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Abstract

In this study, a novel control design of nonlinear systems has been presented to solve the problem of matched and unmatched perturbations (disturbances) with unknown upper bounds. The proposed controller combined a backstepping control approach with a barrier function integral sliding mode controller (BS-BFISMC). The control design utilized the powerful backstepping control algorithm to decompose the nonlinear system into subsystems to provide global stability, while the barrier function ISMC has been adopted to cope with perturbation, avoid the pre-knowledge of upper bound in the perturbation, and mitigate the chattering effect in the control signal. A stability analysis based on Lyapunov has been conducted for the proposed BS-BFISMC to prove the ultimate boundedness of tracking errors. Another control scheme, which is concerned with the problem of unmatched perturbation, has been also presented: a robust continuous backstepping-based controller (CRBS). A comparison study has been made between the proposed BS-BFISMC and three control structures (conventional backstepping, SMC, and CRBS). Numerical simulation based on MATLAB programing software has been conducted to verify the effectiveness of the proposed control approach for a nonlinear system subjected to both matched and unmatched non-vanishing disturbance and parametric uncertainty. The performance and robustness characteristics of the proposed control scheme and other controllers have been evaluated and compared in a quantitative manner.

Keywords

Backstepping control unmatched perturbation barrier function integral sliding mode unknown upper bound

Introduction

Perturbed systems are widely faced in industrial and medical applications. Examples of such systems include manipulators, hydraulic drives as well as electromechanical systems like induction, synchronous, and DC motors. However, controlling such systems constitutes a vital challenge due to the uncertainty and ambiguity introduced by perturbations. These perturbations, whether they are in the form of uncertainty, disturbance, or a combination of both of them, are crucial in determining the robustness characteristics of the system.^1,2 In the sense of nonlinear control, perturbations are classified, according to their effectiveness in the equilibrium set, as vanishing and non-vanishing perturbations; also, according to their impact in the controller column space, as matched and unmatched perturbations. The worst condition that the perturbation exists is the non-vanishing unmatched perturbation, where the best situation any controller can lead to is uniform ultimate boundedness. The race in control theory is to design a controller that offers the least ultimate bound.³ An additional challenge for nonlinear systems subjected to non-vanishing unmatched perturbation is the unknown upper bound of the perturbation which represents the motivation of this study.

Controlling systems affected by unmatched perturbations with unknown bounds is a challenging problem waiting to be solved. Many attempts are made to cope with the effects of unmatched disturbances by using disturbance observers, adaptive estimators,⁴ high-order sliding mode controllers, and approximated sliding mode controllers. These control methods require a prerequisite of disturbance structure. The sliding mode control approach, for example, must know the upper bound of the perturbation, while the adaptive and nonlinear disturbance observers need a slow variation of perturbation.⁵

The backstepping control is an effective control algorithm for a class of complex systems with nonlinear dynamic characteristics. The basic concept of backstepping control is to step back through the system channels by breaking down a complex control problem into a series of simpler, interconnected control tasks. By iteratively designing virtual controllers for each of these channels, backstepping control became a suitable approach for stabilizing nonlinear systems in strict-feedback form.^3,6 As a nonlinear control technique, the backstepping control strategy has some robustness limitations to handle a wide range of uncertainties, nonlinearities, and disturbances. The tracking performance and asymptotic stability can be achieved for the whole system by stabilizing each subsystem individually. To enhance flexibility and robustness, the backstepping control can be combined with other control techniques like disturbance rejection control or sliding mode control.⁷

Sliding mode control (SMC) is a nonlinear control method, which can control the dynamic system insensitively to the matched perturbations. By defining a sliding variable with relative degree one, two phases of control occur: the reaching phase and the sliding phase. In the reaching phase, the controller will attract the system trajectory to the sliding manifold to begin the sliding phase.⁸ The sliding phase is where the trajectory rides on the sliding manifold and the control is insensitive to the matched perturbations. The good performance acquired from the SMC is on the price of the chattering appearance in the control signal. However, due to chattering phenomena, the SMC cannot be used as a virtual controller in the backstepping control; as it is difficult to track a virtual controller corrupted by high-frequency chattering during the intermediate step of the backstepping control algorithm.⁹ The other concern with SMC is the sensitivity to perturbations in the reaching phase. As such, in order to apply SMC in the backstepping control approach, the chattering problem has to be solved and the reaching phase must be eliminated.^5,10

Integral sliding mode control (ISMC), or what is called in some references “disturbance remover,” is an extension to the conventional SMC where the reaching phase is eliminated. The sliding phase will begin from the first moment of controller operation; consequently, the insensitivity to the matched perturbation will also begin from the first moment of controller operation. However, the chattering will still be the profile of the control action. Also, it is worth mentioning that ISMC requires auxiliary dynamics to be added to the system dynamics to achieve the aforementioned performance characteristics.^11,12

Barrier function-based SMC extension is firstly proposed in the scheme of adaptive sliding mode control to deal with the over-determined adaptive gain problem. Barrier function-based SMC proposes a boundary layer around zero of the sliding variable, and whenever the sliding variable enters that bound, it will never leave it.^13,14 Taking advantage of ISMC, where the sliding variable begins at zero and keeps it for all time, it is possible to incorporate the barrier function for ISMC where an invariant set is defined in the boundary layer around zero of the sliding variable.^15,16 Due to the continuous and differentiable nature of the resulting control law and the inherent adaptive mechanism of the BFISMC that rests on the condition of knowing the upper bound of the perturbation, one may apply this as a virtual controller within the steps of the backstepping control algorithm. Hence, the problem of unmatched non-vanishing perturbed systems can be dealt with. Accordingly, a pre-specified small uniform ultimate bound can be acquired by using a backstepping-based barrier function integral sliding mode controller.

In nonlinear control theory, attention has been paid to backstepping controllers with unmatched disturbance. Previous works have addressed this control concern by conducting different control approaches like adaptive higher order SMC disturbance estimation,¹⁷ nonlinear disturbance observer-based backstepping,¹⁸ approximated SMC-based backstepping,¹⁹ ISMC mixed $H_{\infty}$ control,²⁰ and quasi-continuous HOSM hierarchal controller.²¹ Also, there are relevant and recent works that are devoted to unmatched perturbation systems. Observer-based adaptive control scheme based on SMC and adaptive backstepping techniques is designed to stabilize the rotor angle and speed for a synchronous machine connected to an infinite bus.²² In Ref.,²³ a backstepping control scheme optimized by an adaptive SMC and an adaptive nonlinear PI structure has been proposed for the electrohydraulic elastic manipulator, which includes an electrohydraulic servo system and variable stiffness mechanism. In Ref.,²⁴ combined backstepping control and SMC have been designed based on extended disturbance observer to accurately estimate the complex disturbance term for integrated missile guidance control system. In Ref.,²⁵ a backstepping integral sliding model control based on extended state observer (ESO) to estimate the unmatched uncertainties for speed control of reversible cold strip rolling mill. In Ref.,²⁶ high-order ESO and block backstepping sliding mode for quad-rotor aircraft were presented. In Ref.,²⁷ a high-order sliding mode and backstepping controller have been fused for tracking control of a class of unknown nonlinear systems with a strict feedback form. In Ref.,²⁸ an extended multiple sliding surface controller based on a lumped perturbation observer is proposed for a system with matched and unmatched uncertainties. In Ref.,²⁹ a combination of a backstepping algorithm and adaptive super-twisting sliding mode control has been developed for altitude and velocity tracking control of air-breathing hypersonic vehicles. In Ref.,³⁰ control of double-rod hydraulic servo systems in the presence of largely unknown matched and mismatched disturbances is proposed by applying backstepping-based extended state observers for each channel of load dynamics and pressure dynamics for the hydraulic system. In Ref.,³¹ a nonlinear disturbance observer-based adaptive backstepping SMC has been presented to control wheel slip tracking of a car-like robot. In Ref.,³² a class of adaptive nonlinear disturbance observers is developed to estimate the unmatched perturbations with the SMC for integrated missile guidance and control system. In Ref.,³³ a robust adaptive neural network output feedback controller has been presented for DC–DC buck converter system using backstepping control in the presence of perturbations. In Ref.,³⁴ a control design based on backstepping and super-twisting control is developed for robot manipulators with matched and mismatched uncertainties. In Ref.,³⁵ Backstepping control based on neural network approximation is applied with a nonsingular fast terminal sliding mode controller for control of spacecraft formation flight. In Ref.,³⁶ Backstepping control based on the Adaptive function approximation technique was presented for haptic robot. Combining a backstepping sliding mode with time delay estimation for the problem of high-precision joint position trajectory tracking in the presence of matched and unmatched perturbations of the humanoid NAO robot was solved in Ref.³⁷ In Ref.,³⁸ a full-order terminal SMC backstepping-like method is proposed for the rectifier side and the inverter side of the soft open point. In Ref.,³⁹ an adaptive multiple-surface sliding mode technique is applied. Disturbance-observer-based adaptive finite-time variable surface control for the PMSM with unmatched perturbation based on a backstepping framework is presented in Ref.⁴⁰ Backstepping based on reduced order extended state observer controller is proposed in Ref.⁴¹

Despite that all previous researchers in the literature have conducted various control structures to tackle the problem of unmatched perturbation, they could not address the problem of unknown upper bound in the perturbation. This was the motivation key of this study, where a novel control design has been proposed to effectively solve this problem.

It is interesting to combine the barrier function-based integral sliding mode control and backstepping control to achieve the global ultimate boundedness of a controlled system suffering from unmatched perturbation with unknown bounds. In this study, the tracking control problem of the nonlinear system subjected to matched and unmatched perturbations with unknown upper bounds will be solved by proposing a novel backstepping-based barrier function integral sliding mode controller (BS-BFISMC). Also, in addition to the conventional backstepping controller and the sliding mode controller, the suggested controller will be compared with a robust controller that aims to manage the unmatched perturbation in the framework of backstepping control. The latter controller is termed a robust continuous backstepping-based controller (CRBS). Therefore, the contribution of this study is manifold and can be summarized by the following salient points:

1. Design a novel backstepping-based barrier function integral sliding mode control (BFISMC) based on the Lyapunov stability. This novel design will provide the following salient control features:

□ No need for perturbations upper bound.

□ No need for constraint on robust control gain (gain-less or adaptive gain).

□ Chattering-free controller; as there is no need for filters to eliminate high-frequency chattering in the control signal.

□ Pre-specified estimation error bound.

□ Perturbation estimation error convergence from $t \geq 0$ .

2. Assessment of estimation and compensation error based on matched and unmatched perturbations using Lyapunov stability analysis, which leads to specifying the uniform ultimate bounds individually.

3. Conduct a comparison study between the proposed control design and the other three controllers: conventional backstepping control, sliding mode control, and robust continuous backstepping-based controller.

The last point in the contribution requires the development of a backstepping control algorithm, nonlinear robust continuous backstepping-based control and the sliding mode control based on Lyapunov stability analysis for the system subjected to matched and unmatched perturbations.

The proposed controller can handle a wide range of electromechanical systems such as DC motors and I.M motors and systems in the strict feedback form subjected to unmatched perturbations with unknown upper bounds such as manipulators and crane systems.^42–48

In addition to this section, the rest of the article comes with five sections. In Section 2, the formulation of the control problem and assumptions are presented, Section 3 presents the control design based on Lyapunov stability analysis, Section 4 conducts numerical simulation and discussion of simulation results using an illustrative nonlinear dynamic system, and finally, Section 5 highlights the concluded points.

Problem formulation

For the system of the following dynamics in the strict feedback form

\begin{matrix} {\overset{\cdot}{x}}_{1} = f_{1}' (t, x_{1}) + x_{2} + d_{1} (t) \\ {\overset{\cdot}{x}}_{2} = f_{2}' (t, x) + u + d_{2} (t) \end{matrix}

Where $f_{1}' (t, x_{1}) = f_{1} (t, x_{1}) + Δ f_{1} (t, x_{1})$ is a smooth nonlinear function with parametric uncertainty, $Δ f_{1} (t, x_{1})$ represents the unmatched uncertainty while, $f_{1} (t, x_{1})$ represent the nominal part, and $d_{1} (t)$ represents the unmatched disturbance, let $w_{1} = Δ f_{1} (t, x_{1}) + d_{1} (t)$ represents the unmatched perturbation, and similarly for $f_{2}' (t, x_{1}) = f_{2} (t, x_{1}) + Δ f_{2} (t, x_{1})$ , $w_{2} = Δ f_{2} (t, x_{1}) + d_{2} (t)$ where $Δ f_{2} (t, x_{1})$ and $d_{2} (t)$ represent the matched uncertainty and disturbance respectively and $w_{2}$ represent the matched perturbation, then the above dynamic system can be written as

{\overset{\cdot}{x}}_{1} = f_{1} (t, x_{1}) + x_{2} + w_{1}

(1)

{\overset{\cdot}{x}}_{2} = f_{2} (t, x) + u + w_{2}

(2)

where the state variables $x_{1}$ , $x_{2}$ are available, $u$ represent the control action, $f_{1} (t, x_{1}), f_{2} (t, x)$ are smooth nominal known nonlinear functions with certain parameters, $w_{2}$ and $w_{1}$ represent the matched and unmatched perturbations respectively and they are bounded and their first derivatives are also bounded.

The control aim is to design a robust controller based on backstepping approach such that the: $| x_{1} | \leq ρ (δ)$ , where $x_{1}$ may generalized as an tracking error or a regulating control problem, $ρ$ is a $κ$ -class function which can be made arbitrary small via a suitable selection of the controller parameters and $δ$ is the upper bound of the perturbations or their first derivative (such bounds are assumed to exist but not necessary to be known).

Definition 1 ³ :

A continuous function $ρ : [0, c) \to [0, \infty)$ is said to belong to class $κ$ if it is strictly increasing and $ρ (0) = 0$ , also, it is said to belong to class $κ_{\infty}$ if $c = \infty$ and $ρ (x) \to \infty$ as $x \to \infty$ .

A continuous function $B_{κ_{L}} (p, q) : [0, c) \times [0, \infty) \to [0, \infty)$ is said to belong to class $κ_{L}$ if ∀ fixed $q$ , $B_{κ_{L}} (p, q) \in$ class $κ$ w.r.t $p$ , and if ∀ fixed $p$ , $B_{κ_{L}} (p, q) \to 0$ as $q \to \infty$ .

Theorem 1 ³ : For the system of equations (1) and (2), a Lyapunov candidate $V : R^{2} \to R$ satisfy: $f_{l} ‖ (x) ‖ \leq V \leq f_{u} ‖ (x) ‖$ , $\frac{\partial V}{\partial t} + \frac{\partial V}{\partial x_{i}} f_{i} \leq - W ‖ (x) ‖, \forall x \geq μ > 0$ , $\forall t > 0$ and $\forall x \in R^{2}$ , where $f_{l} ‖ (x) ‖, f_{u} ‖ (x) ‖ \in class κ_{\infty}$ , $W (x)$ is positive definite function; considering a surface level $c > 0 \exists μ < f_{u}^{- 1} (f_{l} (c))$ , $\exists B_{κ_{L}} \in class κ_{L}$ and $\forall x (t_{0})$ satisfy $‖ x (t_{0}) ‖ \leq f_{u}^{- 1} (f_{l} (c))$ , $\exists T > 0$ such that

‖ x ‖ \leq B_{κ_{L}} (‖ x (t_{0}) ‖, t - t_{0}), \forall t_{0} \leq t \leq t_{0} + T

‖ x ‖ \leq f_{l}^{- 1} (f_{u} (μ)), \forall t \geq t_{0} + T

The proof of Theorem 1 can be found in Ref.³

Controller design

Backstepping controllers’ extensions to deal with an unmatched disturbance with unknown upper bound have acquired a lot of interest in the nonlinear control theory community. In what follows, the backstepping controller will be designed for systems affected by the non-vanishing matched and unmatched perturbation with unknown upper bound. A solution to this situation will be proposed as a comparison analysis among two controllers in addition to the conventional backstepping, the proposed controllers are named, robust continuous backstepping-based controller (CRBS), and the novel backstepping-based BFISMC (BS-BFISMC). The ultimate bound of the error and the control effort will be the key characteristics to specify the usefulness of the controller.

Backstepping control

The basic concept behind backstepping control involves step back through the system channels, breaking down a complex control problem effectively into a series of simpler interconnected control tasks.^46,47 By iteratively designing a virtual controller for each of these channels, backstepping control became a suitable approach for stabilizing nonlinear systems in what is called strict-feedback form.³

Assumptions:

The unmatched perturbation is bounded:

| w_{1} | \leq α

(3)

The lumped perturbations are defined as:

ζ_{1} = a w_{1} + {\overset{\cdot}{f}}_{1} (t, x_{1}) + w_{2}

(4)

‖ D_{1} ‖_{2} = \sqrt{{| w_{1} |}^{2} + {| ζ_{1} |}^{2}}

(5)

Proposition 1: For the system of the equations (1) and, (2), satisfy assumptions of equations (3–5) using the control law of:

u = - f_{2} (t, x) - (1 - a^{2}) x_{1} - 2 a z_{2}

(6)

will drive the states to a uniform ultimate bound of: $‖ (x, z) ‖ \leq \frac{1}{θ} ‖ D_{1} ‖_{2}$ ,where $‖ D_{1} ‖_{2}$ is given in equation (5).

Proof: For the system described by equations (1) and (2), one can define the virtual controller $x_{2}^{*}$ for the sub-system of equation (1) as:

x_{2}^{*} = - a x_{1} - f_{1} (t, x_{1})

(7)

Generally, the virtual controller contains two parts: the first part ( $- a x_{1})$ , where $a > 0 \in R$ is a design parameter which induces an exponential convergence rate of $x_{1}$ to the zero. The second part of the virtual controller represents the additive counterpart of all unwanted terms in the upper-stepped channels. Nevertheless, according to,³ $x_{2}$ must be forced to follow the virtual controller $x_{2}^{*}$ . So, using Lyapunov of the unperturbed individual channel of equation (1) with replacement of $x_{2}$ by $x_{2}^{*}$ of equation (7) as,

V_{x_{1}} = \frac{1}{2} x_{1}^{2}

(8)

\begin{matrix} {\overset{\cdot}{V}}_{x_{1}} = x_{1} {\overset{\cdot}{x}}_{1} = x_{1} (f_{1} (t, x) + x_{2}^{*}) = x_{1} (- a x_{1}) \\ = - {ax}_{1}^{2} < 0 \forall x_{1} \neq 0 \end{matrix}

(9)

Now let the error of the second channel defined as $z_{2} = x_{2} - x_{2}^{*}$ and that means

{\overset{\cdot}{x}}_{1} = - a x_{1} + z_{2} + w_{1}

(10)

The interesting matter of the choice of virtual controller is that all the controller terms must be certain and that’s the reason why the backstepping is sensitive to the non-vanishing unmatched perturbation.

Now, the error dynamics can be considered as ${\overset{\cdot}{z}}_{2} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{2}^{*}$ then

{\overset{\cdot}{z}}_{2} = f_{2} (t, x) + u - a^{2} x_{1} + a z_{2} + ζ_{1}

(11)

where $ζ_{1}$ the lumped perturbations is defined as in the equation (4), whereby defining a Lyapunov candidate of the form

V (x_{1}, z_{2}) = V_{x_{1}} + \frac{1}{2} z_{2}^{2} = \frac{1}{2} (x_{1}^{2} + z_{2}^{2})

(12)

The time derivative of equation (12) can be obtained as

\overset{\cdot}{V} (x_{1}, z_{2}) = x_{1} {\overset{\cdot}{x}}_{1} + z_{2} {\overset{\cdot}{z}}_{2}

(13)

by substitute equations (1) and (11) in the equation (13) one can obtain

\begin{matrix} \overset{\cdot}{V} (x_{1}, z_{2}) = - {ax}_{1}^{2} + x_{1} w_{1} + z_{2} (f_{2} (t, x) + (1 - a^{2}) x_{1} \\ + a z_{2} + u + ζ_{1}) \end{matrix}

(14)

then according to the backstepping control theory, the control action can be chosen as in equation (6), and that’s leads to,

\begin{matrix} \overset{\cdot}{V} (x_{1}, z_{2}) = - a (x_{1}^{2} + z_{2}^{2}) + x_{1} w_{1} + z_{2} ζ_{1} \\ \leq - a ‖ (x, z) ‖_{2}^{2} + \sqrt{{w_{1}}^{2} + ζ_{1}^{2}} ‖ (x, z) ‖_{2} \end{matrix}

(15)

Reconsidering equation (5), then according to,³ Let $0 < θ < 1$ then,

\overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (1 - θ) ‖ (x, z) ‖_{2}^{2} + ‖ D_{1} ‖_{2} ‖ (x, z) ‖_{2} - a θ ‖ (x, z) ‖_{2}^{2}

(16)

\overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (1 - θ) ‖ (x, z) ‖_{2}^{2}, \forall ‖ (x, z) ‖_{2} > \frac{1}{a θ} ‖ D_{1} ‖_{2}

(17)

Hence, according to Theorem 1, the ultimate bound is

‖ (x, z) ‖_{2} \leq \frac{1}{a} ‖ D_{1} ‖_{2}

(18)

▪

The ultimate bound of equation (18) reflects the challenging problem of this study and the backstepping control could not successfully manage the upper system independently. However, rigorous work is required to guarantee this issue. The ultimate bound can be satisfied by the backstepping control approach using the design parameter $a$ , and the upper bound of $‖ D_{1} ‖_{2}$ which cannot be arbitrarily minimized.

Robust continuous backstepping-based controller (CRBS)

Considering the unmatched upper bound in the virtual controller is the key feature of the current enhancement to the backstepping control design to result in what is called backstepping-based robust continuous virtual control law (CRBS).⁴⁹

Assumptions:• The unmatched perturbation is bounded, recall equation (3), but $α > 0 \in R$ must be known.

• The lumped perturbations are defined as:

ζ_{2} = w_{2} + (a + \frac{α^{2}}{2 ϵ}) w_{1} + {\overset{\cdot}{f}}_{1} (t, x)

(19)

with $| ζ_{2} | \leq β$ where $β > 0 \in R$ must be known, the overall perturbation norm will be

‖ D_{2} ‖_{2} = \sqrt{{| w_{1} |}^{2} + {| ζ_{2} |}^{2}}

(20)

Proposition 2: For the system of the equations (1) and (2), using the control law of the form:

u = u_{1} + u_{2}

(21)

where,

u_{1} = - f_{2} (t, x) - (1 - {(a + \frac{α^{2}}{2 ε})}^{2}) x_{1} - (2 a + \frac{α^{2}}{2 ε}) z_{2}

(22)

and,

u_{2} = - \frac{β^{2} e_{2}}{2 ε}

(23)

will drive the states to a uniform ultimate bound of: $‖ (x, z) ‖_{2} \leq \sqrt{\frac{ε}{a}}$ .

Proof:

From equation (1), the virtual controller of the upper sub-system is defined as⁴⁹:

x_{2}^{*} = - a x_{1} - f_{1} (t, x_{1}) - \frac{x_{1} α^{2}}{2 ε}

(24)

where $a > 0 \in R and 1 >> ε > 0 \in R$ . are design parameters. The same assessment of equations (7–9) then the error dynamics of the upper channel can be rewritten as

{\overset{\cdot}{x}}_{1} = - a x_{1} + z_{2} + w_{1} - \frac{x_{1} α^{2}}{2 ε}

(25)

and hence,

\begin{matrix} {\overset{\cdot}{z}}_{2} = f_{2} (t, x) + u + w_{2} + a {\overset{\cdot}{x}}_{1} + {\overset{\cdot}{f}}_{1} (t, x_{1}) + \frac{{\overset{\cdot}{x}}_{1} α^{2}}{2 ε} \\ {\overset{\cdot}{z}}_{2} = f_{2} (t, x) + u + w_{2} - {(a + \frac{α^{2}}{2 ε})}^{2} x_{1} + (a + \frac{α^{2}}{2 ε}) z_{2} \\ + (a + \frac{α^{2}}{2 ε}) w_{1} + {\overset{\cdot}{f}}_{1} (t, x) \end{matrix}

(26)

Using the lumped perturbations of equation (19) and controller of equations (21–23), then,

{\overset{\cdot}{z}}_{2} = - a z_{2} - x_{1} + ζ_{2} - \frac{β^{2} z_{2}}{2 ε}

(27)

Using the same Lyapunov candidate of equation (12) ends to:

\begin{matrix} \overset{\cdot}{V} (x_{1}, z_{2}) = x_{1} (- a x_{1} + z_{2} + w_{1} - \frac{x_{1} α^{2}}{2 ε}) \\ + z_{2} (- a z_{2} - x_{1} + ζ_{2} - \frac{β^{2}}{2 ε} z_{2}) \end{matrix}

(28)

\overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (x_{1}^{2} + z_{2}^{2}) + x_{1} | w_{1} | - \frac{x_{1}^{2} α^{2}}{2 ε} + z_{2} | ζ_{2} | - \frac{β^{2} z_{2}^{2}}{2 ε}

(29)

Consider the fact of (Young’s inequality) of $x^{2} + y^{2} \geq 2 xy$ , then

\begin{matrix} \frac{x_{1}^{2} α^{2}}{2 ε} + \frac{ε^{2}}{2 ε} \geq 2 \frac{x_{1} α ε}{2 ε} \geq x_{1} α \geq x_{1} | w_{1} | \\ \frac{ε}{2} \geq x_{1} | w_{1} | - \frac{x_{1}^{2} α^{2}}{2 ε} \end{matrix}

(30)

And similarly for the lower sub-system

\frac{ε}{2} \geq z_{2} | ζ_{2} | - \frac{z_{2}^{2} β^{2}}{2 ε}

(31)

Then the total Lyapunov candidate time derivative will be assessed to the uniform ultimate boundedness as

\begin{matrix} \overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (x_{1}^{2} + z_{2}^{2}) + \frac{ε}{2} + \frac{ε}{2} \\ \overset{\cdot}{V} (x_{1}, z_{2}) \leq - a ‖ (x, z) ‖_{2}^{2} + ε \end{matrix}

(32)

Let $0 < θ < 1$ then,

\begin{matrix} \overset{\cdot}{V} (x_{1}, z_{2}) \leq - a ‖ (x, z) ‖_{2}^{2} + ε + a θ ‖ (x, z) ‖_{2}^{2} - a θ ‖ (x, z) ‖_{2}^{2} \\ \overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (1 - θ) ‖ (x, z) ‖_{2}^{2} - a θ ‖ (x, z) ‖_{2}^{2} + ε \\ \overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (1 - θ) ‖ (x, z) ‖_{2}^{2} \forall ‖ (x, z) ‖_{2} > \sqrt{\frac{ε}{a θ}} \end{matrix}

(33)

Then according to Theorem 1, the ultimate bound is

‖ (x, z) ‖_{2} \leq ρ (\frac{ε}{a})

(34)

where $ρ (\frac{ε}{a}) = \sqrt{\frac{ε}{a}}$ , then,

‖ (x, z) ‖_{2} \leq \sqrt{\frac{ε}{a}}

(35)

▪

The ultimate bound of equation (35) represents a considerable achievement where the bound depends on some prescribed parameter $ε$ . That magic solution was a trade-off that costs the over-estimation of the control action which relates to the upper bound of the perturbations which also, required to be known.

Backstepping-based BFISMC controller

There are two issues in fusing ISMC as virtual control with a backstepping approach. The first one is the discontinuous control action due to ISMC, while the unknown perturbation upper bound is the second issue in such integration. Barrier function and integral sliding mode control may collaborate to solve these issues by combining them into the backstepping regime. This results in what so called “backstepping-based barrier function integral sliding mode control (BS-BFISMC).” Firstly, the concept of Barrier Function (BF) will be introduced.¹³

Definition 2: For some $ϵ > 0,$ which is known and fixed, the BF can be defined as an even continuous function $k_{b f} : x \in (- ϵ, ϵ) \to k_{b f} (x) \in [b, \infty]$ strictly increasing on $[0, ϵ)$ .

$lim_{| x | \to ϵ} k_{bf} (x) = + \infty,$

$k_{bf} (x)$ has a unique minimum at zero and $k_{bf} (0) = b \geq 0$ .

In this study, a class of semi-definite BF is considered as $k_{bf} (x) = \frac{| x |}{ϵ - | x |}$ .

The main result of the present work will be presented after the next proposition, which derives the ultimate bound for a perturbed single input single output (SISO) system, where the controller is designed based on ISMC with the use of a barrier function.

The scalar SISO system is given by:

\overset{\cdot}{y} = f (t, y) + u + w

(36)

Where $y \in R$ , $| w | < μ_{1}$ and $\overset{\cdot}{w} < μ_{2}$ .is bounded perturbation.

Proposition 3: Consider the following ISMC uses barrier function for the system of equation (36),

u = - f (t, y) - ay - \frac{λ s}{ϵ - | s |}

(37)

where $s$ is the sliding variable, and $a > 0 \in R, λ > 0 \in R$ and $1 << ϵ > 0 \in R$ are design parameters

s = y + r, r (t_{0}) = - y (t_{0}) \Rightarrow s (t_{0}) = 0 \forall t \geq t_{0}

(38)

And $r$ is the auxiliary variable such that:

\overset{\cdot}{r} = ay

(39)

Then the closed loop system $\forall t \geq t_{0}$ is given by

\overset{\cdot}{y} = - ay + \overset{\cdot}{s}

(40)

where,

| s | \leq \frac{ϵ μ_{1}}{λ + μ_{1}}

(41)

And also,

| \overset{\cdot}{s} | \leq \frac{ϵ μ_{2}}{λ}

(42)

As result, the output $y$ is input to state stable (ISS) with respect to $s$ and the ultimate bound given by

| y | \leq \frac{ϵ μ_{2}}{a λ}

(43)

Proof : by differentiating $s$ in equation (38) and using the controller of equation (37) and the auxiliary dynamics of equation (39), $\overset{\cdot}{s}$ becomes

\overset{\cdot}{s} = w - \frac{λ s}{ϵ - | s |}

(44)

Accordingly, the system model is given in the following form $\forall t \geq t_{0}$

\overset{\cdot}{y} = - ay + w - \frac{λ s}{ϵ - | s |} = - ay + \overset{\cdot}{s}

(45)

where $\overset{\cdot}{s}$ is considered as the perturbation term in equation (45). To this end we need to estimate the bound on $\overset{\cdot}{s}$ and then prove that the system of equation (45) is ISS. To get the bound on $\overset{\cdot}{s} \forall t \geq t_{0}$ , we use the following Lyapunov candidate

H_{1} = \frac{{\overset{\cdot}{s}}^{2}}{2}, {\overset{\cdot}{H}}_{1} = \overset{\cdot}{s} \overset{\cdot\cdot}{s}

(46)

where, $\overset{\cdot}{s} = w - \frac{λ s}{ϵ - | s |}$ then, $\overset{\cdot\cdot}{s} = - \frac{ϵ λ \overset{\cdot}{s}}{{(ϵ - | s |)}^{2}} + \overset{\cdot}{w}$ , then

{\overset{\cdot}{H}}_{1} = - \frac{ϵ λ {\overset{\cdot}{s}}^{2}}{{(ϵ - | s |)}^{2}} + \overset{\cdot}{s} \overset{\cdot}{w} \leq - \frac{ϵ λ {\overset{\cdot}{s}}^{2}}{{(ϵ - | s |)}^{2}} + | \overset{\cdot}{s} | μ_{2}

(47)

But $s$ is also bounded $\forall t \geq t_{0}$ as can be proved by selecting the following Lyapunov function

H_{2} = \frac{s^{2}}{2}, {\overset{\cdot}{H}}_{2} = s \overset{\cdot}{s} = s (w - \frac{λ s}{ϵ - | s |}) \leq - \frac{λ s^{2}}{ϵ - | s |} + | s | μ_{1}

(48)

{\overset{\cdot}{H}}_{2} \leq - (1 - θ) \frac{λ s^{2}}{ϵ - | s |} - θ \frac{λ s^{2}}{ϵ - | s |} + | s | μ_{1}

(49)

where, $0 < θ < 1$ , Then

{\overset{\cdot}{H}}_{2} \leq - (1 - θ) \frac{λ s^{2}}{ϵ - | s |}, \forall | s | > \frac{ϵ μ_{1}}{θ λ + μ_{1}}

(50)

To show that the bound above is valid $\forall t \geq t_{0}$ and consequently $| s | < ϵ$ will not leave $ϵ \forall t \geq t_{0}$ , it is enough to observe that $s = 0$ at $t = t_{0}$ , therefore, the set $| s | < ϵ$ is invariant set. Back to equation (47) which becomes

\begin{matrix} {\overset{\cdot}{H}}_{1} \leq - \frac{λ {\overset{\cdot}{s}}^{2}}{ϵ} + | \overset{\cdot}{s} | μ_{2} \leq - (1 - θ) \frac{λ {\overset{\cdot}{s}}^{2}}{ϵ} - θ \frac{λ {\overset{\cdot}{s}}^{2}}{ϵ} + | \overset{\cdot}{s} | μ_{2} \\ {\overset{\cdot}{H}}_{1} \leq - (1 - θ) \frac{λ {\overset{\cdot}{s}}^{2}}{ϵ}, \forall | \overset{\cdot}{s} | > \frac{ϵ μ_{2}}{λ θ} \end{matrix}

(51)

Then

| \overset{\cdot}{s} | \leq \frac{ϵ μ_{2}}{λ}

(52)

Eventually, to estimate the ultimate bound on $y$ the following Lyapunov is used

\begin{matrix} V_{y} = \frac{y^{2}}{2}, {\overset{\cdot}{V}}_{y} = y \overset{\cdot}{y} = - a y^{2} + y \overset{\cdot}{s} \leq - a (1 - θ) y^{2} \\ + a θ y^{2} + | y | | \overset{\cdot}{s} | \end{matrix}

(53)

{\overset{\cdot}{V}}_{y} \leq - a y^{2}, \forall | y | > \frac{| \overset{\cdot}{s} |}{a θ}

(54)

Then using equation (52)

| y | \leq \frac{ϵ μ_{2}}{a λ}

(55)

▪

Regarding the results of Proposition 3 the backstepping-based BFISMC can be designed and assessed in the following proposition.

Assumptions:

The unmatched perturbation is bounded, recall equation (3), but $α > 0 \in R$ be unknown, and $| {\overset{\cdot}{w}}_{1} | < α_{1} > 0 \in R$ be unknown.

The lumped perturbations are defined as:

ζ_{3} = w_{2} + a w_{1} + {\overset{\cdot}{f}}_{1} (t, x) + {\overset{\cdot}{\hat{w}}}_{1}

(56)

with $| {\overset{\cdot}{ζ}}_{3} | \leq β_{1}$ where $β_{1} > 0 \in R$ be unknown, the overall perturbation norm will be

‖ D_{3} ‖_{2} = \sqrt{{| α_{1} |}^{2} + {| β_{1} |}^{2}}

(57)

Proposition 4: For the system of the equations (1) and (2), using the control law of the form:

u = u_{1} + u_{2} + u_{3}

(58)

u_{1} = - f_{2} (t, x) + a^{2} x_{1} - 2 a z_{2} + a {\hat{w}}_{1}

(59)

{\hat{w}}_{1} = \frac{λ_{1} s_{1}}{ϵ_{1} - | s_{1} |}

(60)

u_{2} = - \frac{λ_{2} s_{2}}{ϵ_{2} - | s_{2} |}

(61)

u_{3} = - x_{1}

(62)

with the sliding variables $s_{1} = x_{1} + r_{1}$ , $s_{2} = z_{2} + r_{1}$ and the auxiliary dynamics of:

{\overset{\cdot}{r}}_{1} = a x_{1} - z_{2}

(63)

{\overset{\cdot}{r}}_{2} = a z_{2} - u_{3}

(64)

and the design parameters $a > 0 \in R$ , $λ_{1} > 0 \in R$ , $1 >> ϵ_{1} > 0 \in R$ , $1 >> ϵ_{2} > 0 \in R$ , and $λ_{2} > 0 \in R$ with assumption of $\frac{ϵ}{λ} = \frac{ϵ_{1}}{λ_{1}} = \frac{ϵ_{2}}{λ_{2}}$ , will drive the state to a controllable uniform ultimate bound of: $(x, z) \leq \frac{ϵ}{a λ} {D_{3}}_{2}$ which can be made arbitrary small by suitable selection of the control parameters $ϵ, λ$ .

Proof: Reconsider equation (3) of

{\overset{\cdot}{x}}_{1} = f_{1} (t, x_{1}) + x_{2} + w_{1}

(64)

A corporation design of the Proposition 1 backstepping control procedure and the result of the Proposition 3 of BFISMC. Let $z_{2} = x_{2} - x_{2}^{*}$ and that’s mean

x_{2}^{*} = - a x_{1} - f_{1} (t, x_{1}) - {\hat{w}}_{1}

(65)

where ${\hat{w}}_{1}$ as in equation (38). then Considering and analysis same to the procedure of equations (36–55) regarding the sliding variable $s_{1}, s_{1}$ , and auxiliary dynamic of equations (63) and (64), then

{\overset{\cdot}{x}}_{1} = - a x_{1} + z_{2} + {\overset{\cdot}{s}}_{1}

(66)

| s_{1} | \leq \frac{ϵ_{1} α_{1}}{λ_{1} + α_{1}}

(67)

| {\overset{\cdot}{s}}_{1} | \leq \frac{ϵ_{1} α_{1}}{λ_{1}}

(68)

and,

{\overset{\cdot}{z}}_{2} = f_{2} (t, x) - a {\hat{w}}_{1} - a^{2} x_{1} + a z_{2} + u + ζ_{3}

(69)

Considering control action of equations (60–62) leads to:

{\overset{\cdot}{z}}_{2} = - a z_{2} - x_{1} + {\overset{\cdot}{s}}_{2}

(70)

| s_{2} | \leq \frac{ϵ_{2} | ζ_{3} |}{λ_{2} + | ζ_{3} |}

(71)

| {\overset{\cdot}{s}}_{2} | \leq \frac{ϵ_{2} | {\overset{\cdot}{ζ}}_{3} |}{λ_{2}}

(72)

For more one time, consider the Lyapunov candidate of equation (12) then:

\overset{\cdot}{V} (x_{1}, z_{2}) = x_{1} (- a x_{1} + z_{2} + {\overset{\cdot}{s}}_{1}) + z_{2} (- a z_{2} - x_{1} + {\overset{\cdot}{s}}_{2})

(73)

\overset{\cdot}{V} (x_{1}, z_{2}) = - a (x_{1}^{2} + z_{2}^{2}) + {\overset{\cdot}{s}}_{1} x_{1} + {\overset{\cdot}{s}}_{2} z_{2}

(74)

The ultimate bound of the perturbation estimation error $| {\overset{\cdot}{s}}_{1} |$ and $| {\overset{\cdot}{s}}_{2} |$ of equations (68) and (72) are substituted in the above equation, then

\overset{\cdot}{V} (x_{1}, z_{2}) \leq - a ‖ (x, z) ‖_{2}^{2} + \frac{ϵ}{λ} ‖ D_{3} ‖_{2} ‖ (x, z) ‖_{2}

(75)

where, $‖ D_{3} ‖_{2}$ defined in equation (57) and let $0 < θ < 1$ then

\begin{matrix} \overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (1 - θ) ‖ (x, z) ‖_{2}^{2} + \frac{ϵ}{λ} ‖ D_{3} ‖_{2} ‖ (x, z) ‖_{2} \\ - a θ ‖ (x, z) ‖_{2}^{2} \\ \overset{\cdot}{V} (x_{1}, z_{2}) \leq - a (1 - θ) ‖ (x, z) ‖_{2}^{2} \forall ‖ (x, z) ‖_{2} > \frac{ϵ}{a λ θ} ‖ D_{3} ‖_{2} \end{matrix}

(76)

Then, according to Theorem 1 the ultimate bound is

‖ (x, z) ‖ \leq ρ (\frac{ϵ}{λ} ‖ D_{3} ‖_{2})

where,

ρ (\frac{ϵ}{λ} {‖ D_{3} ‖}_{2}) = \frac{ϵ}{λ} ‖ D_{3} ‖_{2}

(77)

▪

Also, it is worth mentioning that due to the use of BS + BFISMC, the equivalent control principle must be modified due to the continuous barrier function employment with the suggested controller as considering equations (67) and (71) where, $\forall ϵ \to 0 \Rightarrow s_{1} \to s_{1 eq} .$ And that holds for $s_{2} \to s_{2 eq}$ . And accordingly, equations (68) and (72) will tend to $| {\overset{\cdot}{s}}_{1} | \to 0$ , $| {\overset{\cdot}{s}}_{2} | \to 0$ which means the nullity of the perturbation estimation error. Not so far, the BS + BFISMC controller possesses the benefits of both the backstepping, by dealing with the overall system as a combination of multiple subsystems and then gathering the global stabilization regime in one stability analysis, and the advantages of BFISMC, of no need for the perturbation upper bound and chattering free control action.

Proposition 4 contains five design parameters for the proposed controller to be given values by the designer according to the application characteristics. These parameters are $a > 0 \in R$ which represents the exponential decay of each error sub-dynamic of the transformed system and it is up to the designer to specify the required convergence rate. The other design parameters are $λ_{1} > 0 \in R$ , and $1 >> ϵ_{1} > 0 \in R$ , which represent the unmatched barrier function control law parameters, and $1 >> ϵ_{2} > 0 \in R$ , and $λ_{2} > 0 \in R$ which represent the lumped matched barrier function control law parameters, these parameters are selected without a constraint and the only correspondence may affect their selection is the controller hardware. The assumption of $\frac{ϵ}{λ} = \frac{ϵ_{1}}{λ_{1}} = \frac{ϵ_{2}}{λ_{2}}$ will indeed decrease the design parameters from five to only three.

Simulation results

The effectiveness of the proposed control approach, the following case study has been considered for simulations within MATLAB environment.

\begin{matrix} {\overset{\cdot}{x}}_{1} = a_{1} x_{1}^{2} + a_{2} \sin (x_{1}) + x_{2} + d_{1} \\ {\overset{\cdot}{x}}_{2} = a_{3} x_{1}^{3} + a_{4} x_{2} + u + d_{2} \end{matrix}

where, $a_{1} = a_{1 o} + δ a_{1}$ , $a_{2} = a_{2 o} + δ a_{2}$ , $a_{3} = a_{3 o} + δ a_{3}, a_{4} = a_{4 o} + δ a_{4}$ , $a_{1} = a_{1 o} + δ a_{1}$ , and

\begin{matrix} d_{1} = 1.5 + \sin (0.6 π t) + 0.1 \sin (4 π t) \\ d_{2} = \sin (2 π t) + 0.1 \sin (6 π t) \end{matrix}

For the simulation, the system nominal parameters $a_{10} = a_{20} = a_{30} = a_{40} = 1$ , and the uncertainty of the system parameters will be $10 %$ . Hence, $w_{1} = δ a_{1} x_{1}^{2} + δ a_{2} \sin (x_{1}) + d_{1}$ , $w_{2} = δ a_{3} x_{1}^{3} + δ a_{4} x_{2} + d_{2}$ .

The considered system is an unstable and nonlinear system subjected to matched and unmatched disturbances and parametric uncertainty.

The design parameters are selected as: $ϵ_{1} = 0.02$ , $λ_{1} = 10$ , $ϵ_{2} = 2$ , $λ_{2} = 1000$ , $ε = 0.02$ , $α = 3$ , $β = 15$ , and $a = 5$ . The initial condition $x_{1} (0) = - 2$ and $x_{2} (0) = 0$ and the reference to be tracked is $x_{1 d} = 0$ .

Figure 1 shows the behavior of state variable $x_{1}$ for a controlled system based on a robust continuous-based backstepping controller (CRBS), SMC, the backstepping controller, and the proposed backstepping-based barrier function integral sliding mode controller (BS-BFISMC). It is clear from the figure that the backstepping (BS) controller and SMC have the worst behavior among the other controllers and it lacks robustness characteristics, while the CRBS has better performance than BS and SMC. However, in the case of CRBS, the behavior at a steady state shows considerable ripple and the response oscillates off the reference value. One can easily see that the response based on BS-BFISMC. Moreover, one can argue that the transient profile based on CRBS is faster than that based on BS-BFISMC due to the higher negative controller gain. The BS-BFISMC could successfully compensate for the unmatched perturbation (disturbance) without prior knowledge of its upper bound. Table 1 records some samples of transient responses belonging to controllers under competition. The table indicates that the steady state of the proposed controller’s response is confined within the smallest bound as compared to others. In other words, the proposed controller could enforce the response to finally converge to the smallest bound in spite that the upper limit of uncertainty is not known priori.

Figure 1.

The behavior of state variable $x_{1}$ .

Table 1.

Reports of transient characteristics for state variable $x_{1}$ .

Controller	Time instant (s)
Controller	1	3	6	9	18
BS Controller	0.9	0.24	0.17	0.2	0.8
CRBS	0.025	0.005	0.007	0.005	0.023
BS-BFISMC	−0.02	0	0	−0.003	−0.001
SMC	0.38	0.12	0.06	0.05	0.41

Figure 2 shows the behaviors of state variable $x_{2}$ , which represents the virtual controls of suggested and proposed controllers. According to the figure, the virtual control based on BS-BFISMC tries to mimic the equivalent control behavior without using a filtering tool. In addition, the presence of ISMC provides the controller (BS-BFISMC) the privilege of rejecting the perturbation from $t \geq 0$ . It is evident from the figure that the CRBS starts with a high magnitude virtual control value due to the high initial gain control.

Figure 2.

The behavior of state variable $x_{2}$ .

Figure 3 shows the control actions resulting from the four controllers. It is clear from the figure that the control action due to backstepping control is higher than the three other controllers. Moreover, the CRBS gives spiky control action at the startup moment which may reach $3.7 \times 10^{5}$ . This can be attributed to the direct relation between the designed control law of the upper sub-system and the matched and unmatched perturbation upper bound, which may not be valid to effort such control action from the actuator. However, the control action of CRBS decreases shortly after startup and its profile coincides with that of BS-BFISMC in the steady state time. Table 2 evaluates the control action at some samples of time and the table indicates that the less control efforts are given by CRBS and BS-BFISMC.

Figure 3.

The behaviors of control efforts due to four controllers.

Table 2.

Evaluation of control efforts $u$ for the three controllers.

Controller	Time instant (s)
Controller	2	10	18
BS Controller	8	0	6
CRBS	1	−1.5	2.5
BS-BFISMC	1	−1.5	2.5
SMC	1	−1.5	2.7

Figure 4 shows the behavior of error $z_{2}$ for the lower sub-system. The effect of matched and unmatched perturbations is obvious on the error of the backstepping controller, where the perturbations prevent the state $x_{2}$ from following the virtual controller. However, CRBS shows less error due to the high-gain behavior of this controller. The BS-BFISMC shows the least error among the three controllers where the behavior is considerably enhanced since the proposed control approach could compensate the perturbations without the need of an upper bound.

Figure 4.

Behaviors of error $e_{2}$ for the four controllers.

Figure 5 shows the behavior of sliding variable $s_{1}$ for the upper channel of the system based on BS-BFISMC. On the other hand, the behavior of sliding variable $s_{2}$ for the lower channel of the system based on SMC and BS-BFISMC is shown in Figure 6. The boundedness of the sliding variable from $t \geq 0$ is the privilege acquired from the proposed controller. Figure 7 shows the behavior of unmatched perturbation $w_{1}$ and its estimate ${\hat{w}}_{1}$ based on the BS + BFISMC controller. One can conclude that the BS + BFISMC controller could successfully estimate the unmatched perturbation from the first instance $t \geq 0$ . Figure 8 shows the estimation error between unmatched perturbation $w_{1}$ and its estimate ${\hat{w}}_{1}$ , where the error asymptotically converges to zero according to the analysis that has been conducted earlier.

Figure 5.

The behavior of sliding variable $s_{1}$ based on BS-BFISMC.

Figure 6.

The behavior of sliding variable $s_{2}$ based on SMC and BS-BFISMC.

Figure 7.

The behavior of actual uncertainty $w_{1}$ and estimated ${\hat{w}}_{1}$ based on BS-BFISMC.

Figure 8.

The behavior of estimation error ( $w_{1} - {\hat{w}}_{1}$ ) based on BS-BFISMC.

Conclusions

This article has presented a solution to the control problem of nonlinear systems with bounded matched and unmatched perturbations (disturbances) with unknown upper bound. Based on the backstepping control approach, a novel backstepping-based barrier function integral sliding mode controller (BS-BFISMC) has been developed for the control of nonlinear systems with bounded matched and unmatched perturbations with an unknown upper bound. According to Lyapunov-stability analysis, the design mechanism utilized the concept of a backstepping control approach by dealing with the overall system as a combination of sub-systems the global stabilization is proven. The contributed feature of BS-BFISMC is that it can cope with perturbation of unknown upper bound and produce chattering-free control action. A comparison has been made between the proposed BS-BFISMC and the other three controllers (BS Controller, SMC controller and CRBS). In the sense of convergence, all states based on all controllers converge to uniform ultimate boundedness. However, the final ultimate boundedness differs from one controller to another. It has been shown that the proposed controller results in the least ultimate bound. Moreover, the BS-BFISMC reaches the ultimate bound with less bound limits. In addition, the control action consumption can be sorted from lower to higher as BS-BFISMC controller, CRBS, SMC and BS Controller.

Footnotes

A.1- Appendix (SMC)

Assumptions:

(A1)

| w_{1} | \leq α

(A2)

ζ_{3} = (a + c) w_{1} + {\overset{\cdot}{f}}_{1} (t, x_{1}) + w_{2}

(A3)

u = - f_{2} (t, x) + (ca + a^{2}) x_{1} - (a + c) z_{2} - k_{3} \tanh (s / τ)

will drive the states to a uniform ultimate bound of: $| x_{1} | \leq \frac{α}{c + a}$ ,where $α$ is given in equation (A1).

(A4)

z_{2} = x_{2} + a x_{1} + f_{1} (t, x_{1})

which implies

(A5)

x_{2} = z_{2} - a x_{1} - f_{1} (t, x_{1})

and

(A6)

{\overset{\cdot}{x}}_{1} = - a x_{1} + z_{2} + w_{1}

The time derivative of equation (A4), ends to

(A7)

{\overset{\cdot}{z}}_{2} = f_{2} (t, x) + u - a^{2} x_{1} + a z_{2} + (a + c) w_{1} + {\overset{\cdot}{f}}_{1} (t, x_{1}) + w_{2}

The following sliding is defined

(A8)

s = z_{2} + c x_{1}

The time derivative of the sliding variable of equation (A8) becomes

(A9)

\overset{\cdot}{s} = f_{2} (t, x) - (ca + a^{2}) x_{1} + (a + c) z_{2} + u + ζ_{3}

The control objective is converted to stabilize the sliding variable. Generally, using Lyapunov analysis of the sliding variable dynamics of equation (A9) as,

(A10)

V_{s} = \frac{1}{2} s^{2}

(A11)

{\overset{\cdot}{V}}_{x_{1}} = s \overset{\cdot}{s} = s (f_{2} (t, x) - (ca + a^{2}) x_{1} + (a + c) z_{2} + u + ζ_{3})

Using the Control action of equation (A3), equation (A11) can be rewritten as

(A12)

{\overset{\cdot}{V}}_{x_{1}} = s (- k \tanh (s / τ) + ζ_{3}) \leq - | s | (k - | ζ_{3} |)

Choosing large enough $k$ means $k - | ζ_{3} | = μ_{3} > 0 \in R$ , that’s mean $s = 0$ in finite time, and that’s yields

(A13)

z_{2} = - c x_{1}

Substitute equations (A13) in (6), then

(A14)

{\overset{\cdot}{x}}_{1} = - (a + c) x_{1} + w_{1}

The ultimate bound of equation (A14) can be assesses by the following Lyapunov candidate of the form

(A15)

V_{x_{1}} = \frac{1}{2} x_{1}^{2}

The time derivative of equation (A15) can be obtained as

(A16)

{\overset{\cdot}{V}}_{x_{1}} = x_{1} {\overset{\cdot}{x}}_{1} = - (a + c) x_{1}^{2} + x_{1} w_{1}

then according to,³ Let $0 < θ < 1$ then,

(A17)

{\overset{\cdot}{V}}_{x_{1}} \leq - (a + c) (1 - θ) x_{1}^{2} - (a + c) θ x_{1}^{2} + | x_{1} | α

Or,

(A18)

{\overset{\cdot}{V}}_{x_{1}} \leq - (a + c) (1 - θ) x_{1}^{2}, \forall | x_{1} | > α / (a + c) θ

Hence, according to Theorem 1, the ultimate bound is

(A19)

| x_{1} | > α / (a + c)

▪

The ultimate bound of equation (A19) reflects the challenging problem of this study and even a robust controller like SMC could fail to successfully manage the upper system in asymptotic manner.

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 iDs

Akram Hashim Hameed

Amjad Jaleel Humaidi

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

Krstic

Kanellakopoulos

Kokotovic

. Nonlinear and Adaptive Control Design. New York: Wiley, 1995.

Al-Samarraie

Al-Nadawi

LYK

Mishary

, et al. Electronic throttle valve control design based on sliding mode perturbation estimator. J Iraqi J Comput Commun Control Eng 2015; 15(2): 65–74.

Khalil

. Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall, 2002.

Humaidi

Hameed

. PMLSM position control based on continuous projection adaptive sliding mode controller. Syst Sci Control Eng 2018; 6(3): 242–252.

AL-Samarraie

. A Chattering free sliding mode observer with application to DC motor speed control. In 2018 third scientific conference of electrical engineering (SCEE), Baghdad, Iraq, 2018. IEEE, pp. 259–264.

Humaidi

Hameed

. Design of block-backstepping controller to ball and arc system based on zero dynamic theory. J Eng Sci Technol 2018; 13(7): 2084–2105.

Humaidi

Hameed

Ibraheem

. Design and performance study of two sliding mode backstepping control schemes for roll channel of delta wing aircraft. In 2019 6th international conference on control, decision and information technologies (CoDIT), Paris, France, 23 April 2019: IEEE, pp. 1215–1220.

Humaidi

Hameed

. Design and comparative study of advanced adaptive control schemes for position control of electronic throttle valve. Information 2019; 10(2): 65.

Humaidi

Hameed

. “Design and performance investigation of block-backstepping algorithms for ball and arc system,” 2017 IEEE International Conference on Power, Control, Signals and Instrumentation Engineering (ICPCSI), Chennai, India, 2017, pp. 325–332.

10.

Shtessel

Edwards

Fridman

, et al. Sliding mode control and observation. New York: Springer, 2014.

11.

Utkin

Guldner

Shi

Sliding mode control in electro-mechanical systems. Boca Raton: CRC Press, 2017.

12.

Al-Samarraie

. Positively invariant sets in sliding mode control theory with application to servo actuator system with friction. J Iraqi J Comput Commun Control Eng 2010; 10(1): 121–134.

13.

Obeid

Fridman

Laghrouche

, et al. Barrier function-based adaptive sliding mode control. Automatica 2018; 93: 540–544.

14.

MohammadRidha

Al-Samarraie

. A barrier function-based integral sliding mode control of heart rate during treadmill exercise. Math Model Eng Probl 2023; 10(1): 179–188.

15.

Obeid

Fridman

Laghrouche

, et al. Barrier function-based adaptive integral sliding mode control. In 2018 IEEE conference on decision and control (CDC), Florida, 2018. IEEE, pp. 5946–5950.

16.

Hassan

Al-Samarraie

. Optimal feedback control for HVAC systems: an integral sliding mode control approach based on barrier function. Math Model Eng Probl 2023; 10(3): 1053–1062.

17.

Guo

Chang

Guo

, et al. Adaptive twisting sliding mode algorithm for hypersonic reentry vehicle attitude control based on finite-time observer. ISA Trans 2018; 77: 20–29.

18.

Rashad

El-Badawy

Aboudonia

. Sliding mode disturbance observer-based control of a twin rotor MIMO system. ISA Trans 2017; 69: 166–174.

19.

Mattei

Monaco

. Robust backstepping control of missile lateral and rolling motions in the presence of unmatched uncertainties. In 2012 IEEE 51st IEEE conference on decision and control (CDC), Hawaii, 2012. IEEE, pp. 2878–2883.

20.

Castanos

Fridman

. Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans Automat Contr 2006; 51(5): 853–858.

21.

Estrada

Frid

. Integral HOSM semiglobal controller for finite-time exact compensation of unmatched perturbations. IEEE Trans Automat Contr 2010; 55(11): 2645–2649.

22.

Canabal

Loukianov

Castañeda

JMC

, et al. Adaptive power system stabilizer with sliding mode for electric power systems. Int J Electr Power Energy Syst 2023; 145: 143–157.

23.

Tran

Ahn

. Adaptive backstepping sliding mode control for equilibrium position tracking of an electrohydraulic elastic manipulator. IEEE Trans Ind Electron 2020; 67(5): 3860–3869.

24.

Wang

Jiao

Hou

. Multi: sliding mode backstepping method for integrated missile guidance and control system. In Proceedings of the 2019 4th international conference on automation, control and robotics engineering, Shenzhen, China, 2019, pp. 1–6.

25.

Ding

Liu

Shao

, et al. Backstepping sliding mode control for the speed and tension system of reversible cold strip rolling mill based on extended state observers. In 2020 39th Chinese control conference (CCC), Shenzhen, China, 2020. IEEE, pp. 2078–2083.

26.

Jiang

Zhao

, et al. Trajectory tracking control of quadrotor aircraft based on improved ESO backstepping sliding mode. In 2020 Chinese Automation Congress (CAC), Shenzhen, China, 2020. IEEE, pp. 5963–5968.

27.

Liu

Yang

Cao

, et al. High order sliding mode backstepping control for a class of unknown pure feedback nonlinear systems. IEEE Access 2020; 8: 227528–227537.

28.

Thanh

HLNN

Mung

, et al. Perturbation observer-based robust control using a multiple sliding surfaces for nonlinear systems with influences of matched and unmatched uncertainties. Mathematics 2020; 8(8): 1371.

29.

Wang

Jiang

. Adaptive backstepping sliding mode control of air-breathing hypersonic vehicles. Int J Aerosp Eng 2020; 2020: 1–11.

30.

Yang

. Dual extended state observer-based backstepping control of electro-hydraulic servo systems with time-varying output constraints. Trans Inst Meas Contr 2020; 42(5): 1070–1080.

31.

Adiguzel

Mumcu

. Adaptive backstepping sliding mode control for ABS with nonlinear disturbance observer. Electr 2021; 21(1): 121–128.

32.

Guo

Wang

, et al. Sliding mode control for systems subjected to unmatched disturbances/unknown control direction and its Application. Int J Robust Nonlinear Control 2021; 31(4): 1303–1323.

33.

Jin

Lü

Che

, et al. Robust adaptive radial-basis function neural network-based backstepping control of a class of perturbed nonlinear systems with unknown system parameters. Int J Robust Nonlinear Control 2021; 31(11): 5101–5117.

34.

Kali

Saad

Benjelloun

. Backstepping super-twisting for robotic manipulators with matched and unmatched uncertainties. In 2021 18th international multi-conference on systems, signals & devices (SSD), Monastir, Tunisia, 2021. IEEE, pp. 1154–1159.

35.

Yue

Fan

Chen

, et al. Spacecraft formation control based on adaptive backstepping terminal sliding mode. In 2021 40th Chinese control conference (CCC), Shanghai, China, 2021. IEEE, pp. 7868–7873.

36.

Brahmi

El-Monajjed

Driscoll

. Novel adaptive backstepping control of uncertain electrically driven haptic robot for surgical training systems. Int J Control 2022; 95(6): 1432–1455.

37.

Kali

Saad

Boland

, et al. Walking control using TDE-based backstepping SM of position-commanded NAO biped robot with matched and unmatched perturbations. J Contr Autom Electr Syst 2022; 33(6): 1633–1642.

38.

Zhou

, et al. Full-order terminal sliding-mode control for soft open point. Energies 2022; 15(14): 4999.

39.

Ferrara

Incremona

Vecchio

. Adaptive multiple-surface sliding mode control of nonholonomic systems with matched and unmatched uncertainties. IEEE Transact Automati Control, 2023; 69(1): 614–621.

40.

Zhang

, et al. Disturbance-observer-based adaptive finite-time dynamic surface control for PMSM with time-varying asymmetric output constraint. Asian J Control 2023; 25(5): 3752–3775.

41.

Shi

Zhang

, et al. On backstepping control for a class of multiple uncertain systems with reduced-order ESO. Int J Control 2023; 96(2): 309–320.

42.

Mahdi

Yousif

Oglah

, et al. Adaptive synergetic motion control for wearable knee-assistive system: a rehabilitation of disabled patients. Actuators 2022; 11(7): 176.

43.

Humaidi

Kadhim

Gataa

. Development of a novel optimal backstepping control algorithm of magnetic impeller-bearing system for artificial heart ventricle pump. Cybernetics and Systems 2020; 51(4): 521–541.

44.

, et al. Kinematic coupling-based trajectory planning for rotary crane system with double-pendulum effects and output constraints. J Field Robot 2023; 40(2): 289–305.

45.

, et al. Time-polynomial-based optimal trajectory planning for double-pendulum tower crane with full-state constraints and obstacle avoidance. IEEE/ASME Trans Mechatron 2023; 28(2): 919–932.

46.

Jaleel

Talaat

Hameed

, et al. Design of adaptive observer-based backstepping control of cart-pole pendulum system. In: Proceeding of IEEE International Conference on Electrical, Computer and Communication Technologies (ICECCT 2019), Coimbatore, India, 2019, pp.1–5.

47.

Amjad

Hameed

. Robustness enhancement of MRAC using modification techniques for speed control of three phase induction motor. J Electr Syst, 2017; 13(4): 723–741.

48.

, et al. Optimal trajectory planning strategy for underactuated overhead crane with pendulum-sloshing dynamics and full-state constraints. Nonlinear Dyn 2022; 109(2): 815–835.

49.

Ren

. Backstepping variable structure control of nonlinear systems with unmatched uncertainties. IFAC Proc Volumes 1999; 32(2): 2737–2741.

50.

Alawad

Humaidi

Alaraji

. “Sliding mode-based active disturbance rejection control of assistive exoskeleton device for rehabilitation of disabled lower limbs”. Anais da Academia Brasileira de Ciências 2023; 95(2): 214–228

A novel control solution to nonlinear systems of unmatched perturbations with unknown bounds

Abstract

Keywords

Introduction

Problem formulation

Controller design

Backstepping control

Robust continuous backstepping-based controller (CRBS)

Backstepping-based BFISMC controller

Simulation results

Conclusions

Footnotes

A.1- Appendix (SMC)

Declaration of conflicting interests

Funding

ORCID iDs

Data availability statement

References