A Sliding Mode Control Algorithm with Elementary Compensation for Input Matrix Uncertainty in Affine Systems

Abstract

This paper aims to develop a sliding mode control (SMC) approach with elementary compensation for input matrix uncertainty in affine systems. As a multiplicative uncertainty regarding the control inputs, input matrix uncertainty adversely modifies the control effort and even further causes the instability of systems. To solve this issue, a sliding mode control algorithm is developed based on a two-step design strategy. The first step is to design a general sliding mode controller for the system without input matrix uncertainty. In the second step, a control term is specially designed to compensate for input matrix uncertainty. In order to realize the elementary compensation for input matrix uncertainty, this term is obtained by solving a nonlinear vector equation which is derived from the Lyapunov function inequality. Theorems and lemmas based on the convex cone theory are proposed to guarantee the existence and uniqueness of the solution to the vector equation. Additionally, an algorithmic process is proposed to solve the vector equation efficiently. In the simulation part, the proposed controller is applied to two systems with different structures and compared with two state-of-the-art SMC algorithms. The comprehensive simulation results demonstrate that the proposed method is able to provide the closed-loop system with a competitive performance in terms of convergence level, overshoot reduction and chattering suppression.

1. Introduction

Sliding mode control (SMC) is an effective approach to affine uncertain systems due to its advantages such as strong robustness, fast response and small overshoot [1,2]. These benefits stem from the property that the system is completely insensitive to matched uncertainties during sliding mode [3,4,5,6,7]. However, such a desirable characteristic depends on a proper control algorithm which is capable of enforcing the sliding mode. Thus, designing a controller to ensure the enforcement of sliding mode is a crucial subject for the SMC method.

Matched uncertainty is a challenging issue that poses a threat to the enforcement of the sliding mode and brings difficulties in the controller design process. In general, there are two types of matched uncertainties, each of which affects the system on different levels: (1) the additive uncertainty influences the system in a parallel way with the control inputs. (2) Input matrix uncertainty, which is multiplicative with respect to the control inputs, refracts the control effort by modifying the gain of the control loop; additionally, the occurrence of the sliding mode will be destroyed if the norm of input matrix uncertainty exceeds a certain value; additionally, for multi-input systems, input matrix uncertainty implies the presence of uncertainty in the control direction [8]. It has been discovered that input matrix uncertainty is more difficult to handle and numerous control strategies have been proposed to suppress its effect [8,9,10,11]. For instance, Bejarano-Rincón et al. [8] design a sliding mode controller for MIMO nonlinear systems with uncertain control direction; nevertheless, the coefficient of each discontinuous control is excessively large as it bounds above the sum of input uncertainties and other system components. One representative method is proposed by Cao and Xu [10], which develops a compensation term for input matrix uncertainty by iteratively relaxing the inequality regarding the derivative of the Lyapunov function. The efficacy of this approach has led to its widespread adoption by many researchers: Chang [12] proposes a control law for the linear MIMO system based on the information of the full-order compensator and the system outputs and addresses input matrix uncertainty with Cao and Xu’s strategy; Shtessel et al. [13] present a unit-vector approach to control the linear systems with matched uncertainties, where input matrix uncertainty is handled by Cao and Xu’s method; Zhang et al. [14] put forward a robust control design based on a new switched robust integral sliding mode (SRISM) for switched systems with uncertain input matrix and compensate for the uncertainties using Cao and Xu’s strategies; more recently, Lin and Hu [15] propose a robust SMC to achieve chaos synchronization for master-slave chaotic systems with matched/mismatched disturbances and handle input uncertainty with Cao and Xu’s scheme.

Nevertheless, the aforementioned methods exhibit a high degree of conservativeness, i.e., the use of triangle inequality results in the continuous amplification of the Lyapunov inequality and further causes an overlarge magnitude of the control gain. Therefore, the excessive control gain will result in amplified amplitude of chattering [16,17]. To address this issue, several continuous SMC strategies have been proposed recently [18,19,20]. For example, Martínez-Fuentes et al. [19] propose a Lipschitz continuous sliding mode controller (LCSMC), which ensures the global finite-time convergence to the sliding manifold despite an unknown control coefficient. Gutiérrez-Oribio et al. [20] propose two continuous second-order sliding mode controllers for a class of underactuated mechanical systems with two degrees of freedom, which are robust against the uncertainty in the control coefficient. However, only the single-input systems are studied in the literature studies above. Recently, Feng et al. [21] proposed a chattering-reduction sliding mode control method for affine uncertain systems, in which a control term is specially designed to compensate for input matrix uncertainty and the term is obtained by solving a nonlinear equation, so that the magnitude of control is effectively decreased and chattering is further suppressed.

Despite the fact that many studies have shown effective compensation for input matrix uncertainty, to the best of our knowledge, few studies have centered on elementarily compensating for input matrix uncertainty. Nasiri et al. [9] propose an adaptive sliding mode control scheme for an affine class of multi-input multi-output (MIMO) nonlinear systems with uncertain control distribution gain; however, the bounds of the input uncertainties are quite strict for a large number of systems. In practical applications, where the input matrix has a physical meaning and specific information for each component, it is critical to be informed about the elements of input matrix uncertainty. In the spacecraft attitude control system, for example, the uncertainty in the control gain matrix represents the faults of actuators and the elements of the matrix represent the execution effectiveness loss of the corresponding actuator (a certain reaction flywheel) [22]. The most common strategy for dealing with uncertainties in the gain matrix is to employ the observer solely for fault estimation and incorporate the gain adaption scheme to realize the controller design. However, the implementation complexity is high and the fault diagnosis introduces an unavoidable time delay, which will inevitably degrade the control performance.

In order to reduce the design conservativeness and simplify the practical implementation, this paper proposes a novel sliding mode algorithm to control the affine systems with input matrix uncertainty. The proposed algorithm is based on a two-step design strategy. More specifically, in the first step, a general SMC is designed for the system without input matrix uncertainty. In the second step, a control term is specifically developed to compensate for input matrix uncertainty by element. By solving a nonlinear vector equation derived from the derivative of the Lyapunov function, the compensation term for input matrix uncertainty is obtained. A theorem and several lemmas based on the convex cone theory are proposed to demonstrate the existence and uniqueness of the solution to the vector equation. In addition, an algorithmic process is proposed to solve the vector equation efficiently. As a result, the main contributions of this work are listed as follows,

• A two-step design strategy is utilized to develop the controller: the first step aims at the additive matched uncertainties; the second step focuses on addressing input matrix uncertainty.
• In the proposed method, a new compensation term is specially developed to suppress the input matrix uncertainty. This control term is determined by solving a nonlinear vector equation. By this means, the uncertainty in the input matrix can be compensated for by the element. A theorem and several lemmas based on the principle of the convex cone set are proposed to guarantee the existence and uniqueness of the solution to the vector equation.
• A practical algorithmic process is proposed to obtain the unique solution to the vector equation efficiently.
• The proposed method is applied to two systems with different structures. The simulations are conducted by comparing the proposed SMC strategy with the SMC method proposed by Cao and Xu [10] and the SMC method proposed by Feng et al. [21]. The simulation results show that the proposed method can successfully suppress the effect of input matrix uncertainty and is effective in chattering reduction.
• Compared with SMC studies that integrated with intelligent control approaches such as observers, fuzzy logics and adaptive control, the proposed method has much simpler structure and many fewer control parameters and is useful for practical implementations.

The rest of this paper is organized as follows. Section 2 is devoted to system, sliding surfaces descriptions. In Section 3, the proposed control algorithm with theoretical analyses and the calculation process are given. Section 4 presents two numerical examples with simulation results and comparisons with other state-of-the-art methods. Section 5 draws the conclusion.

Notation 1. 

The notation used in this paper is fairly standard. Specially, ${\mathbb{R}}^{+}$ denotes the set of real numbers and ${\mathbb{R}}^n$ denotes the set of $n\times 1$ real column vectors. Furthermore, ${\left(·\right)}^T$ is written for transpose, $\left|·\right|$ denotes the absolute value for a real number and the absolute value of each element in a matrix. $∥·∥$ denotes the Euclidean norm and its induced matrix norm, ${∥·∥}_1$ denotes the Taxicab norm. $\left[n\right]$ denotes the set $\left\{1,\dots ,n\right\}$. If $\mathit{x}\in {\mathbb{R}}^n$ is a vector, ${\left[x\right]}_i$ denotes the i-th element of $\mathit{x}$, $i\in \left[n\right]$. If $\mathit{A}\in {\mathbb{R}}^{n\times m}$ is a matrix, ${\left[A\right]}_{i,j}$ denotes the element in the i-th row and j-th column of $\mathit{A}$ and $i\in \left[n\right]$, $j\in \left[m\right]$. ${\sum }^n$ denotes a matrix set ${\sum }^n=\left\{S\mid S=diag\left\{{\zeta }_1,\dots ,{\zeta }_n\right\},{\zeta }_i\in \{-1,1\}\right\}$, which contains $2^n$ members of $n\times n$ matrix. $S\left(\mathit{x}\right):{\mathbb{R}}^n\to {\sum }^n$ denotes matrix function in which each element fills the sign of the corresponding component in $\mathit{x}$, i.e., $S\left(\mathit{x}\right)=diag\left\{sign\left({\left[x\right]}_i\right)\right\}$ and $sign\left({\left[x\right]}_i\right)=\left\{\begin{array}{c}1,{\left[x\right]}_i\ge 0\\ -1,{\left[x\right]}_i<0.\end{array}\right.$

2. Preliminaries of Sliding Mode Controller Design

Consider the following affine uncertain systems [10,23]:

$$\begin{array}{c}\hfill \dot{\mathit{x}}\left(t\right)=\mathit{f}(\mathit{x},t)+\mathit{B}(\mathit{x},t)\left[\left(\mathit{I}+\mathsf{\Delta }\mathit{B}(\mathit{x},t)\right)\mathit{u}\left(t\right)+\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right],\end{array}$$

(1)

where $t\in [t_0,\infty )$ represents time, $\mathit{x}\in {\mathbb{R}}^n$ is a vector of measurable states, $\mathit{u}\in {\mathbb{R}}^m$ is a vector of control inputs, $\mathit{f}(\mathit{x},t)$ and $\mathit{B}(\mathit{x},t)$ are known vector and matrix with appropriate size and their elements are piecewise continuous functions with respect to $\mathit{x}$ and t [13]. $\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)$ is an unknown vector representing the additive matched uncertainties and $\mathsf{\Delta }\mathit{B}(\mathit{x},t)$ represents the input matrix uncertainty.

Assumption 1 ([24]). 

Assume that $\mathit{B}(\mathit{x},t)$ has full rank, i.e., rank$\left(\mathit{B}\right(\mathit{x},t\left)\right)=m$.

Assumption 2 ([10,23]). 

For all $\left(\mathit{x},t\right)\in {\mathbb{R}}^n\times {\mathbb{R}}^{+}$, $\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)$ is bounded by a known nonlinear function as $∥\mathsf{\Delta }{\mathit{f}}_m\left(\mathit{x},t\right)∥\le {\rho }_m\left(\mathit{x},t\right)$.

Assumption 3 ([10]). 

For all $\left(\mathit{x},t\right)\in {\mathbb{R}}^n\times {\mathbb{R}}^{+}$, $\mathsf{\Delta }\mathit{B}\left(\mathit{x},t\right)\in {\mathbb{R}}^{m\times m}$ satisfies

$$\begin{array}{cc}\hfill & \left|{\left[\mathsf{\Delta }B(\mathit{x},t)\right]}_{i,j}\right|\le {\left[\overline{\mathsf{\Delta }B}\right]}_{i,j},\forall i\in \left[m\right],j\in \left[m\right]\hfill \\ \hfill & \parallel \overline{\mathsf{\Delta }\mathit{B}}\parallel \le {\rho }_b<1.\hfill \end{array}$$

(2)

where ${\left[\mathsf{\Delta }B(\mathit{x},t)\right]}_{i,j}$ is the element in the i-th row and the j-th column of $\mathsf{\Delta }\mathit{B}(\mathit{x},t)$, ${\left[\overline{\mathsf{\Delta }B}\right]}_{i,j}$ is the upper bound of $\left|{\left[\mathsf{\Delta }B(\mathit{x},t)\right]}_{i,j}\right|$ and is the element in the i-th row and the j-th column of $\overline{\mathsf{\Delta }\mathit{B}}$. ${\rho }_b$ is a known positive constant which bounds the upper bound matrix $\overline{\mathsf{\Delta }\mathit{B}}$.

In this paper, two kinds of sliding surfaces are investigated as follows.

• If system (1) can be put in the regular canonical form [24,25,26]:

  $$\begin{array}{cc}\hfill {\dot{\mathit{x}}}_1\left(t\right)& ={\mathit{f}}_1({\mathit{x}}_1,{\mathit{x}}_2),\hfill \\ \hfill {\dot{\mathit{x}}}_2\left(t\right)& ={\mathit{f}}_2({\mathit{x}}_1,{\mathit{x}}_2)+{\mathit{B}}_2({\mathit{x}}_1,{\mathit{x}}_2)\left(\left(\mathit{I}+\mathsf{\Delta }\mathit{B}({\mathit{x}}_1,{\mathit{x}}_2)\right)\mathit{u}\left(t\right)+\mathsf{\Delta }{\mathit{f}}_m({\mathit{x}}_1,{\mathit{x}}_2)\right),\hfill \end{array}$$

  (3)

  where ${\mathit{x}}_1\in {\mathbb{R}}^{n-m}$, ${\mathit{x}}_2\in {\mathbb{R}}^m$, ${\mathit{x}}^T=\left({\mathit{x}}_1^T,{\mathit{x}}_2^T\right)$. ${\mathit{B}}_2({\mathit{x}}_1,{\mathit{x}}_2)\in {\mathbb{R}}^m$ and $det\left({\mathit{B}}_2({\mathit{x}}_1,{\mathit{x}}_2)\right)\ne 0$. ${\mathit{f}}^T(\mathit{x},t)=\left({\mathit{f}}_1^T({\mathit{x}}_1,{\mathit{x}}_2),{\mathit{f}}_2^T({\mathit{x}}_1,{\mathit{x}}_2)\right)$. ${\mathit{B}}^T(\mathit{x},t)=\left({\mathbf{0}}_{m\times (n-m)},{\mathit{B}}_2^T({\mathit{x}}_1,{\mathit{x}}_2)\right)$, where ${\mathbf{0}}_{m\times (n-m)}$ is the zero matrix of size $m\times (n-m)$ [27]. Then, a linear sliding surface with the following form is adopted [24,28]:

  $$\mathit{\sigma }\left(t\right)={\mathit{x}}_2\left(t\right)+{\mathit{G}}_1{\mathit{x}}_1\left(t\right)=\mathit{Gx}\left(t\right)=\mathbf{0},$$

  (4)

  where ${\mathit{G}}_1\in {\mathbb{R}}^{m\times (n-m)}$ is a diagonal positive definite matrix. $\mathit{G}=\left[\begin{array}{c}{\mathit{G}}_1,{\mathit{I}}_{m\times m}\hfill \end{array}\right]\in {\mathbb{R}}^{m\times n}$, where ${\mathit{I}}_{m\times m}$ is an $m\times m$ identity matrix.
• Otherwise, the integral sliding surface is adopted as [29]

  $$\mathit{\sigma }\left(t\right)=\mathit{Gx}\left(t\right)-\mathit{Gx}\left(0\right)-{\int }_0^t\mathit{G}\left(\mathit{f}(\mathit{x},\tau )+\mathit{B}(\mathit{x},\tau )\mathit{\kappa }(\mathit{x},\tau )\right)d\tau =\mathbf{0},$$

  (5)

  where $\mathit{G}\in {\mathbb{R}}^{m\times n}$ is a constant matrix; $\mathit{\kappa }(\mathit{x},t)$ is a nominal control under which the system $\dot{\mathit{x}}\left(t\right)=\mathit{f}(\mathit{x},t)+\mathit{B}(\mathit{x},t)\mathit{\kappa }(\mathit{x},t)$ is globally asymptotically stable [10,23].

The derivative of sliding surface (4) and (5) can be integrated into one equation, Equation (6), as follows:

$$\dot{\mathit{\sigma }}\left(t\right)={\mathit{Gf}}_{\sigma }(\mathit{x},t)+\mathit{GB}\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)+\mathit{G}\mathit{B}(\mathit{x},t)\left((\mathit{I}+\mathsf{\Delta }\mathit{B}(\mathit{x},t\left)\right)\mathit{u}\left(t\right)\right),$$

(6)

where $\mathit{GB}(\mathit{x},t)$ is uniformly invertible and

$${\mathit{f}}_{\sigma }(\mathit{x},t)=\left\{\begin{array}{cc}\mathit{f}(\mathit{x},t),\hfill & \mathrm{if} \mathit{\sigma }\left(t\right) \mathrm{is} \mathrm{linear}\hfill \\ -\mathit{B}(\mathit{x},t)\mathit{\kappa }(\mathit{x},t),\hfill & \mathrm{if} \mathit{\sigma }\left(t\right) \mathrm{is} \mathrm{integral}\hfill \end{array}\right..$$

(7)

Item (7) covers the different results corresponding to the linear surface and integral surface, respectively. Therefore, it can be directly used in the following derivation of the controller design. Namely, The controller design strategy proposed in this paper is applicable to both linear and integral sliding surfaces.

3. Sliding Mode Controller Design

In this section, we present the sliding mode controller design for system (1). The design proceeds in two step: In the first step, a general sliding mode controller is developed for the system without considering input matrix uncertainty; in the second step, a control term is designed specially to compensate for input matrix uncertainty. Therefore, the final controller for system (1) comprises two parts and is formulated as

$$\mathit{u}\left(t\right)={\mathit{u}}_d\left(t\right)+{\mathit{u}}_c\left(t\right),$$

(8)

where ${\mathit{u}}_d\left(t\right)$ is the general sliding mode controller for a system only with uncertainty $\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)$ and ${\mathit{u}}_c\left(t\right)$ is the compensation control term for $\mathsf{\Delta }\mathit{B}(\mathit{x},t)$.

3.1. General Control Law

The first step aims at the following system only with additive uncertainty $\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)$:

$$\dot{\mathit{x}}\left(t\right)=\mathit{f}(\mathit{x},t)+\mathit{B}(\mathit{x},t)\left(\mathit{u}\left(t\right)+\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right).$$

(9)

Since $\mathit{GB}(\mathit{x},t)$ is uniformly invertible, ${\mathit{u}}_d\left(t\right)$ can be designed as [10]

$${\mathit{u}}_d\left(t\right)=-{\left(\mathit{G}\mathit{B}(\mathit{x},t)\right)}^{-1}\mathit{G}{\mathit{f}}_{\sigma }(\mathit{x},t)-\rho (\mathit{x},t){\mathit{v}}_0$$

(10)

where $\mathit{v}={\left(\mathit{GB}(\mathit{x},t)\right)}^T\mathit{\sigma }$, ${\mathit{v}}_0=\left\{\begin{array}{cc}\mathit{v}/\parallel \mathit{v}\parallel ,& \parallel \mathit{v}\parallel >0\hfill \\ 0,& \parallel \mathit{v}\parallel =0\hfill \end{array}\right.$ and $\rho (\mathit{x},t)$ satisfies

$$\rho (\mathit{x},t)>{\rho }_m(\mathit{x},t).$$

(11)

To verify that ${\mathit{u}}_d\left(t\right)$ in (10) is able to enforce the sliding mode with regard to system (9), the following Lyapunov function is selected [10]:

$$V_{\sigma }=1/2{\mathit{\sigma }}^T\mathit{\sigma }.$$

(12)

Differentiating $V_{\sigma }$ with respect to time along with (9)–(11), one obtains

$$\begin{array}{cc}\hfill {\dot{V}}_{\sigma }& ={\mathit{\sigma }}^T\left({\mathit{Gf}}_{\sigma }(\mathit{x},t)+\mathit{GB}(\mathit{x},t)\left(-{\left(\mathit{G}\mathit{B}(\mathit{x},t)\right)}^{-1}\mathit{G}{\mathit{f}}_{\sigma }(\mathit{x},t)-\rho (\mathit{x},t){\mathit{v}}_0+\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right)\right)\hfill \\ \hfill & ={\mathit{v}}^T\left(-\rho (\mathit{x},t){\mathit{v}}_0+\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right)\hfill \\ \hfill & \le -\rho (\mathit{x},t)\parallel \mathit{v}\parallel +∥\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)∥\parallel \mathit{v}\parallel \hfill \\ \hfill & \le {\rho }_m(\mathit{x},t)\parallel \mathit{v}\parallel -\rho (\mathit{x},t)\parallel \mathit{v}\parallel <0.\hfill \end{array}$$

(13)

Hence, the enforcement of the sliding mode is guaranteed by the control law ${\mathit{u}}_d\left(t\right)$. Alternatively, ${\mathit{u}}_d\left(t\right)$ can be designed with another form [3]:

$${\mathit{u}}_d\left(t\right)=-{\left(\mathit{GB}(\mathit{x},t)\right)}^{-1}\left({\mathit{Gf}}_{\sigma }(\mathit{x},t)+\rho (\mathit{x},t)sign\left(\sigma \right)\right),$$

(14)

where $\rho (\mathit{x},t)>∥\mathit{GB}(\mathit{x},t)∥{\rho }_m(\mathit{x},t)$.

Choosing Lyapunov function as (12) and differentiating the function with respect to time along with (9), (11) and (14), one obtains:

$$\begin{array}{cc}\hfill {\dot{V}}_{\sigma }=& {\mathit{\sigma }}^T\left({\mathit{Gf}}_{\sigma }(\mathit{x},t)+\mathit{GB}(\mathit{x},t)\left(-{\left(\mathit{G}\mathit{B}(\mathit{x},t)\right)}^{-1}\left({\mathit{Gf}}_{\sigma }(\mathit{x},t)+\rho (\mathit{x},t)sign\left(\mathit{\sigma }\right)\right)\right)\right)\hfill \\ \hfill & +{\mathit{\sigma }}^T\mathit{GB}(\mathit{x},t)\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\hfill \\ \hfill =& {\mathit{\sigma }}^T\left(-\rho (\mathit{x},t)sign\left(\mathit{\sigma }\right)+\mathit{GB}(\mathit{x},t)\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right)\hfill \\ \hfill \le & -{\rho (\mathit{x},t)\parallel \mathit{\sigma }\parallel }_1+∥\mathit{\sigma }∥·∥\mathit{GB}(\mathit{x},t)\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)∥\hfill \\ \hfill \le & -\rho (\mathit{x},t)\parallel \mathit{\sigma }\parallel +∥\mathit{\sigma }∥·∥\mathit{GB}(\mathit{x},t)∥{\rho }_m(\mathit{x},t)<0.\hfill \end{array}$$

(15)

Therefore, ${\mathit{u}}_d\left(t\right)$ in (14) can also guarantee the enforcement of the sliding mode with respect to system (9).

3.2. Compensation Term

However, ${\mathit{u}}_d\left(t\right)$ in (10) and (14) cannot guarantee the enforcement of the sliding mode with respect to system (1), since its control effect will be refracted by $\mathsf{\Delta }\mathit{B}(\mathit{x},t)$. To be specific, considering Lyapunov function (12), differentiating it with respect to time along with the system (1) and general control law ${\mathit{u}}_d\left(t\right)$ (10) (or (14)), the following inequality is obtained:

$$\begin{array}{cc}\hfill {\dot{V}}_{\sigma }=& {\mathit{\sigma }}^T\left({\mathit{Gf}}_{\sigma }(\mathit{x},t)+\mathit{GB}(\mathit{x},t)\left(\mathit{I}+\mathsf{\Delta }\mathit{B}\right){\mathit{u}}_d\left(t\right)+\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right)\hfill \\ \hfill =& {\mathit{\sigma }}^T\left({\mathit{Gf}}_{\sigma }(\mathit{x},t)+\mathit{GB}(\mathit{x},t){\mathit{u}}_d\left(t\right)+\mathit{GB}(\mathit{x},t)\mathsf{\Delta }\mathit{B}{\mathit{u}}_d\left(t\right)+\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right)\hfill \\ \hfill <& {\mathit{\sigma }}^T\left(\mathit{GB}(\mathit{x},t)\mathsf{\Delta }\mathit{B}{\mathit{u}}_d\left(t\right)\right)={\mathit{v}}^T\mathsf{\Delta }\mathit{B}(\mathit{x},t){\mathit{u}}_d\left(t\right).\hfill \end{array}$$

(16)

It is evident that a residual term ${\mathit{v}}^T\mathsf{\Delta }\mathit{B}(\mathit{x},t){\mathit{u}}_d\left(t\right)$ appears in the last line of inequality ${\dot{V}}_{\sigma }$ and destroys the negative definiteness of ${\dot{V}}_{\sigma }$. To address this issue, the compensation term ${\mathit{u}}_c\left(t\right)$ is needed.

To design the compensation term ${\mathit{u}}_c\left(t\right)$, differentiating the Lyapunov function ${\dot{V}}_{\sigma }$ in (12) with respect to time based on (1) and (8), one obtains

$$\begin{array}{cc}\hfill {\dot{V}}_{\sigma }=& {\mathit{\sigma }}^T\left({\mathit{Gf}}_{\sigma }(\mathit{x},t)+\mathit{GB}(\mathit{x},t)\left(\mathit{I}+\mathsf{\Delta }\mathit{B}\right)\left({\mathit{u}}_d\left(t\right)+{\mathit{u}}_c\left(t\right)\right)+\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right)\hfill \\ \hfill <& {\mathit{v}}^T\mathsf{\Delta }\mathit{B}(\mathit{x},t){\mathit{u}}_d\left(t\right)+{\mathit{v}}^T(\mathit{I}+\mathsf{\Delta }\mathit{B}(\mathit{x},t)){\mathit{u}}_c\left(t\right)\hfill \\ \hfill =& {\mathit{v}}^T{\mathit{u}}_c\left(t\right)+{\mathit{v}}^T\mathsf{\Delta }\mathit{B}(\mathit{x},t)\left({\mathit{u}}_d\left(t\right)+{\mathit{u}}_c\left(t\right)\right).\hfill \end{array}$$

(17)

Let

$$\psi ={\mathit{v}}^T{\mathit{u}}_c\left(t\right)+{\mathit{v}}^T\mathsf{\Delta }\mathit{B}(\mathit{x},t)\left({\mathit{u}}_d\left(t\right)+{\mathit{u}}_c\left(t\right)\right).$$

(18)

It is apparent that if $\psi$ is nonpositive, the negative definiteness of ${\dot{V}}_{\sigma }$ is guaranteed. According to Assumption 3, $\psi$ can be further derived as follows:

$$\begin{array}{cc}\hfill \psi & ={\mathit{v}}^T{\mathit{u}}_c+{\mathit{v}}^T\mathsf{\Delta }\mathit{B}\mathit{u}\hfill \\ \hfill & \le {\mathit{v}}^T{\mathit{u}}_c+\sum _{i=1}^m\left|{\left[v\right]}_i\right|\left(\sum _{j=1}^m\left|{\left[\mathsf{\Delta }B\right]}_{i,j}\right|\left|{\left[u\right]}_j\right|\right)\hfill \\ \hfill & \le {\mathit{v}}^T{\mathit{u}}_c+\sum _{i=1}^m{\left[v\right]}_{i}sign\left({\left[v\right]}_i\right)\left(\sum _{j=1}^m{\left[\overline{\mathsf{\Delta }B}\right]}_{i,j}sign\left({\left[u\right]}_j\right){\left[u\right]}_j\right)\hfill \\ \hfill & ={\mathit{v}}^T{\mathit{u}}_c+{\mathit{v}}^{T}S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}S\left(\mathit{u}\right)\mathit{u}\hfill \\ \hfill & ={\mathit{v}}^T\left({\mathit{u}}_c+S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}S\left(\mathit{u}\right)\mathit{u}\right).\hfill \end{array}$$

(19)

Let

$${\mathit{u}}_c+S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}S\left(\mathit{u}\right)\mathit{u}=\mathbf{0},$$

(20)

then $\psi$ is nonpositive. To be clear, change (20) into the following form

$$\left(S\left(\mathit{u}\right)+S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}\right)\left|\mathit{u}\right|={\mathit{u}}_d.$$

(21)

Since ${\mathit{u}}_d$ has been designed in the first step, the only unknown in (21) is $\mathit{u}$ (if $\mathit{u}$ is determined, ${\mathit{u}}_c$ is determined). Then, the key problems are the following: (1) Whether there exists solution $\mathit{u}$ to Equation (21)? (2) If the solution exists, is it unique?

Before handling the above two problems, change (21) into a the following compact form:

$$\left(S+H\right)\mathit{p}={\mathit{u}}_d,$$

(22)

where $S=S\left(\mathit{u}\right)$, $\mathit{p}=\left|\mathit{u}\right|\ge \mathbf{0}$. According to Assumption 3, $H=S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}$ and satisfies

$$∥H∥=∥S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}∥\le ∥\overline{\mathsf{\Delta }\mathit{B}}∥<1.$$

Based on the definition of S, there is a total of $2^m$ different selections of S. Let the i-th selection of S be $S_i,i\in \left[2^m\right]$ and let $D=S+H$; then, there exists $2^m$ corresponding to D and the i-th selection of D can be denoted by $D_i=S_i+H,i\in \left[2^m\right]$. Thus, the following definitions are given

Definition 1. 

Regarding $D_i$, define the following closed convex cone of real vector space ${\mathbb{R}}^m$ as

$$C_i=\left\{D_i\mathit{p}\mid \forall \mathit{p}\ge \mathbf{0}\right\}.$$

(23)

Correspondingly, define the interior open region of $C_i$ as

$$C_i^o=\left\{D_i\mathit{p}\mid \forall \mathit{p}>\mathbf{0}\right\},$$

(24)

and define the conical surface of $C_i$ as

$${\widehat{C}}_i=C_i-C_i^o.$$

(25)

The objective of the following text is to solve $S\mathit{p}$ from (22) which is equivalent to finding the unique solution $\mathit{u}$ to Equation (21).

Based on the definition of the convex cone, the following Theorem 1 analyzes the existence and uniqueness of the solution to Equation (22). In order to support the proof of Theorem 1, three lemmas describing the properties of the convex cone $C_i$, the interior open region $C_i^o$ and the conical surface ${\widehat{C}}_i$ are first proposed.

Lemma 1. 

If $∥H∥<1$, $C_i^o\cap C_j^o=Ø$ for all $i\ne j,i,j\in \left[2^m\right]$.

Proof. 

Proof of contradiction is adopted here. Assume that an m-dimensional vector $\mathit{y}$ belongs to both $C_i^o$ and $C_j^o,i\ne j$, there exist two positive $\mathit{p}$s, denoted by ${\mathit{p}}_a$ and ${\mathit{p}}_b$, such that $\mathit{y}=D_i{\mathit{p}}_a=D_j{\mathit{p}}_b$. Then

$$S_i{\mathit{p}}_a-S_j{\mathit{p}}_b=S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}\left({\mathit{p}}_b-{\mathit{p}}_a\right).$$

(26)

If ${\mathit{p}}_a={\mathit{p}}_b$, $S_i=S_j$ can be derived from (26), then

$$D_i=S_i+S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}=S_j+S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}=D_j.$$

$C_i^o$ and $C_j^o$ are the same cone.

If ${\mathit{p}}_a\ne {\mathit{p}}_b$, the following inequality holds:

$$∥S_i{\mathit{p}}_a-S_j{\mathit{p}}_b∥\le ∥S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}∥∥{\mathit{p}}_b-{\mathit{p}}_a∥<∥{\mathit{p}}_b-{\mathit{p}}_a∥.$$

(27)

However, the following inequality also holds:

$$\left|S_i{\mathit{p}}_a-S_j{\mathit{p}}_b\right|\ge \left|{\mathit{p}}_b-{\mathit{p}}_a\right|,$$

(28)

since ${\mathit{p}}_a$ and ${\mathit{p}}_b$ are positive. Then, it is easy to obtain that

$$∥S_i{\mathit{p}}_a-S_j{\mathit{p}}_b∥\ge ∥{\mathit{p}}_b-{\mathit{p}}_a∥,$$

(29)

which derives the contradiction with the previous inequality (27). This contradiction indicates that $\mathit{y}$ cannot possibly simultaneously belong to $C_i^o$ and $C_j^o$ if $i\ne j$. Consequently, Lemma 1 is proved. □

Lemma 2. 

Any closed convex cone $C_i$ does not have an independent conical surface; namely, if an m-dimensional vector $\mathit{y}\in {\widehat{C}}_i$, there exists another conical surface ${\widehat{C}}_j,i\ne j$ that $\mathit{y}\in {\widehat{C}}_j$.

Proof. 

Suppose an m-dimensional vector $\mathit{y}\in {\widehat{C}}_i$. Then, $\mathit{y}=D_i\mathit{p}$ and there is at least one element of $\mathit{p}$ being zero. Therefore, there exists $S_i$ and $S_j$, $i\ne j$ so that $S_i\mathit{p}=S_j\mathit{p}$. It follows that $\mathit{y}\in {\widehat{C}}_j$. Consequently, the conical surface ${\widehat{C}}_i$ is not independent. Lemma 2 is proved. □

Lemma 3. 

If $∥H∥<1$, $C={\bigcup }_{i=1}^{2^m}{C}_i$ is equal to ${\mathbb{R}}^m$.

Proof. 

$C={\bigcup }_{i=1}^{2^m}{C}_i$ is the union of finite closed convex cones; thus, C is a closed cone. Correspondingly, the conical surface of C should be the union of the independent surfaces of $C_i,i\in \left[2^m\right]$; however, there is no independent conical surface for any one $C_i$ according to Lemma 2; therefore, the surface of C is an empty set. Since ${\mathbb{R}}^m$ itself is a closed cone with no conical surface, uniquely, $C={\mathbb{R}}^m$. □

Based on the above three lemmas, the following theorem is proposed in order to ensure the existence and uniqueness of the solution to Equation (22).

Theorem 1. 

For arbitrarily given vector ${\mathit{u}}_d\in {\mathbb{R}}^m$, the solution $S\mathit{p}$ of (22) exists and is unique if $∥H∥<1$.

Proof. 

If ${\mathit{u}}_d=\mathbf{0}$, since $∥H∥<1$, $D=S+H$ in (22) is nonsingular, the unique solution to the equation $D\mathit{p}=\mathbf{0}$ exists, which is $\mathit{p}=\mathbf{0}$.

If ${\mathit{u}}_d\ne \mathbf{0}$, the following proof will be divided into two parts: proof of existence and proof of uniqueness. □

Proof of Existence. 

According to Lemma 3, any ${\mathit{u}}_d\in {\mathbb{R}}^m$ belongs to at least one closed convex cone. Suppose a ${\mathit{u}}_d$ belongs to the convex cone $C_i$, then ${\mathit{u}}_d=D_i\mathit{p}$ holds with $\mathit{p}\ge \mathbf{0}$. Hence, the solution $S_i\mathit{p}$ exists. □

Proof of Uniqueness. 

If ${\mathit{u}}_d\in C_i^o$, ${\mathit{u}}_d\notin C_j^o,\forall j\ne i$, according to Lemma 1. Then, ${\mathit{u}}_d$ can be uniquely expressed as ${\mathit{u}}_d=D_i\mathit{p}=\left(S_i+H\right)\mathit{p}$ with $\mathit{p}>\mathbf{0}$ and $\mathit{u}=S_i\mathit{p}$ is unique.

If ${\mathit{u}}_d\in {\widehat{C}}_i$, at least one element in $\mathit{p}$ is zero according to the definition of conical surface. Based on Assumption 2, there exist at least two selections of S, denoted by $S_i$ and $S_j$, $i\ne j$, such that $S_i\mathit{p}=S_j\mathit{p}$. It follows that $S\mathit{p}$ is unique. □

According to Theorem 1, there exists a unique solution to Equation (22) if $∥\mathit{H}∥<1$. Namely, if $∥S\left(\mathit{v}\right)\overline{\mathsf{\Delta }\mathit{B}}∥<1$, there exists a solution $\mathit{u}$ to Equation (21) and this solution is unique. Subsequently, the key point is to solve the solution from (21). Algorithm 1 presents an algorithmic process, in which $\mathit{u}$ can be obtained by the traversal of the selections of S. Usually, the low dimension of the control input $\mathit{u}$ speeds up the traversal process. Additionally, the compensation control term ${\mathit{u}}_c$ can be obtained by $\mathit{u}-{\mathit{u}}_d$.

Algorithm 1 Calculation of the control law $\mathit{u}$

Input: m, $\overline{\mathsf{\Delta }\mathit{B}}$ Output: $\mathit{u}$;   1: Design ${\mathit{u}}_d$ and calculate H;   2: $i=1$;   3: $flag=True$;   4: while  $flag$  do   5:     Calculate $\mathit{p}={\left(S_i+H\right)}^{-1}{\mathit{u}}_d$;   6:     if sign($\mathit{p}$) $>\mathbf{0}$ and $i\le 2^m$ then   7:         $flag=False$;   8:         $\mathit{u}=S_i\mathit{p}$;   9:     else 10:         $i=i+1$; 11:     end if 12: end while 13: return $\mathit{u}$;

Remark 1. 

In Assumption 3, $\overline{\mathsf{\Delta }\mathit{B}}$ is bounded in Euclidean norm. In fact, $\overline{\mathsf{\Delta }\mathit{B}}$ can also be bounded in $L_1$ norm and $L_{\infty }$ norm, which are all suitable for Theorem 1.

Remark 2. 

In the proposed method, ${\mathit{u}}_d$ and ${\mathit{u}}_c$ are designed in separate two cascade steps. Such a design strategy reduces the difficulties of the control design for uncertain system (1). Furthermore, ${\mathit{u}}_d$ and ${\mathit{u}}_c$ are designed based on the Lyapunov function and Lyapunov stability theorem, so ${\mathit{u}}_d$ can also be obtained by any other control method following the Lyapunov stability theorem, such as the Backstepping method, the LMI robust method, etc, not limited to SMC.

4. Stability Analysis

The stability of the system in sliding mode is discussed under two scenarios.

• If system (1) can be transformed into the regular canonical form in (3), linear sliding surface is defined as (4). During the sliding mode,

  $${\mathit{x}}_2=-{\mathit{G}}_1{\mathit{x}}_1.$$

  (30)
  Taking (4) into the subsystem of (3), i.e., ${\dot{\mathit{x}}}_1={\mathit{f}}_1({\mathit{x}}_1,{\mathit{x}}_2)$, one obtains

  $${\dot{\mathit{x}}}_1={\mathit{f}}_1({\mathit{x}}_1,-{\mathit{G}}_1{\mathit{x}}_1).$$

  (31)
  Since ${\mathit{G}}_1$ is a diagonal positive definite matrix, ${\mathit{x}}_2=-{\mathit{G}}_1·\mathbf{0}=\mathbf{0}$. Supposing that the origin of system (31), i.e., ${\mathit{x}}_1=\mathbf{0}$ is globally asymptotically stable, if not considering $\mathsf{\Delta }\mathit{B}(\mathit{x},t)$, there exists a gain $\mathit{K}(·)$ providing a global asymptotic stabilization of the origin with respect to (3) by means of the control [24,25,30]:

  $${\mathit{u}}_r\left(t\right)=-{\left(\mathit{GB}\left(\mathit{x},t\right)\right)}^{-1}\mathit{Gf}\left(\mathit{x},t\right)-\mathit{K}sign\left(\mathit{\sigma }\right)$$

  (32)
  One can observe that ${\mathit{u}}_d\left(t\right)$ in (14) has the same form as ${\mathit{u}}_r\left(t\right)$ in (32) and $\mathit{K}$ in (32) is equal to ${\left(\mathit{GB}\left(\mathit{x},t\right)\right)}^{-1}\rho (\mathit{x},t)$ in (14). Therefore, ${\mathit{u}}_d\left(t\right)$ in (14), which ensures the enforcement of sliding mode, is able to provide a global asymptotic stabilization of the origin with respect to (3) without $\mathsf{\Delta }\mathit{B}(\mathit{x},t)$ [25].
  Since $\mathsf{\Delta }\mathit{B}(\mathit{x},t)$ also satisfies the matched condition and the compensation control term proposed in this paper can suppress its effect and guarantee the system reachability, in conclusion, the global asymptotic stabilization of system (3) can be achieved by combining the general control ${\mathit{u}}_d$ in (14) and the compensation term ${\mathit{u}}_c$ proposed in this paper.
• If integral sliding surface (5) is adopted here, considering Assumption 3, as well as the non-singularity of $\mathit{GB}(\mathit{x},t)$, the equivalent control can be solved from equation $\dot{\mathit{\sigma }}=\mathbf{0}$ [10]:

  $${\mathit{u}}_{eq}\left(t\right)={\left(\mathit{I}+\mathsf{\Delta }\mathit{B}(\mathit{x},t)\right)}^{-1}\left(\mathit{\kappa }(\mathit{x},t)-\mathsf{\Delta }{\mathit{f}}_m(\mathit{x},t)\right).$$

  (33)
  Then, the expression for the sliding motion can be obtained by substituting the value of ${\mathit{u}}_{eq}\left(t\right)$ from (1) into (33), which yields

  $$\dot{\mathit{x}}\left(t\right)=\mathit{f}(\mathit{x},t)+\mathit{B}(\mathit{x},t)\mathit{\kappa }(\mathit{x},t),$$

  (34)
  One can find the origin of system (1) in the sliding mode is globally asymptotically stable [10], since $\mathit{\kappa }(\mathit{x},t)$ can globally asymptotically stabilize system (34).

5. Numerical Simulation Results

To evaluate the effectiveness of the proposed method, we consider two systems with different structures: a nonlinear plant and a two-link robot manipulator. An SMC proposed by Cao and Xu [10] and another SMC proposed by Feng et al. [21] are used to compare the proposed method. In the following text, we denote Cao and Xu’s method cSMC, denote the method proposed by Feng et al. fSMC and denote the proposed method pSMC.

5.1. Nonlinear Plant with Matched Uncertainties

Consider a nonlinear plant with the following parameters:

$$\mathit{f}=\left[\begin{array}{ccc}1& -2& 3\\ -4& 5& -6\\ 7& -8& 9\end{array}\right]\mathit{x}+\left[\begin{array}{c}0\\ 0\\ 0.2x_3{sin}^2\left(x_1\right)\end{array}\right],$$

$$\mathit{B}=\left[\begin{array}{cc}1& -2\\ -3+0.1{cos}^3\left(x_2\right)& 4\\ 5& 6+0.1cos\left(x_3^2\right)\end{array}\right],$$

$$\mathsf{\Delta }{\mathit{f}}_m=\left[\begin{array}{c}0.1x_1^2+0.1\\ 0.1x_1^2+0.2x_2^3+0.1x_3^4+0.1\end{array}\right],$$

$$\mathsf{\Delta }\mathit{B}=-{\rho }_b\left[\begin{array}{cc}sin\left(x_2\right)& 0.15\ast cos\left(x_3\right)\\ 0.01\ast {sin}^3\left(x_1\right)& {cos}^2\left(x_1\right)\end{array}\right].$$

where $\mathit{x}={\left[\begin{array}{ccc}{x}_1& x_2& x_3\end{array}\right]}^T\in {\mathbb{R}}^3$ is the state vector. The initial value $\mathit{x}\left(0\right)={\left[\begin{array}{ccc}1& 1& -1\end{array}\right]}^T$. The control variable $\mathit{u}={\left[\begin{array}{cc}{u}_1& u_2\end{array}\right]}^T\in {\mathbb{R}}^2$. ${\rho }_m$ is the upper bound of the Euclidean norm of $\mathsf{\Delta }{\mathit{f}}_m$.

The simulation is performed based on the actual control system. The control signal is calculated by a digital controller based on the current states at each sampling time and is kept unchanged until the next sampling time. Since the parameters of the controlled system are all continuous functions of time, the Runge–Kutta (RK4) method is applied in the simulation. The sampling interval is $T_s$; the trial duration is T.

Since it is difficult to find a diffeomorphism to transfer the system into the regular form, integral sliding surface is utilized here. Select the integral sliding surface as

$$\mathit{\sigma }=\mathit{G}\mathit{x}-\mathit{G}\mathit{x}\left(0\right)-{\int }_0^t(\mathit{G}\mathit{f}+\mathit{G}\mathit{B}\mathit{\kappa })d\tau =\mathbf{0},$$

(35)

where $\mathit{\sigma }={\left[\begin{array}{cc}{\sigma }_1& {\sigma }_2\end{array}\right]}^T\in {\mathbb{R}}^2$ is the sliding variables. $\mathit{G}$ is constant matrix; $\mathit{GB}$ is non-singular. $\mathit{\kappa }=-\mathit{Kx}$ and $\mathit{K}$ is constant matrix. The system parameter values are listed in

Table 1. Parameter Values.

Parameter	Value
${\rho }_m$	$0.2x_1^2+0.2{\left|x_2\right|}^3+0.1x_3^4+0.2$
${\rho }_b$	${\rho }_b<1$
$T_s$	1 (ms)
T	15 (s)
$\mathit{G}$	$\left[\begin{array}{ccc}0& 0.1& 0.2\\ 0.4& -0.1& 0.2\end{array}\right]$
$\mathit{K}$	$\left[\begin{array}{ccc}1.7902& -1.8765& 2.4429\\ -0.0444& 0.2676& 0.0344\end{array}\right]$

. These settings are cited from [10] for a fair comparison. Based on the stability analysis in [10], the value of $\mathit{K}$ is able to ensure the global asymptotical stability of system $\dot{\mathit{x}}=\mathit{f}+\mathit{B}\mathit{\kappa }$.

The first simulation is implemented to demonstrate the influence of input matrix uncertainty on the system. Figure 1 shows the state variable performances in the cases of ${\rho }_b=0.1$ and ${\rho }_b=0.9$ by using cSMC [10]. The control law of cSMC is formulated as [10]

$$\mathit{u}\left(t\right)=\mathit{\kappa }-\rho {\mathit{v}}_0,$$

(36)

where

$$\rho >{\displaystyle \frac{1}{1-{\rho }_b}}\left({\rho }_b∥\mathit{\kappa }∥+{\rho }_m\right).$$

Based on the second case (integral sliding surface) in Section 4, the control law of cSMC can ensure the uncertain system is globally asymptotically stable.

It can be found that three state variables $x_1$, $x_2$ and $x_3$ show obvious oscillations with large amplitude when ${\rho }_b=0.9$, versus much slighter oscillations when ${\rho }_b=0.1$. It is more clear from the insets over 2 s to 2.09 s that the amplitude and frequency of the oscillations in state variables are higher when ${\rho }_b=0.9$. Similarly, in Figure 2, the magnitude of control variables $u_1$ and $u_2$ are much larger when ${\rho }_b=0.9$ than those when ${\rho }_b=0.1$. The reason is that a larger input matrix uncertainty brings a more severe disturbance to the control. Therefore, a more powerful control with larger magnitude of control is needed to suppress the disturbance, which will necessarily increase the chattering amplitude.

The second simulation is conducted to compare the effects of cSMC, fSMC and pSMC under ${\rho }_b=0.9$. The control law of cSMC is formulated as (36). The control law of fSMC is formulated as [21]

$$\begin{array}{cc}\hfill & {\mathit{u}}_d=\mathit{\kappa }-\rho {\mathit{v}}_0\hfill \\ \hfill & {\mathit{u}}_c=-\left(\sqrt{{\beta }^2{\mu }^2+\beta }-\beta \mu \right)∥{\mathit{u}}_d∥{\mathit{v}}_0,\hfill \end{array}$$

(37)

where $\rho >{\rho }_m$, $\beta ={\displaystyle \frac{{\rho }_b^2}{1-{\rho }_b^2}}$, $\mu =\left\{\begin{array}{cc}{\mathit{v}}_0^T{\mathit{u}}_d\left(t\right)/∥{\mathit{u}}_d\left(t\right)∥,\hfill & ∥{\mathit{u}}_d\left(t\right)∥>0\hfill \\ 0,\hfill & ∥{\mathit{u}}_d\left(t\right)∥=0,\hfill \end{array}\right.$.

For pSMC, ${\mathit{u}}_d$ is the same as ${\mathit{u}}_d$ in (37). Then, the control law of pSMC can be obtained according to the algorithmic process in Algorithm 1, where the upper bound matrix $\overline{\mathsf{\Delta }\mathit{B}}=\left[\begin{array}{cc}{\rho }_b& 0.15\ast {\rho }_b\\ 0.01\ast {\rho }_b& {\rho }_b\end{array}\right]$ and $∥\overline{\mathsf{\Delta }\mathit{B}}∥=0.9742<1$, which satisfies Assumption 3. Based on the second case (integral sliding surface) in Section 4, all three methods can ensure that the uncertain system is globally asymptotically stable.

Figure 3 shows the performances of control variables $u_1$ and $u_2$ under cSMC, fSMC and pSMC. One can find that the oscillations of both $u_1$ and $u_2$ under pSMC are obviously the slightest among the three methods, versus fSMC and cSMC with much more severe oscillations existing in the trajectories. From the insets over 14 s to $14.09$ s, it is evident that pSMC outperforms fSMC and cSMC in reducing both the magnitude and frequency of the control impulse. Such an impulse can be viewed as an adverse impact on the system, which leads to chattering. To measure the impulse and statistically analyze the impact, an index termed Mean Jump Value (MJV) is introduced. Assume that the value of $u_i,i=1,2$ at the j-th sampling point is $u_i\left(j\right),j=1,2,\dots ,n=T/T_s$, the MJV across the interval $T_a$ to $T_b$ is defined as ${\varphi }_i(T_a,T_b)$ and computed according to

$${\varphi }_i(T_a,T_b)=\frac{1}{(T_b-T_a)/T_s-1}\sum _{j=T_a/T_s}^{T_b/T_s-1}\left|u_i(j+1)-u_i\left(j\right)\right|,$$

(38)

where $0\le T_a<T_b\le T$, $T_a$ and $T_b$ can be divisible by $T_s$. A small MJV indicates small impulse of the control, which causes lighter chattering.

The MJVs of the control variables in the first five seconds ($T_a=0$ s, $T_b=5$ s) and for the entire time ($T_a=0$ s, $T_b=15$ s) with two methods are listed in

Table 2. MJVs of cSMC, fSMC and pSMC at time intervals [0 s, 5 s], [10 s, 15 s] and [0 s, 15 s].

	${\mathit{\varphi }}_1(0,5)$	${\mathit{\varphi }}_2(0,5)$	${\mathit{\varphi }}_1(10,15)$	${\mathit{\varphi }}_2(10,15)$	${\mathit{\varphi }}_1(0,15)$	${\mathit{\varphi }}_2(0,15)$
cSMC	37.1501	23.4337	21.5019	6.9049	26.6317	12.4571
fSMC	8.9872	4.0914	20.5888	6.6641	16.6333	5.8035
pSMC	4.2052	1.1294	6.4453	2.2433	5.6850	1.8903
cSMC/pSMC	8.8343	20.7494	3.3361	3.0780	4.6846	6.5898
fSMC/pSMC	2.1372	3.6227	3.1944	2.9707	2.9258	3.0701

.

Table 2 presents the MJV comparisons among cSMC, fSMC and pSMC in the first five seconds, the last five seconds and the entire trial time. A clear advantage of pSMC is shown: during the first five seconds, the MJVs under cSMC are more than 8 times and 20 times larger than those under pSMC for $u_1$ and $u_2$, respectively. The MJVs under fSMC are more than 2 times and 3 times larger than those under pSMC, respectively. During the last five seconds, the MJVs under cSMC and fSMC are approximately three times larger than those under pSMC. Throughout the entire trial time, the MJVs for $u_1$ and $u_2$ under cSMC are greater than one-fourth and one-sixth proportional to those under pSMC, respectively. The MJVs under fSMC are approximately three times larger than those under pSMC. Figure 4a,b depict the comparisons on sliding variables ${\sigma }_1$ and ${\sigma }_2$, respectively, which illustrates the smaller magnitude of trajectories under pSMC in comparison to those under cSMC and fSMC. Figure 5 presents the performances of the state variables under all three methods. One can observe that chattering under cSMC and fSMC are obviously more severe, particularly for state variables $x_2$ in Figure 5b and $x_3$ in Figure 5c. A clearer result can be demonstrated in the insets, which cover the time period from 14 s to $14.09$ s.

Finally, the boundary layer solution is applied to ${\mathit{v}}_0$ in the discontinuous control of cSMC, fSMC and pSMC as follows:

$${\mathit{v}}_0=\left\{\begin{array}{cc}{\mathit{v}}_0,\hfill & \parallel \mathit{v}\parallel >\epsilon \hfill \\ {\displaystyle \frac{\mathit{v}}{\epsilon }},\hfill & \parallel \mathit{v}\parallel \le \epsilon ,\hfill \end{array}\right.$$

where $\epsilon =0.011$.

Figure 6 depicts the comparisons of the control variables among the three methods. It is evident that under cSMC, there is no oscillation present in the lines of variables, whereas under cSMC and fSMC, oscillations are present in the entire trajectories. Figure 7 presents the trajectories of the state variables and confirms the superiority of pSMC: there is no chattering in the state trajectories under pSMC, whereas chattering is present in every state variable under cSMC and fSMC. Based on these results, In conclusion, pSMC outperforms cSMC and fSMC in reducing chattering.

5.2. A Two-Link Robot Manipulator

The dynamic equation of the two-link robot manipulator shown in Figure 8 is given by [9,24] and formulated as

$$\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)=\mathit{\tau },$$

(39)

where $\mathit{M}\left(\mathit{q}\right)$ is the moment matrix of inertia, $\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}$ is the Coriolis centripetal torque vector and $\mathit{G}\left(\mathit{q}\right)$ is the the gravitational torque vector. $\mathit{q}={\left[\begin{array}{cc}{q}_1\hfill & q_2\hfill \end{array}\right]}^T$ represents the angular positions; $\mathit{\tau }$ is the vector of applied torques.

For convenience, $s_i=sin\left(q_i\right)$, $c_i=cos\left(q_i\right)$ for $i=1,2$. ${\mathit{q}}_d={\left[\begin{array}{cc}{q}_{1d}\hfill & q_{2d}\hfill \end{array}\right]}^T$ is the target angular positions. The control objective is to control the robot angular positions to track the given target trajectories. Define the vectors of state, target and input as

$$\begin{array}{c}\mathit{x}={\left[\begin{array}{cccc}{q}_1\hfill & q_2\hfill & {\dot{q}}_1\hfill & {\dot{q}}_2\hfill \end{array}\right]}^T,\end{array}$$

(40)

$$\begin{array}{c}{\mathit{x}}_d={\left[\begin{array}{cccc}{q}_{1d}\hfill & q_{2d}\hfill & {\dot{q}}_{1d}\hfill & {\dot{q}}_{2d}\hfill \end{array}\right]}^T,\end{array}$$

(41)

$$\begin{array}{c}\mathit{u}=\mathit{\tau }.\end{array}$$

(42)

Then, the dynamic equation of the two-link manipulator in state-space form can be written as

$$\dot{\mathit{x}}=\mathit{f}\left(\mathit{x}\right)+\mathit{B}\left(\mathit{x}\right)\mathit{u},$$

(43)

where

$$\begin{array}{cc}\hfill {}_1& ={\left[x\right]}_3,{\left[f\left(\mathit{x}\right)\right]}_2={\left[x\right]}_4,\hfill \\ \hfill {\left[f\right(\mathit{x}\left)\right]}_3& =\frac{\left(s_1{c}_3-c_1{s}_3\right)\times \left[m_2{l}_1{l}_2\left(s_1{s}_3+c_1{c}_3\right)x_2^2-m_2{l}_2^2{x}_4^2\right]}{l_1{l}_2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]}\hfill \\ & +\frac{\left[\left(m_1+m_2\right)l_{2}g{s}_1-m_2{l}_{2}g{s}_3\left(s_1{s}_3+c_1{c}_3\right)\right]}{l_1{l}_2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]}\hfill \end{array},$$

$$\begin{array}{cc}\hfill {}_4=& \frac{\left(s_1{c}_3-c_1{s}_3\right)\times \left[-\left(m_1+m_2\right)l_1{x}_2^2+m_2{l}_1{l}_2\left(s_1{s}_3+c_1{c}_3\right)x_4^2\right]}{l_1{l}_2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]}\hfill \\ & +\frac{\left[-\left(m_1+m_2\right)l_{1}g{s}_1\left(s_1{s}_3+c_1{c}_3\right)+\left(m_1+m_2\right)l_{1}g{s}_3\right]}{l_1{l}_2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]},\hfill \end{array}$$

${\left[B\left(\mathit{x}\right)\right]}_{1,1}={\left[B\left(\mathit{x}\right)\right]}_{1,2}={\left[B\left(\mathit{x}\right)\right]}_{2,1}={\left[B\left(\mathit{x}\right)\right]}_{2,2}=0$,

$${\left[B\left(\mathit{x}\right)\right]}_{3,1}=\frac{m_2{l}_2^2}{m_2{l}_1^2{l}_2^2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]},$$

$${\left[B\left(\mathit{x}\right)\right]}_{3,2}=\frac{-m_2{l}_1{l}_2\left(s_1{s}_3+c_1{c}_3\right)}{m_2{l}_1^2{l}_2^2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]},$$

$${\left[B\left(\mathit{x}\right)\right]}_{4,1}=\frac{-m_2{l}_1{l}_2\left(s_1{s}_3+c_1{c}_3\right)}{m_2{l}_1^2{l}_2^2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]},$$

$${\left[B\left(\mathit{x}\right)\right]}_{4,2}=\frac{\left(m_1+m_2\right)l_1^2}{m_2{l}_1^2{l}_2^2\left[\left(m_1+m_2\right)-m_2{\left(s_1{s}_3+c_1{c}_3\right)}^2\right]}.$$

where $m_1$ and $m_2$ are masses of the links and $l_1$ and $l_2$ are lengths of the links.

In fact, uncertainties exist in the system. The additive matched uncertainty and input matrix uncertainty are

$$\mathsf{\Delta }{\mathit{f}}_m=3{\left[\begin{array}{cc}{\left[f\right]}_3\hfill & {\left[f\right]}_4\hfill \end{array}\right]}^T+{\left[\begin{array}{cc}0.015\hfill & 0.015\hfill \end{array}\right]}^T,$$

$$\mathsf{\Delta }\mathit{B}={\rho }_b\left[\begin{array}{cc}{s}_1& c_1^2\\ 0& s_3^2\end{array}\right].$$

The upper bound of $\mathsf{\Delta }{\mathit{f}}_m$ is ${\rho }_m=3\left(\left|{\left[f\right]}_3\right|+\left|{\left[f\right]}_4\right|\right)+0.03$, the upper bound matrix $\overline{\mathsf{\Delta }\mathit{B}}={\rho }_b\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$. With consideration of uncertainties, system (43) can be written as

$$\dot{\mathit{x}}=\mathit{f}+\mathit{B}\left(\left(\mathit{I}+\mathsf{\Delta }\mathit{B}\right)\mathit{u}+\mathsf{\Delta }{\mathit{f}}_m\right).$$

(44)

Since system (44) can be transformed into the following regular form [25,26,30]:

$$\begin{array}{cc}\hfill {\dot{\mathit{x}}}_1& ={\mathit{x}}_2,\hfill \\ \hfill {\dot{\mathit{x}}}_2& ={\mathit{f}}_2({\mathit{x}}_1,{\mathit{x}}_2)+{\mathit{B}}_2({\mathit{x}}_1,{\mathit{x}}_2)\left(\left(\mathit{I}+\mathsf{\Delta }\mathit{B}({\mathit{x}}_1,{\mathit{x}}_2)\right)\mathit{u}+\mathsf{\Delta }{\mathit{f}}_m({\mathit{x}}_1,{\mathit{x}}_2)\right)\hfill \end{array}$$

(45)

where ${\mathit{x}}_1={\left[\begin{array}{cc}{q}_1\hfill & q_2\hfill \end{array}\right]}^T$, ${\mathit{x}}_2={\left[\begin{array}{cc}{\dot{q}}_1\hfill & {\dot{q}}_2\hfill \end{array}\right]}^T$, ${\mathit{f}}_2({\mathit{x}}_1,{\mathit{x}}_2)={\left[\begin{array}{cc}{\left[f\right]}_2\hfill & {\left[f\right]}_4\hfill \end{array}\right]}^T$ and ${\mathit{B}}_2({\mathit{x}}_1,{\mathit{x}}_2)={\left[\begin{array}{cc}{\left[B\right]}_{2,1}\hfill & {\left[B\right]}_{2,2}\hfill \\ {\left[B\right]}_{4,1}\hfill & {\left[B\right]}_{4,2}\hfill \end{array}\right]}^T$. Then, define the linear sliding surface as

$$\sigma ={\mathit{x}}_2+{\mathit{G}}_1{\mathit{x}}_1=\mathit{Gx}.$$

(46)

The initial state is ${\mathit{x}}_0={\left[\begin{array}{cccc}0.01\hfill & 0.01\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$. The target state is ${\mathit{x}}_d={\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$.

The simulation is performed based on the actual control system. The control signal is calculated by a digital controller based on the current states at each sampling time and is kept unchanged until the next sampling time. Since the parameters of the controlled system are all continuous functions of time, the Runge–Kutta (RK4) method is applied in the simulation. The sampling interval is $T_s$; the trial duration is T. The parameters’ values are listed in

Table 3. Parameter Values.

Parameter	Value
$m_1$	1 (kg)
$m_2$	1 (kg)
$l_1$	1 (m)
$l_2$	1 (m)
g	9.8 (m/s${}^2$)
T	6 (s)
$T_s$	1 (ms)
${\mathit{G}}_1$	$\left[\begin{array}{cc}-5& 0\\ 0& -5\end{array}\right]$
$\mathit{G}$	$\left[\begin{array}{cccc}-5& 0& 1& 0\\ 0& -5& 0& 1\end{array}\right]$

.

In the first simulation, we attempt to investigate the influence of input matrix uncertainty on the system. Under cSMC [10], the cases of ${\rho }_b=0$ and ${\rho }_b=0.607$ are compared. For ${\rho }_b=0.607$, $∥\overline{\mathsf{\Delta }\mathit{B}}∥\le {\rho }_b\ast ∥\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]∥=0.9821<1$, which satisfies Assumption 3. The control law of cSMC is

$$\begin{array}{cc}\hfill {\mathit{u}}_d& =-{\left(\mathit{GB}\right)}^{-1}\left(\mathit{Gf}+\rho sign\left(\mathit{\sigma }\right)\right)\hfill \\ \hfill {\mathit{u}}_c& =-\frac{{\rho }_b}{1-{\rho }_b}∥{\mathit{u}}_d\left(t\right)∥{\mathit{v}}_0\hfill \end{array}$$

(47)

where

$$\rho >∥\mathit{GB}∥{\rho }_m.$$

Based on the first case in Section 4, cSMC is able to ensure the global asymptotical stability of the system with uncertainties.

Figure 9 depicts the performances of state variables $q_1$ and $q_2$ in the case of ${\rho }_b=0.607$ and ${\rho }_b=0$. One can observe that obvious chattering and severe overshoots exist before 2 s in the state trajectories when ${\rho }_b=0.607$. The states converge to zero without chattering and overshoot when ${\rho }_b=0$. Figure 10 plots the control performances when ${\rho }_b=0.607$ and ${\rho }_b=0$. It is evident that before 2 s, the magnitude and oscillations of the control variables when ${\rho }_b=0.607$ are larger than those when ${\rho }_b=0$. In conclusion, input matrix uncertainty has adverse effects on systems and leads to the large magnitude of control gain, which aggravates the chattering effect.

The second simulation is conducted to investigate the performance of pSMC by comparing with cSMC and fSMC. ${\rho }_b=0.6$ and $∥\overline{\mathsf{\Delta }\mathit{B}}∥\le {\rho }_b\ast ∥\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]∥=0.9708<1$, which also satisfies Assumption 3.

The control law of fSMC is formulated as

$$\begin{array}{cc}\hfill & {\mathit{u}}_d=-{\left(\mathit{GB}\right)}^{-1}\left(\mathit{Gf}+\rho sign\left(\mathit{\sigma }\right)\right)\hfill \\ \hfill & {\mathit{u}}_c=-\left(\sqrt{{\beta }^2{\mu }^2+\beta }-\beta \mu \right)∥{\mathit{u}}_d∥{\mathit{v}}_0,\hfill \end{array}$$

(48)

where $\rho >∥\mathit{GB}∥{\rho }_m$, $\beta ={\displaystyle \frac{{\rho }_b^2}{1-{\rho }_b^2}}$, $\mu =\left\{\begin{array}{cc}{v}_0^T{\mathit{u}}_d/∥{\mathit{u}}_d∥,\hfill & ∥{\mathit{u}}_d∥>0\hfill \\ 0,\hfill & ∥{\mathit{u}}_d∥=0,\hfill \end{array}\right.$.

For pSMC, ${\mathit{u}}_d$ is the same as ${\mathit{u}}_d$ in cSMC and fSMC; $\mathit{u}$ can be calculated according to the process Algorithm 1. Based on the first case in Section 4, all three methods can ensure the global asymptotical stability of the system.

Figure 11 shows the comparisons of the system states $q_1$ and $q_2$ among cSMC, fSMC and pSMC. One can observe that compared to cSMC and fSMC, pSMC demonstrates superior performance in terms of convergence level, overshoot reduction and chattering suppression. To be specific, for cSMC, both $q_1$ and $q_2$ exhibit severe chattering and large overshoots before 2 s, whereas no chattering and overshoots exist in the state trajectories under fSMC and pSMC. The insets over the period from 5 s to $5.09$ s visualize that pSMC outperforms fSMC and cSMC in reducing the steady-state errors.

Figure 12 shows the comparisons of the control variables $u_1$ and $u_2$ under cSMC, fSMC and pSMC. As illustrated from the original images and the insets over 5 s to $5.09$ s, the oscillations in the variable trajectories under pSMC are featured with smaller magnitude and notably lower frequency in comparison with the oscillations under cSMC and fSMC. The consistent results are reported in

Table 4. MJVs of cSMC, fSMC and pSMC.

	${\mathit{\varphi }}_1(0,6)$	${\mathit{\varphi }}_2(0,6)$
cSMC	7.5517	12.0323
fSMC	3.8651	5.7159
pSMC	0.6051	26.2142
cSMC/pSMC	12.4810	55.1825
fSMC/pSMC	6.3881	26.2142

. It is shown that the MJVs of $u_1$ and $u_2$ under cSMC are more than 12 times and 55 times greater than those under pSMC, respectively, and the MJVs under fSMC are more than 6 times and 26 times larger than those under pSMC, respectively. Similarly, in Figure 13, larger magnitude and higher frequency of the oscillations are shown in the performances of sliding variables ${\sigma }_1$ and ${\sigma }_2$ under cSMC and fSMC than those under pSMC. Consequently, pSMC is more effective than cSMC and fSMC at mitigating the steady-state errors, while simultaneously reducing the magnitude and frequency of oscillations in the discontinuous control, which aids in the suppression of chattering.

In the third simulation, the saturation function $sat(\mathit{\sigma }/\epsilon )$ is utilized to replace the discontinuous controls in cSMC, fSMC and pSMC for continuous approximation, where $\epsilon =0.0005$.

Figure 14 and Figure 15 depict the performance comparisons of state variables and control variables among three control methods. As shown in Figure 15, only under pSMC, trajectories do not exhibit oscillations, whereas obvious oscillations appear in the entire lines under cSMC and fSMC. Figure 14 demonstrates that the continuous approximation in control reduces chattering in all three methods. However, cSMC still exhibits some minor chattering in the state trajectories and the steady-state error under pSMC remains the smallest of the three approaches. Based on the above analysis, it can be concluded that pSMC is able to eliminate chattering with smaller $\epsilon$.

Finally, simulation is implemented in the case where the initial point ${\mathit{x}}_0$ is selected far from the target state ${\mathit{x}}_d=\mathbf{0}$. Choose ${\mathit{x}}_0={\left[\begin{array}{cccc}0.1\hfill & 0.1\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$. The saturation function $sat(\mathit{\sigma }/\epsilon )$ is applied to three control methods for continuous approximation, where $\epsilon =0.0005$.

Figure 16 presents the performances of state variables $q_1$ and $q_2$ under cSMC, fSMC and pSMC. The comparisons indicates that pSMC maintains the same high quality performances, including good convergence to the desired states and the absence of chattering. However, larger steady-state errors are present in the trajectories under fSMC and chattering exists in the trajectories before 2 s under cSMC. Figure 17 illustrates the comparisons of control variables among three methods. Evidently, with the exception of a slight oscillation at the very beginning of the line in $u_1$, there is no oscillation present in the control variables under pSMC. Both cSMC and fSMC display continuous oscillations throughout the entire trial time. In conclusion, pSMC is effective in improving convergence, reducing overshoot and suppressing chattering.

6. Conclusions

In this paper, a novel SMC method is proposed for affine systems with input matrix uncertainty. To facilitate the controller design, the proposed method builds on a two-step strategy. In the first step, a general control law is adopted for the system without considering input matrix uncertainty; in the second step, a novel control term is designed to specially address input matrix uncertainty. This term is able to compensate for every element of the uncertain input matrix and can be obtained by solving a nonlinear vector equation. A theorem and three lemmas are proposed to guarantee the existence and the uniqueness of the equation solution and an algorithmic process is proposed to solve the vector equation. Numerical simulations on two affine uncertain systems illustrate that the proposed method can provide desired closed-loop system performances in the presence of input uncertainties. Furthermore, the simulation results revealed that the proposed method (pSMC) outperforms the SMC approach proposed by Cao and Xu (cSMC) and by Feng et al. (fSMC) in terms of improving convergence, reducing overshoot and alleviating the chattering problem.