i} θ_{i} \leq \frac{ν_{i}}{2} {\tilde{θ}}_{i}^{2} + \frac{ν_{i}}{2} θ_{i}^{2}

(60)

which results to

ν_{i} {\tilde{θ}}_{i} {\hat{θ}}_{i} \leq - \frac{ν_{i}}{2} {\tilde{θ}}_{i}^{2} + \frac{ν_{i}}{2} θ_{i}^{2}

(61)

while

\frac{ν_{i}}{2} {\tilde{θ}}_{i}^{2} = \frac{ν_{i}}{4} {\tilde{θ}}_{i}^{2} + \frac{ν_{i}}{4} {\tilde{θ}}_{i}^{2}

and

\frac{ν_{i}}{4} {\tilde{θ}}_{i}^{2} = \frac{ν_{i}}{2} (\frac{1}{2} {\tilde{θ}}_{i}^{2})

. Equation (61) is rewritten as

ν_{i} {\tilde{θ}}_{i} {\hat{θ}}_{i} \leq - \frac{ν_{i}}{4} {\tilde{θ}}_{i}^{2} - \frac{ν_{i}}{2} (\frac{1}{2} {\tilde{θ}}_{i}^{2}) + \frac{ν_{i}}{2} θ_{i}^{2}

(62)

Applying Lemma 1 and considering

Υ_{1} = 1, Υ_{2} = \frac{1}{2} {\tilde{θ}}_{i}^{2}

c_{1} = 1 - δ, c_{2} = δ

, and

ϑ = δ^{\frac{δ}{δ - 1}}

, we get

{| \frac{1}{2} {\tilde{θ}}_{i}^{2} |}^{δ} \leq (1 - δ) δ^{\frac{δ}{δ - 1}} + δ^{\frac{δ}{δ - 1} - \frac{1 - δ}{δ}} | \frac{1}{2} {\tilde{θ}}_{i}^{2} |

(63)

After simplification, it can obtain

{| \frac{1}{2} {\tilde{θ}}_{i}^{2} |}^{δ} \leq (1 - δ) ϑ + | \frac{1}{2} {\tilde{θ}}_{i}^{2} |

(64)

where

0 < δ < 1

. Equation (64) is written as

- | \frac{1}{2} {\tilde{θ}}_{i}^{2} | \leq (1 - δ) ϑ - {| \frac{1}{2} {\tilde{θ}}_{i}^{2} |}^{δ}

(65)

By substituting (65) into (62), it yields

ν_{i} {\tilde{θ}}_{i} {\hat{θ}}_{i} \leq - \frac{ν_{i}}{4} {\tilde{θ}}_{i}^{2} - \frac{ν_{i}}{2} {| \frac{1}{2} {\tilde{θ}}_{i}^{2} |}^{δ} + \frac{ν_{i}}{2} (1 - δ) ϑ + \frac{ν_{i}}{2} θ_{i}^{2}

(66)

Substituting (66) into (59), defining

δ = \frac{α + 1}{2}

and after some simplifications, we have

{\dot{V}}_{n} \leq - \sum_{i = 1}^{n} [η_{i} σ_{i}^{2} + \frac{ν_{i}}{4} λ_{\min} (μ_{i}) {\tilde{θ}}_{i}^{2}] - \sum_{i = 1}^{n} [κ_{i} σ_{i}^{2 δ} + \frac{ν_{i}}{4} λ_{\min} (μ_{i}) {| \frac{1}{2} {\tilde{θ}}_{i}^{2} |}^{δ}] + \sum_{i = 1}^{n} \frac{ν_{i}}{2} (1 - δ) ϑ + \frac{ν_{i}}{2} θ_{i, j}^{2}

(67)

where applying Lemma 2 and with respect to the Lyapunov function, it can achieve

{\dot{V}}_{n} \leq - a V_{n} - b V_{n}^{δ} + ϖ_{n}

(68)

where

a = \min {2 η_{i}, λ_{\min} (μ_{i}) \frac{ν_{i}}{2}}

ω_{n} = \sum_{i = 1}^{n} \frac{ν_{i}}{2} (1 - δ) ϑ + \frac{ν_{i}}{2} θ_{i}^{2}

, and

b = \min {2^{δ} κ_{i}, λ_{\min}^{δ} (μ_{i}) \frac{ν_{i}}{2}}

. Considering Lemma 4, it can obtain

V_{n} \leq \min {\frac{ϖ_{n}}{(1 - ξ) a}, {(\frac{ϖ_{n}}{(1 - ξ) b})}^{ρ^{\frac{1}{δ}}}}

(69)

where the reaching time is bounded as

T_{r} \leq \max {T_{1} + \frac{1}{ξ a (1 - δ)} l n \frac{b + ξ a V^{1 - δ} (T_{1})}{b}, T_{1} + \frac{1}{a (1 - δ)} l n \frac{ξ b + a V^{1 - δ} (T_{1})}{ξ b}}

(70)

According to the Lyapunov functional

V_{n} = \sum_{i = 0}^{n} (0.5 σ_{i}^{2} + 0.5 μ_{i} {\tilde{θ}}_{i}^{2})

and equation (69), we have

0.5 σ_{i}^{2} \leq V_{n} \leq \min {\frac{ϖ_{n}}{(1 - ξ) a}, {(\frac{ϖ_{n}}{(1 - ξ) b})}^{ρ^{\frac{1}{δ}}}}

(71)

Therefore, we obtain

| σ_{i} | \leq \min {\sqrt{\frac{2 ϖ_{n}}{(1 - ξ) a}}, \sqrt{2 {(\frac{ϖ_{n}}{(1 - ξ) b})}^{ρ^{\frac{1}{δ}}}}}

(72)

From equations (32) and (33) and Lemma 4, we have

| e_{i} | \leq | σ_{i} | + | E_{i} | \leq \min {\sqrt{\frac{2 ϖ_{n}}{(1 - ξ) a}} + \sqrt{\frac{k_{p 0}}{2 k_{0}}}, \sqrt{2 {(\frac{ϖ_{n}}{(1 - ξ) b})}^{ρ^{\frac{1}{δ}}}} + \sqrt{\frac{k_{p 0}}{2 k_{0}}}}

(73)

Thus, the trajectories of the tracking errors converge to the origin in the finite time. This theorem is proved completely.

Remark 1

As it can be observed from the above-designed control strategy, the advantages of the proposed method are the decrease of high-order derivative of the virtual control input because of employment of a command filter system and removal of differential signals produced by the command-filter system due to the proposition of the compensation error system. The disadvantage of the suggested method is the existence of a little chattering in the control signals.

Remark 2

For the improvement and decrease of the chattering phenomenon in the command filter (22–24) and control input (43), the hyperbolic tangent function $\tanh$ is recommended instead of the discontinuous $sign$ function.

5. Simulation and experimental results

5.1 Simulation results

In this section, first, the block diagram of the control strategy based on the finite-time backstepping ASMC procedure is shown in Figure 2. In addition, the values of the parameters related to the RIP as well as the design parameters are given in Tables 1 and 2, respectively. Simulation results are compared with the existed method in Huang et al. (2019b) (which is represented by [1] in the simulation figures). From Figure 3, it can be observed that trajectories related to the position of the arm and its velocity are stabilized in the finite time. Also, the stability of the pendulum and its velocity are accomplished as shown in Figure 4. Hence, the stability control of the RIP system is done based on the finite-time backstepping by means of ASMC. According to Figures 5 and 6, it is found that trajectories of the tracking error and compensating error system are converged to zero in the finite time. Besides, according to Figure 7, the trajectories of sliding surfaces are converged to the origin in the finite time by means of the finite-time backstepping ASMC method. The applied torque as control input of the RIP system is displayed in Figure 8, which possess no chattering phenomenon. Estimation of the upper bound of model uncertainties and external disturbance is shown in Figure 9, so it can be said that the approximation operation is done completely. Finally, with respect to Figure 10, the output of command filter $x_{2}^{c}$ is tracked by virtual control input $u$ properly.

Table 1.

RIP system’s parameters.

Parameters	Value	Parameters	Value
$m_{1} (k g)$	0.056	$m_{2}$ (kg)	0.022
$l_{1} (m)$	0.16	$l_{2} (m)$	0.16
$c_{1} (m)$	0.08	$c_{2} (m)$	0.08
$J_{1} (Nm/rads)$	0.001569	$J_{2} (Nm/rads)$	0.0001785

Table 2.

Control parameters.

$[q_{1}, {\dot{q}}_{1}, q_{2}, {\dot{q}}_{2}] = [π, 0, - π / 15,0]$	$[{\hat{θ}}_{1}, {\hat{θ}}_{2}] = [0.01, 0.01]$
$Θ_{i} = 0.01 sin (0.01 t) + 0.01 cos (0.01 t), i = 1, \dots, 4$	$L_{1} = 0.5, L_{2} = 2, L_{3} = 0.25, L_{4} = 1$
$a_{1} = 100, a_{2} = 100$	$η_{1} = 2, η_{2} = 2$
$κ_{1} = 0.1, κ_{2} = 0.1$	$ν_{1} = 0.1, ν_{2} = 0.1$
$μ_{1} = 0.01, μ_{2} = 0.01$	$α = 3 / 5$

5.2 Experimental results

In this part, some experimental outcomes are implemented on the real electro-mechanical engineering control system which is run by TeraSoft company in our research center at National Yunlin University of Science and Technology. The components of this system are shown in Figure 11. Moreover, this system has a support package in MATLAB^® software as an embedded coder toolbox, which supports the processors Texas Instruments C2000. The laboratory environment for implementation of the suggested method on the real RIP system is displayed in Figure 12. The applied voltage for the motor in the control of the RIP system is calculated by the subsequent equation

e = \frac{R_{m}}{K_{t}} [τ_{1} + \frac{K_{t}^{2}}{R_{m}} {\dot{q}}_{1}]

(74)

where

R_{m}

and

K_{t}

are the motor armature resistance and motor torque constants, respectively.

The following experimental results are performed in two different cases. In the first case, the external disturbances are ignored, while the impact of the external disturbances are considered in the second case. In Figure 13(a), the position and velocity of the arm are displayed, while Figure 13(b) shows the results in the existence of external disturbances which have been entered to the system in different directions as specified in the figure. It can be understood that the stability of the system is suitable under exterior perturbations. In Figure 14(a), the position and velocity related to the pendulum are illustrated, while Figure 14(b) presents the results in the appearance of the external disturbance with various direction. It can be observed that the pendulum is stabilized completely using the proposed method against exterior perturbation. In Figure 15, time responses of the applied voltage in the DC motor is displayed. Hence, the validation of the suggested method is proved. According to these experimental results, it can be observed that the recommended method possesses high efficiency in practice.

6. Conclusion

In this study, an adaptive command filter based backstepping sliding mode control technique has been recommended for finite-time stabilization control of the perturbed and uncertain rotary-inverted-pendulum system. Whereas the complex calculation leads to the difficulty in the control strategy, a command filter based on the virtual control input has been considered. Consequently, the output of the command filter is replaced instead of the differential virtual control signal. In addition, the command-filtered backstepping sliding mode control technique is adopted to prove the finite-time reaching of the sliding surface to the origin. Finally, the efficiency of the suggested method has been demonstrated using simulation and experimental results. The extension of the advised control technique for event trigger-based nonlinear dynamical systems in the presence of model uncertainty, input saturation, and exterior disturbance will be investigated in the future research work.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Saleh Mobayen

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