H∞ observer-based sliding mode control for uncertain discrete-time singularly perturbed systems

Abstract

This paper studies the sliding mode control (SMC) problem of discrete-time singularly perturbed systems (DTSPSs) with external perturbations and nonlinear inputs. First, to estimate the state of the original system, a novel state observer is constructed. Under this condition, the observer error system is ensured to be input-to-state stable (ISS) and satisfies the given H∞ performance index. Then, an appropriate sliding mode control law is proposed to satisfy the reachability condition. On this basis, the ISS criterion for the sliding mode dynamics (SMDs) is given, and the controller gain matrix is also obtained. In addition, a feasible algorithm is given to compute the upper bound for the small parameter. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed method.

1 Introduction

Singularly perturbed systems (SPSs) are an important research topic in the field of automatic control and applied mechanics, physics, astronomy and other domains [1–5]. However, in practical engineering problems, uncertainty and external disturbances frequently induce instability and are usually considered as the main factors for system performance degradation [6, 7]. So, the discussion on the topic is meaningful.

As is well known, the control of the SPSs has received considerable attention in the past decades [8–14], while the SMC is one of most effective robust control policies for many kinds of systems, which gives the SMC great research value in the field of industry, mechanical engineering, and aerospace industry [15–19]. Yang and Che in [20] constructed a stable sliding surface based on the reachability condition and derived a criterion by linear matrix inequality (LMI) technology and ε bound estimation method. The criterion guaranteed that the closed-loop system is internally exponentially stable. Reference [21] investigated the slow and fast subsystems using the sliding mode control theory. Then, a sufficient condition was provided such that the full-order closed-loop SPS is asymptotically stable. Recently, the relevant results have been extended to stochastic control [22, 23]. In [22], the finite-time bounded problem for stochastic nonlinear singular systems is addressed, in which an appropriate sliding mode function is constructed such that the system exhibits the finite-time bounded stability with the prescribed H∞ performance.

The discrete-time control has been intensified in recent years due to the control implemented by using computers [24, 25]. The authors of [25] studied simplex SMC for a linear DTSPS by modified simplex method, in which a sliding mode control law is constructed to guarantee reachability. In addition, the upper bound of the parameter is obtained by generalized Nyquist plot. Nevertheless, the study of SMC for DTSPSs is far from complete and more work needs to be done.

On the other hand, it is worth noting that the trajectories of system state are not along stable sliding surface strictly, but move up and down the stable sliding surface and tend to equilibrium point. This phenomenon is an inherent disadvantage for discrete-time systems and is influenced by external factors. The approach of observer is adopted to weaken up the chattering phenomenon. The main features of this method are to reduce the external disturbances and to make up for uncertainty. So, designing an appropriate state observer is a key. Many efforts have been made to solve this problem. See [26–32]. References [26–28] mentioned the observer-based SMC for SPSs. The observer-based SMC for uncertain systems is involved in [29, 30]. Among them, a way for designing state observer and sliding mode control law is provided in [28], and the criteria for stability of observer error system and closed-loop system were presented based on LMI. However, few existing results focus on how to employ observer to investigate the SMC problem for uncertain DTSPSs with input nonlinearity. So, further study needs to be carried out for the problem.

In view of the works above, this paper addresses the H∞ sliding mode control for uncertain DTSPSs with external disturbances and input nonlinearity. First, to estimate the state of the original system, a novel state observer is constructed. Under this condition, with the help of Lyapunov method, the observer error system is ensured to be input-to-state stable (ISS) and satisfies the given H∞ performance index. Then, an appropriate sliding mode control law is proposed to satisfy the reachability condition. On this basis, the ISS criterion for the sliding mode dynamics (SMDs) is given and the controller gain matrix is also obtained. Compared to the previous works, the advantages can be summarized as follows: (1) To overcome the difficulty caused by the small parameter in deriving the sliding mode dynamics, a novel sliding mode surface with the SMC gain matrix is constructed to ensure the desired property of the sliding mode dynamics; (2) The method presented in this paper is more convenient in the application, despite the fact that the system considered here is more complicated. The reason is that the proposed sufficient condition does not depend on the equality constraint or Ricatti equations, which reduces the computation cost imposed by the iterative algorithm; (3) A workable way for addressing the upper bound is provided, which can be converted into an optimal problem. It is noticed that the prescribed upper bound is required in many references when solving, which brings great difficulties to the practical operation. The reason is that it is difficult to choose an appropriate upper bound to guarantee the solvability. However, this case has been avoided in this paper.

2 Problem formulation

Consider the uncertain DTSPSs with external disturbance and input nonlinearity

$$\begin{aligned}& x ( k + 1 ) = ( A_{\varepsilon} + \Delta A )x ( k ) + B_{u\varepsilon} \varphi ( u ) + D_{w\varepsilon} w ( k ), \end{aligned}$$

(1)

$$\begin{aligned}& y ( k ) = Cx ( k ), \end{aligned}$$

(2)

where $x(k) = (x_{1}^{T}(k),x_{2}^{T}(k))^{T}$ is the system state vector; $\varphi ( u )$ is the nonlinear input; $w ( k ) \in \mathbb{R}^{p}$ is an external disturbance input vector; $y ( k ) \in \mathbb{R}^{m}$ is the system output vector; $\varepsilon > 0$ is a singular perturbation parameter; ΔA is a uncertain matrix with appropriate dimension; $A_{\varepsilon}$, $B_{u\varepsilon}$, $D_{w\varepsilon}$, C are known and satisfy the following definition:

$$ A_{\varepsilon} = E_{0} + E_{\varepsilon} A,\qquad B_{u\varepsilon} = E_{\varepsilon} B_{u},\qquad B_{w\varepsilon} = E_{\varepsilon} B_{w}, $$

where A, $B_{u}$, C, $D_{w}$, $E_{\varepsilon}$ and $E_{0}$ are matrices with appropriate dimensions

$$\begin{aligned}& E_{0} = \begin{pmatrix} I & O \\ O & O \end{pmatrix},\qquad E_{\varepsilon} = \begin{pmatrix} \varepsilon I & O \\ O & I \end{pmatrix},\qquad A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}, \\& B_{u} = \begin{pmatrix} B_{u1} \\ B_{u2} \end{pmatrix},\qquad D_{w} = \begin{pmatrix} D_{w1} \\ D_{w2} \end{pmatrix},\qquad C = \begin{pmatrix} C_{1} & C_{2} \end{pmatrix}. \end{aligned}$$

Assumption 2.1

Matrix $B_{u}$ is of full column remark.

Assumption 2.2

The uncertain matrix ΔA is assumed to satisfy the following admissible form:

$$ \Delta A = MF ( k )N, $$

where M, N are real constant matrices, $M = ( M_{1}^{T} \ M_{2}^{T} )^{T}$, $N = ( N_{1} \ N_{2} )$, the time-varying matrix function $F ( k )$ satisfy

$$ F^{T} ( k )F ( k ) \le I. $$

(3)

Assumption 2.3

Nonlinear input function $\varphi ( u )$ satisfies the following property:

$$ u^{T} ( k ) \bigl( \alpha u ( k ) - \varphi ( u ) \bigr) \le 0,\quad u ( k ) \in \mathbb{R}^{q}, $$

where α is considered as a nonzero positive constant, and $\varphi ( 0 ) = 0$.

The H∞ performance index is chosen as

$$ J ( k ) = \sum_{k = 0}^{\infty} z^{T} ( k )z ( k ) - \gamma ^{2}w^{T} ( k )w ( k ). $$

(4)

Some basic lemmas are given before proceeding.

Lemma 2.1

([33])

System $x ( k + 1 ) = f ( x ( k ),u ( k ) )$ is ISS, if there exist a class $\mathcal {K}_{\infty}$ function $\alpha _{1}$, $\alpha _{2}$ and a class $\mathcal {K}$ function ρ, and continuous positive definite functions $W ( x )$ in $\mathbb{R}^{n}$ satisfy the following inequalities:

$$\begin{aligned}& \alpha _{1} \bigl( \Vert x \Vert \bigr) \le V ( x ) \le \alpha _{2} \bigl( \Vert x \Vert \bigr),\\& V \bigl( x ( k + 1 ) \bigr) - V \bigl( x ( k ) \bigr) \le - W \bigl( x ( k ) \bigr),\quad \Vert x \Vert \ge \rho \bigl( \Vert u \Vert \bigr), \end{aligned}$$

where $V:\mathbb{R}^{n} \to \mathbb{R}$ is continuous differentiable.

Lemma 2.2

([17])

Let $\Sigma _{1}$ and $\Sigma _{2}$ be constant matrices with appropriate dimensions. Then for any $F ( k )$ satisfying (5) and a scalar $\delta > 0$, the following inequality holds

$$ \Sigma _{1}F ( k )\Sigma _{2} + \bigl( \Sigma _{1}F ( k )\Sigma _{2} \bigr)^{T} \le \delta \Sigma _{1}\Sigma _{1}^{T} + \delta ^{ - 1} \Sigma _{2}^{T}\Sigma _{2}. $$

Lemma 2.3

([10])

Let $\Pi _{1}$, $\Pi _{2}$, and $\Pi _{3}$ be symmetric matrices with appropriate dimensions, $\eta _{0}$ be nonnegative scalar, then $\Pi _{1} + \eta \Pi _{2} + \eta ^{2}\Pi _{3} \ge 0$, $\forall \eta \in ( 0, \eta _{0} ]$, if the following conditions hold:

$$\begin{aligned}& (1)\quad \Pi _{1} \ge 0; \\& (2)\quad \Pi _{1} \ge 0,\qquad \Pi _{1} + \eta _{0}\Pi _{2} \ge 0;\\& (3)\quad \Pi _{1} \ge 0,\qquad \Pi _{1} + \eta _{0}\Pi _{2} \ge 0,\qquad \Pi _{1} + \eta \Pi _{2} + \eta ^{2}\Pi _{3} \ge 0. \end{aligned}$$

The main objective of this paper is to design an appropriate sliding surface satisfying some performance requirements. On this basis, a sliding control law is further designed to guarantee that the sliding surface $s ( k ) = 0$ is reachable in a finite time. Meanwhile, a sufficient condition needs to be found to guarantee the SMDs ISS.

3 Main results

3.1 The stability analysis of the observer error system

First, a Luenberger-like full-order observer is given by:

$$\begin{aligned}& \hat{x}(k + 1) = A_{\varepsilon} \hat{x}(k) + B_{u\varepsilon} \varphi ( u ) + L ( y - C\hat{x} ), \end{aligned}$$

(5)

$$\begin{aligned}& \hat{y}(k) = C\hat{x}(k), \end{aligned}$$

(6)

where $\hat{x} = (\hat{x}_{1}^{T},\hat{x}_{2}^{T})^{T} \in \mathbb{R}^{n}$ is the reconstructed state of vector. ŷ is the observer output vector. L is an observer gain matrix, $L = ( L_{1}^{T} \ L_{2}^{T} )^{T}$.

Remark 1

L is an undetermined unknown matrix with appropriate dimension. It should be noted that the value of L is not unique and depends on the coefficient of the system.

Let $e(k) = x(k) - \hat{x}(k)$, $z(k) = y(k) - \hat{y}(k)$. Then, the observer error system can be described by

$$\begin{aligned}& e(k + 1) = ( A_{\varepsilon} - LC + \Delta A )e ( k ) + \Delta A\hat{x} ( k ) + D_{w\varepsilon} w(k), \end{aligned}$$

(7)

$$\begin{aligned}& z(k)=Ce(k). \end{aligned}$$

(8)

Next, based on LMI and the Lyapunov method, we give a sufficient condition to ensure that the observer error system is ISS with the given H∞ performance index γ.

Theorem 1

For the observer error system (7)–(8), if there exist scalars $\delta _{i} > 0$ ($i = 1, 2, 3, 4$) and $\sigma > 0$, matrices $P > 0$ and Y, such that the following condition holds:

$$ \phi = \begin{pmatrix} \Lambda _{11} & \Lambda _{12} & P & 0 & 0 & A^{T}P - C^{T}Y^{T} \\ {*} & \Lambda _{13} & 0 & - M^{T}P\bar{D}_{w2} & 0 & M^{T}P \\ {*} & {*} & - 3 / 2I & 0 & 0 & 0 \\ {*} & {*} & {*} & - \gamma ^{2}I & \bar{D}_{w2}^{T}PM & \bar{D}_{w2}^{T}P \\ {*} & {*} & {*} & {*} & - ( \delta _{3} + \delta _{4} )I & 0 \\ {*} & {*} & {*} & {*} & {*} & - P \end{pmatrix} < 0, $$

(9)

where

$$\begin{aligned}& \Lambda _{11} = \bar{A}_{1}^{T} ( YC ) + ( YC )^{T}\bar{A}_{1} + ( \delta _{1} + \delta _{3} )N^{T}N + \sigma ^{4} / 2I + C^{T}C - A_{1}^{T}PA_{1} - P_{2},\\& \Lambda _{12} = P\bar{M}_{1} - A_{1}^{T}P \bar{M},\qquad \Lambda _{13} = - ( \delta _{1} + \delta _{2} )I - M^{T}PM,\\& A_{1} = \begin{pmatrix} A_{11} & A_{12} \\ 0 & 0 \end{pmatrix}, \qquad \bar{A}_{1} = \begin{pmatrix} A_{11} - I & A_{12} \\ 0 & 0 \end{pmatrix}, \qquad P = \begin{pmatrix} P_{11} & 0 \\ 0 & P_{22} \end{pmatrix},\\& P_{2} = \begin{pmatrix} 0 & 0 \\ 0 & P_{22} \end{pmatrix}, \qquad \bar{M}_{1} = \begin{pmatrix} M_{1} \\ 0 \end{pmatrix}, \qquad \bar{D}_{w2} = \begin{pmatrix} 0 \\ D_{w2} \end{pmatrix},\qquad Y = \begin{pmatrix} Y_{1} \\ Y_{2} \end{pmatrix}. \end{aligned}$$

Then, there exists a scalar $\tilde{\varepsilon} > 0$ such that observer error system is ISS with the given H∞ performance index γ for any $\varepsilon \in (0,\tilde{\varepsilon} ]$. In addition, the observer gain matrix is given with the following form:

$$ L = P^{ - T}Y. $$

(10)

Proof

Substituting (10) into (9) and then use the Schur’s Complement, the following inequality can be derived.

$$\begin{aligned}& \begin{pmatrix} \tilde{\Lambda}_{11} & \tilde{\Lambda}_{12} & P & 0 & 0 \\ {*} & \tilde{\Lambda}_{22} & 0 & - M^{T}P\bar{D}_{w2} & 0 \\ {*} & {*} & - 2 / 3I & 0 & 0 \\ {*} & {*} & {*} & - \gamma ^{2}I & \bar{D}_{w2}^{T}PM \\ {*} & {*} & {*} & {*} & \tilde{\Lambda}_{44} \end{pmatrix} \\& \quad {}+ \begin{pmatrix} ( A - LC )^{T} \\ M^{T} \\ 0 \\ \bar{D}_{w2}^{T} \\ 0 \end{pmatrix}P \begin{pmatrix} ( A - LC )^{T} \\ M^{T} \\ 0 \\ \bar{D}_{w2}^{T} \\ 0 \end{pmatrix}^{T} < 0, \\& \tilde{\Lambda}_{11} = \bar{A}_{1}^{T}P^{T} ( LC ) + ( LC )^{T}P\bar{A}_{1} + ( \delta _{1} + \delta _{3} )N^{T}N + \sigma ^{4} / 2I + C^{T}C - A_{1}^{T}PA_{1} - P_{2}, \\& \tilde{\Lambda}_{12} = PM_{1} - A_{1}^{T}PM,\qquad \tilde{\Lambda}_{22} = - ( \delta _{1} + \delta _{2} )I,\qquad \tilde{\Lambda}_{44} = - ( \delta _{3} + \delta _{4} )I. \end{aligned}$$

(11)

Since $P_{11} > 0$ and $P_{22} > 0$, there exists a scalar $\varepsilon _{11} > 0$ satisfying $P_{11} - \varepsilon _{11}^{2}P_{21}P_{22}^{ - 1}P_{21}^{T} \ge 0$ for all $\varepsilon \in (0,\varepsilon _{11}]$. According to Schur’s complement,

$$P_{\varepsilon} = \begin{pmatrix} P_{11} & \varepsilon P_{21} \\ \varepsilon P_{21}^{T} & P_{22} \end{pmatrix} > 0,\qquad \varepsilon \in (0,\varepsilon _{11}]. $$

Define the Lyapunov function with the following form

$$ V_{1}(e) = e^{T}P_{\varepsilon} e. $$

Let $w(k) = 0$, we have

$$\begin{aligned} \Delta V_{1} ( e ) = &e ( k + 1 )^{T}P_{\varepsilon} e ( k + 1 ) - e ( k )^{T}P_{\varepsilon} e ( k ) \\ =& \bigl( ( A_{\varepsilon} - LC + \Delta A )e ( k ) + \Delta A\hat{x} ( k ) \bigr)^{T}P_{\varepsilon} \bigl( ( A_{\varepsilon} - LC + \Delta A )e ( k ) + \Delta A\hat{x} ( k ) \bigr) \\ &{}- e^{T}P_{\varepsilon} e \\ \le& e^{T} ( k ) \bigl\{ ( A_{\varepsilon} - LC )^{T}P_{\varepsilon} ( A_{\varepsilon} - LC ) + 1 / \delta _{1} ( A_{\varepsilon} - LC )^{T}P_{\varepsilon} MM^{T}P_{\varepsilon} ( A_{\varepsilon} - LC ) \\ &{}+ 1 / \delta _{2} ( A_{\varepsilon} - LC )^{T}P_{\varepsilon} MM^{T}P_{\varepsilon} ( A_{\varepsilon} - LC ) + \delta _{1}N^{T}N \\ &{}+ 3 / 2P_{\varepsilon}^{T}P_{\varepsilon} + \sigma ^{4} / 2I - P_{\varepsilon} \bigr\} e ( k ) \\ &{}+ \hat{x}^{T} ( k ) \bigl\{ \delta _{2}N^{T}N + \sigma ^{4}I \bigr\} \hat{x} ( k ) \\ =& e^{T} ( k ) \bigl( \bar{\phi}_{0} + \bar{ \phi}_{1} + \varepsilon \bar{\phi}_{2} + \varepsilon ^{2}\bar{\phi}_{3} \bigr)e ( k ) + \hat{x}^{T} ( k ) \bigl\{ \delta _{2}N^{T}N + \sigma ^{4}I \bigr\} \hat{x} ( k ), \end{aligned}$$

(12)

where

$$\begin{aligned}& \bar{\phi}_{0} = \begin{pmatrix} \bar{\Lambda}_{11} & PM_{1} - A_{1}^{T}PM & P \\ {*} & - ( \delta _{1} + \delta _{2} ) - M^{T}PM & 0 \\ {*} & {*} & - 2 / 3I \end{pmatrix}, \\& \bar{\phi}_{1} = \begin{pmatrix} ( A - LC )^{T} \\ M^{T} \\ 0 \end{pmatrix}P \begin{pmatrix} ( A - LC )^{T} \\ M^{T} \\ 0 \end{pmatrix}^{T},\\& \bar{\phi}_{2} = \begin{pmatrix} \bar{\Sigma}_{11} & A_{1}^{T}PM - ( E_{0} + A_{2} - LC )^{T}P_{3}M & P_{3} \\ {*} & 0 & 0 \\ {*} & {*} & 0 \end{pmatrix}, \\& \bar{\phi}_{3} = \begin{pmatrix} ( A - \bar{L}_{2}C )^{T} \\ \bar{M}_{2}^{T} \\ 0 \end{pmatrix}P_{4} \begin{pmatrix} ( A - \bar{L}_{2}C )^{T} \\ \bar{M}_{2}^{T} \\ 0 \end{pmatrix}^{T},\\& \bar{\Lambda}_{11} = \bar{A}_{1}^{T}P^{T} ( LC ) - ( LC )^{T}P\bar{A}_{1} + \delta _{1}N^{T}N + \sigma ^{4} / 2I - A_{1}^{T}PA_{1} - P_{2},\\& \bar{\Sigma}_{11} = ( E_{0} + A_{2} - LC )^{T}P_{1} ( A - \bar{L}_{2}C ) + ( A - \bar{L}_{2}C )^{T}P_{1}^{T} ( E_{0} + A_{2} - LC ) - P_{3},\\& P_{1} = \begin{pmatrix} P_{11} & 0 \\ P_{21}^{T} & 0 \end{pmatrix}, \qquad P_{3} = \begin{pmatrix} 0 & P_{21} \\ P_{21}^{T} & 0 \end{pmatrix},\qquad P_{4} = \begin{pmatrix} P_{11} & P_{21} \\ P_{21}^{T} & 0 \end{pmatrix},\\& \bar{M}_{2} = \begin{pmatrix} 0 \\ M_{2} \end{pmatrix},\qquad A_{2} = \begin{pmatrix} 0 & 0 \\ A_{21} & A_{22} \end{pmatrix},\qquad \bar{L}_{2} = \begin{pmatrix} 0 \\ L_{2} \end{pmatrix}, \end{aligned}$$

where $\bar{\phi}_{0} < 0$ can be guaranteed by (11). Thus, there exists a scalar $\varepsilon _{12} > 0$ such that $\bar{\phi}_{0} + \bar{\phi}_{1} + \varepsilon \bar{\phi}_{2} + \varepsilon ^{2}\bar{\phi}_{3} < 0$ for any given $\varepsilon \in (0,\varepsilon _{12}]$. Note $\varepsilon _{1}^{ *} = \min \{ \varepsilon _{11}, \varepsilon _{12} \}$. Let $a = \lambda _{\min} ( - \bar{\phi}_{0} - \bar{\phi}_{1} - \varepsilon \bar{\phi}_{2} - \varepsilon ^{2}\bar{\phi}_{3} )$, then $a > 0$ for all $\varepsilon \in ( 0,\varepsilon _{1}^{ *} ]$. Thus,

$$ \Delta V_{1} ( x ) \le - a \bigl\Vert e ( k ) \bigr\Vert ^{2} + \varsigma _{1} \bigl\Vert \hat{x} ( k ) \bigr\Vert ^{2} \le - a ( 1 - \theta ) \bigl\Vert e ( k ) \bigr\Vert ^{2}, \quad \forall \bigl\Vert e ( k ) \bigr\Vert \ge \frac{\sqrt{4\varsigma _{1}a\theta}}{ 2a\theta} \bigl\Vert \hat{x} ( k ) \bigr\Vert , $$

where

$$ 0 < \theta < 1,\qquad \varsigma _{1} = \bigl\Vert \delta _{2}N^{T}N + \sigma ^{4}I \bigr\Vert . $$

The above shows that the observer error system is ISS with respect to estimate state x̂ for any $\varepsilon \in (0,\varepsilon _{1}^{ *} ]$.

Next, we consider the H∞ performance of the closed-loop system under the condition $w ( k ) \in L_{2}$.

$$\begin{aligned}& V \bigl( e ( k + 1 ) \bigr) - V \bigl( e ( k ) \bigr) + z^{T} ( k )z ( k ) - \gamma ^{2}w^{T} ( k )w ( k ) \\& \quad = \begin{pmatrix} e^{T} ( k ) & w^{T} ( k ) \end{pmatrix} \bigl( \phi _{0} + \phi _{1} + \varepsilon \phi _{2} + \varepsilon ^{2}\phi _{3} \bigr) \begin{pmatrix} e^{T} ( k ) & w^{T} ( k ) \end{pmatrix}^{T} \\& \qquad {}+ \hat{x}^{T} ( k ) \bigl\{ ( \delta _{2} + \delta _{4} )N^{T}N + \sigma ^{4}I \bigr\} \hat{x} ( k ), \end{aligned}$$

(13)

where

$$\begin{aligned}& e^{T} ( k ) ( A_{\varepsilon} - LC )^{T}P_{\varepsilon} \Delta Aw ( k ) + e^{T} ( k )\Delta A^{T}P_{\varepsilon} ( A_{\varepsilon} - LC )w ( k ) \\& \quad \le \delta _{3}e^{T} ( k )N^{T}Ne ( k ) + 1 / \delta _{3}w^{T} ( k ) ( A_{\varepsilon} - LC )^{T}P_{\varepsilon} MM^{T}P_{\varepsilon}^{T} ( A_{\varepsilon} - LC )w ( k ) \\& \qquad {}\times \hat{x}^{T} ( k ) ( A_{\varepsilon} - LC )^{T}P_{\varepsilon} \Delta Aw ( k ) + w^{T} ( k )\Delta A^{T}P_{\varepsilon} ( A_{\varepsilon} - LC )\hat{x} ( k ) \\& \quad \le \delta _{4}\hat{x}^{T} ( k )N^{T}N \hat{x} ( k ) + 1 / \delta _{4}w^{T} ( k ) ( A_{\varepsilon} - LC )^{T}P_{\varepsilon} MM^{T}P_{\varepsilon}^{T} ( A_{\varepsilon} - LC )w ( k ), \\& \phi _{0} = \begin{pmatrix} \tilde{\Lambda}_{11} & PM_{1} - A_{1}^{T}PM & P & 0 & 0 \\ {*} & - ( \delta _{1} + \delta _{2} )I - M^{T}PM & 0 & - M^{T}P\bar{D}_{w2} & 0 \\ {*} & {*} & - \frac{2}{3}I & 0 & 0 \\ {*} & {*} & {*} & - \gamma ^{2}I & \bar{D}_{w2}^{T}PM \\ {*} & {*} & {*} & {*} & - ( \delta _{3} + \delta _{4} )I \end{pmatrix},\\& \phi _{1} = \begin{pmatrix} A - LC & M & 0 & \bar{D}_{w2} & 0 \end{pmatrix}^{T}P \begin{pmatrix} A - LC & M & 0 & \bar{D}_{w2} & 0 \end{pmatrix},\\& \phi _{2} = \begin{pmatrix} \bar{\Sigma}_{11} & A_{1}^{T}PM - ( E_{0} + A_{2} - LC )^{T}P_{3}M & P_{3} & ( E_{0} + A_{2} - LC )^{T}P_{1}^{T}M & 0 \\ {*} & 0 & 0 & 0 & 0 \\ {*} & {*} & 0 & 0 & 0 \\ {*} & {*} & {*} & 0 & D_{w}^{T}P_{1}M_{1} \\ {*} & {*} & {*} & {*} & 0 \end{pmatrix},\\& \bar{\Sigma}_{11} = ( E_{0} + A_{2} - LC )^{T}P_{1} ( A - \bar{L}_{2}C ) + ( A - \bar{L}_{2}C )^{T}P_{1}^{T} ( E_{0} + A_{2} - LC ) - P_{3},\\& \phi _{3} = \begin{pmatrix} A - \bar{L}_{2}C & \bar{M}_{2} & 0 & D_{w} & \tilde{M}_{1} \end{pmatrix}^{T}P_{4} \begin{pmatrix} A - \bar{L}_{2}C & \bar{M}_{2} & 0 & D_{w} & \tilde{M}_{1} \end{pmatrix}, \\& \tilde{M}_{1} = \begin{pmatrix} O & M_{1}^{T} \end{pmatrix}^{T} \end{aligned}$$

there exists a scalar $\varepsilon _{13} > 0$, satisfying $\phi _{0} + \phi _{1} + \varepsilon \phi _{2} + \varepsilon ^{2}\phi _{3} < 0$ for any given $\varepsilon \in (0,\varepsilon _{13}]$. Note $\tilde{\varepsilon} = \min \{ \varepsilon _{11}, \varepsilon _{12}, \varepsilon _{13} \}$. Let $b = \lambda _{\min} ( - \phi _{0} - \phi _{1} - \varepsilon \phi _{2} - \varepsilon ^{2}\phi _{3} )$, then $b > 0$ for all $\varepsilon \in ( 0,\tilde{\varepsilon} ]$. Thus,

$$ \Delta V_{1} \bigl( x ( k ) \bigr) \le - b \bigl\Vert e ( k ) \bigr\Vert ^{2} + \varsigma \bigl\Vert \hat{x} ( k ) \bigr\Vert ^{2} \le - b ( 1 - \theta ) \bigl\Vert e ( k ) \bigr\Vert ^{2}, \quad \forall \bigl\Vert e ( k ) \bigr\Vert \ge \frac{\sqrt{4\varsigma b\theta}}{2b\theta} \bigl\Vert \hat{x} ( k ) \bigr\Vert , $$

where

$$ 0 < \theta < 1, \quad \varsigma = \big\| ( \delta _{2} + \delta _{4} )N^{T}N + \sigma ^{4}I \} \big\| , $$

which implies $V ( e ( k + 1 ) ) - V ( e ( k ) ) + z^{T} ( k )z ( k ) - \gamma ^{2}w^{T} ( k )w ( k ) < 0$. Furthermore, the H∞ performance index under zero initial condition can be given by

$$\begin{aligned} J ( k ) =& \sum_{k = 0}^{\infty} z^{T} ( k )z ( k ) - \gamma ^{2}w^{T} ( k )w ( k )\\ \le& \sum_{k = 0}^{\infty} z^{T} ( k )z ( k ) - \gamma ^{2}w^{T} ( k )w ( k ) + \Delta V_{1} ( k )\\ =& V \bigl( e ( \infty ) \bigr) - V \bigl( e ( 0 ) \bigr) + \sum _{k = 0}^{\infty} z^{T} ( k )z ( k ) - \sum _{k = 0}^{\infty} \gamma ^{2}w^{T} ( k )w ( k ) \end{aligned}$$

thus, we have

$$ \sum_{k = 0}^{\infty} z^{T} ( k )z ( k ) \le \sum_{k = 0}^{\infty} \gamma ^{2}w^{T} ( k )w ( k ). $$

(14)

It is obvious that the H∞ performance index can be satisfied. □

Remark 2

The estimate of the upper bound ε̃ for SPSs is an interesting topic. Extensive results for the problem have been proposed [9, 28]. Among them, solving the GEVP is a most common way due to its easier operability and higher accuracy. However, in view of the particularity of inequality (15), the upper bound ε̃ cannot be obtained directly by solving the following GEVP. Based on Lemma 2.3, we will resolve the problem in a special way. See Theorem 2.

Theorem 2

After deriving the observer gain matrix L from (8)–(9), if there exist matrices $P_{11} > 0$, $P_{22} > 0$ and $W_{1}$ satisfying:

$$\begin{aligned}& \begin{pmatrix} P_{11} & W_{{1}}^{T} \\ W_{1} & P_{22} \end{pmatrix} > 0,\qquad W_{1} < \varepsilon P_{21}, \qquad \phi _{0} < 0,\qquad \phi _{0} + \phi _{1} + \varepsilon \phi _{2} + \varepsilon ^{2} \phi _{3} < 0, \end{aligned}$$

(15)

where $\phi _{0}$, $\phi _{1}$, $\phi _{2}$, $\phi _{3}$ are defined in Theorem 1, then the observer error system is ISS with respect to estimate-state with disturbance attenuation level γ for any $\varepsilon \in (0,\tilde{\varepsilon}]$.

To obtain the minimum of disturbance attenuation level γ, a possible approach is to transform the minimum problem into an optimization problem, which can be solved with MATLAB.

3.2 Designing scheme of the sliding surface and reachability analysis

In this section, we first construct a sliding surface, under which the sliding mode control law is derived such that the sliding surface is reachable.

First, an observer-based sliding surface function is considered as:

$$ s ( k ) = \sigma ( k ) + B_{u\varepsilon}^{T}\hat{x} ( k ), $$

(16)

with

$$ \Delta \sigma ( k ) = B_{u\varepsilon}^{T}B_{u\varepsilon} K\hat{x} ( k ) - B_{u\varepsilon}^{T}A_{\varepsilon} \hat{x} ( k ) + B_{u\varepsilon}^{T}\hat{x} ( k ), $$

where $B_{u\varepsilon}^{T}B_{u\varepsilon}$ is non-singular. The gain matrix K is to be obtained later.

$$ s ( k + 1 ) - s ( k ) = \bigl( B_{u\varepsilon}^{T}B_{u\varepsilon} K \bigr)\hat{x} ( k ) + B_{u\varepsilon}^{T}B_{u\varepsilon} \varphi ( u ) + B_{u\varepsilon}^{T}L ( y - C\hat{x} ). $$

By further derivation, the equivalent control law can be obtained as follows

$$ \varphi _{eq} = - K\hat{x} ( k ) - \bigl( B_{u\varepsilon}^{T}B_{u\varepsilon} \bigr)^{ - 1}B_{u\varepsilon}^{T}L \bigl( y - C\hat{x} ( k ) \bigr), $$

(17)

let’s substitute (17) into observer system (5), the SMDs is given by

$$ \hat{x} ( k + 1 ) = ( A_{\varepsilon} - B_{u\varepsilon} K )\hat{x} ( k ) + \tilde{B}L \bigl( y - C\hat{x} ( k ) \bigr), $$

(18)

where $\tilde{B} = I - B_{u\varepsilon} ( B_{u\varepsilon}^{T}B_{u\varepsilon} )^{ - 1}B_{u\varepsilon}^{T}$.

The sliding mode control law is designed as follows:

$$ u ( k ) = - \frac{\vartheta ( \hat{x} )}{\alpha \Vert s ( k ) \Vert }s ( k ), $$

(19)

with $\vartheta ( \hat{x} ) = \beta + ( \Vert K\hat{x} ( k ) \Vert + \Vert ( B_{u}^{T}B_{u} )^{ - 1}B_{u}^{T}L ( y - C\hat{x} ( k ) ) \Vert )$, where $\beta > 0$.

The following theorem guarantees that the sliding surface is reachable.

Theorem 3

With the Assumption 2.1and the control law (19), the trajectories of the observer system (5) are driven onto the sliding surface $s ( k ) = 0$ in a finite time.

Proof

Based on the Assumption 2.1, the matrix $B_{u\varepsilon}^{T}B_{u\varepsilon}$ is non-singular. To verify the reachability, we define the following Lyapunov function:

$$\begin{aligned}& V_{3} ( k ) = \frac{1}{2}s^{T} ( k ) \bigl( B_{u}^{T}B_{u} \bigr)^{ - 1}s ( k ), \\& \Delta V_{3} ( k ) =\frac{1}{2}\Delta s^{T} ( k ) \bigl( B_{u\varepsilon}^{T}B_{u\varepsilon} \bigr)^{ - 1}s ( k + 1 ) + \frac{1}{2}s^{T} ( k ) \bigl( B_{u\varepsilon}^{T}B_{u\varepsilon} \bigr)^{ - 1}\Delta s ( k ) \\& \hphantom{\Delta V_{3} ( k )}\le \frac{1}{2}\varphi ^{T} ( u )s ( k ) + \frac{1}{2} \bigl\{ \bigl\Vert \bigl( K\hat{x} ( k ) \bigr)^{T} \bigr\Vert + \bigl\Vert \bigl[ \bigl( B_{u\varepsilon}^{T}B_{u\varepsilon} \bigr)^{ - 1}L \bigl( y - C\hat{x} ( k ) \bigr) \bigr]^{T} \bigr\Vert \bigr\} \bigl\Vert s ( k ) \bigr\Vert \\& \hphantom{\Delta V_{3} ( k ) =}{} + \frac{1}{2}s^{T} ( k )\varphi ( u ) + \frac{1}{2} \bigl\Vert s^{T} ( k ) \bigr\Vert \bigl\{ \bigl\Vert K\hat{x} ( k ) \bigr\Vert + \bigl\Vert \bigl( B_{u\varepsilon}^{T}B_{u\varepsilon} \bigr)^{ - 1}L \bigl( y - C\hat{x} ( k ) \bigr) \bigr\Vert \bigr\} \\& \hphantom{\Delta V_{3} ( k ) }\le - \beta \bigl\Vert s ( k ) \bigr\Vert . \end{aligned}$$

(20)

Using (19) and the Assumption 2.3, we have

$$ \varphi ^{T} ( u )u ( k ) = - \varphi ^{T} ( u ) \frac{\vartheta ( \hat{x} )}{\alpha \Vert s ( k ) \Vert }s ( k ) \ge \alpha u^{T} ( k )u ( k ), $$

(21)

then

$$ \varphi ^{T} ( u )s ( k ) \le - \vartheta ( \hat{x} ) \bigl\Vert s ( k ) \bigr\Vert , $$

(22)

and

$$ s^{T} ( k )\varphi ( u ) \le - \vartheta ( \hat{x} ) \bigl\Vert s ( k ) \bigr\Vert . $$

(23)

Substituting (22) and (23) into (20) yields

$$ \Delta V_{3} ( k ) \le - \beta \bigl\Vert s ( k ) \bigr\Vert < 0, \quad \text{for } \bigl\Vert s ( k ) \bigr\Vert \ne 0. $$

(24)

This indicates that the sliding surface is reachable. This completes the proof. □

3.3 Sliding mode dynamics analysis

In this subsection, we will propose a criterion to guarantee that the SMDs is ISS with respect to observer error $e ( k )$, from which the controller gain matrix K can be solved.

Theorem 4

There exist a scalar $\bar{\varepsilon} > 0$ such that the SMDs (18) is ISS with respect to observer error $e ( k )$ for any $\varepsilon \in ( 0,\bar{\varepsilon} ]$, if there exist matrices $X_{11} > 0$, $X_{22} > 0$ and G, satisfying

$$ \psi = \begin{pmatrix} X^{T}\bar{A}^{T} + \bar{A}X - B_{u}G - G^{T}B_{u}^{T} & X^{T}\bar{A}_{2}^{T} - G^{T}B_{u2}^{T} \\ {*} & - X_{22} \end{pmatrix} < 0, $$

(25)

where

$$ X = \begin{pmatrix} X_{11} & 0 \\ X_{21} & X_{22} \end{pmatrix}, \qquad \bar{A} = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} - I \end{pmatrix},\qquad \bar{A}_{2} = \begin{pmatrix} A_{21} & A_{22} - I \end{pmatrix}, $$

moreover, the gain matrix is obtained by

$$ K = GX^{ - 1}. $$

(26)

Proof

For (25), the following inequality is obtained by the Schur’s complement

$$ \begin{pmatrix} \bar{A}X + X^{T}A^{T} - B_{u}KX - X^{T}K^{T}B_{u}^{T} & X^{T}\bar{A}_{2} - X^{T}K^{T}B_{u2}^{T} \\ {*} & - X_{22} \end{pmatrix} < 0. $$

(27)

Pre- and post-multiplying inequality (27) by $\operatorname{diag}(X^{ - T},I)$ and $\operatorname{diag}(X^{ - 1}, I)$, respectively, and let

$$X^{ - 1} = Q = \begin{pmatrix} Q_{11} & O \\ Q_{21} & Q_{22} \end{pmatrix}, $$

the inequality (28) is given by

$$ \begin{pmatrix} Q^{T} ( \bar{A} - B_{u}K ) + ( \bar{A} - B_{u}K )^{T}Q & ( \bar{A}_{2} - B_{u2}K )^{T} \\ {*} & - Q_{22}^{ - 1} \end{pmatrix} < 0, $$

(28)

where $Q_2=\left(\begin{array}{cc}O& O\\ O& Q_{22}\end{array}\right)$, the inequality (29) is derived from the (28) by using Schur’s complement

$$ Q^{T} ( \bar{A} - B_{u}K ) + ( \bar{A} - B_{u}K )^{T}Q + ( \bar{A}_{2} - B_{u2}K )^{T}Q_{22} ( \bar{A}_{2} - B_{u2}K ) < 0, $$

(29)

therefore, there exists a scalar $\varepsilon _{21} > 0$ such that $Q_{11} - \varepsilon Q_{12}^{T}Q_{22}^{ - 1}Q_{21} > 0$ for all $\varepsilon \in (0,\varepsilon _{21}]$. Based on Schur’s complement lemma, it yields

$$ Q_{\varepsilon} = \begin{pmatrix} \varepsilon ^{ - 1}Q_{11} & Q_{21} \\ Q_{21}^{T} & Q_{22} \end{pmatrix} > 0. $$

Define the following Lyapunov function:

$$\begin{aligned}& V_{2} ( k ) = x^{T} ( k )Q_{\varepsilon} x ( k ), \\& \Delta V_{2} ( k ) = \hat{x}^{T} ( k + 1 )Q_{\varepsilon} \hat{x} ( k + 1 ) - \hat{x}^{T} ( k )Q_{\varepsilon} \hat{x} ( k ) \\& \hphantom{\Delta V_{2} ( k )}= \bigl( ( A_{\varepsilon} - B_{\varepsilon} K )\hat{x} ( k ) + \tilde{B} ( LC )e ( k ) \bigr)^{T}Q_{\varepsilon} \bigl( ( A_{\varepsilon} - B_{\varepsilon} K )\hat{x} ( k ) + \tilde{B} ( LC )e ( k ) \bigr) \\& \hphantom{\Delta V_{2} ( k )=}{} - \hat{x}^{T} ( k )Q_{\varepsilon} \hat{x} ( k ) \\& \hphantom{\Delta V_{2} ( k )} = \hat{x}^{T} ( \psi _{0} + \varepsilon \psi _{1} )\hat{x} + 2 \hat{x}^{T} ( k ) ( A_{\varepsilon} - B_{\varepsilon} K )^{T}Q_{\varepsilon} \tilde{B} ( LC )e ( k ) \\& \hphantom{\Delta V_{2} ( k )=}{} + e^{T} ( k ) ( \tilde{B}LC )^{T}Q_{\varepsilon} ( \tilde{B}LC )e ( k ), \end{aligned}$$

(30)

where

$$\begin{aligned}& \psi _{0} = Q^{T} ( \bar{A} - B_{u}K ) + ( \bar{A} - B_{u}K )^{T}Q + ( \bar{A}_{2} - B_{u2}K )^{T}Q_{22} ( \bar{A}_{2} - B_{u2}K ),\\& \psi _{1} = ( A - B_{u}K )^{T}Q_{3} ( A - B_{u}K ),\qquad Q_{3} = \begin{pmatrix} Q_{11} & Q_{21} \\ Q_{21}^{T} & 0 \end{pmatrix}. \end{aligned}$$

The inequality (29) indicates that there exist a scalar $\varepsilon _{22} > 0$, satisfying $\psi _{0} + \varepsilon \psi _{1} < 0$ for all $\varepsilon \in ( 0,\varepsilon _{22} ]$. Note $\bar{\varepsilon} = \min \{ \varepsilon _{21}, \varepsilon _{22} \}$, let $d = \lambda _{\min} ( - \psi _{0} - \varepsilon \psi _{1} )$, then $d > 0$ for all $\varepsilon \in ( 0,\bar{\varepsilon} ]$. Thus,

$$\begin{aligned} \Delta V_{2} \bigl( x ( k ) \bigr) \le& - d \bigl\Vert \hat{x} ( k ) \bigr\Vert ^{2} + \mu \bigl\Vert \hat{x} ( k ) \bigr\Vert \bigl\Vert e ( k ) \bigr\Vert + \nu \bigl\Vert e ( k ) \bigr\Vert ^{2} \\ \le& - d ( 1 - \theta ) \bigl\Vert \hat{x} ( k ) \bigr\Vert ^{2}, \quad \forall \bigl\Vert \hat{x} ( k ) \bigr\Vert \ge \frac{\mu + \sqrt{\mu ^{2} + 4\nu d\theta}}{2d\theta} \bigl\Vert e ( k ) \bigr\Vert , \end{aligned}$$

where

$$ 0 < \theta < 1, \quad \mu = 2\sup_{\varepsilon \in ( 0,\bar{\varepsilon} ]} \bigl\{ \bigl\Vert ( A_{\varepsilon} - B_{\varepsilon} K )^{T}P_{\varepsilon} \tilde{B} ( LC ) \bigr\Vert \bigr\} , \qquad \nu = \bigl\Vert ( \tilde{B}LC )^{T}Q_{\varepsilon} ( \tilde{B}LC ) \bigr\Vert . $$

According to Lemma 2.1, the SMDs is ISS with respect to the observer error $e ( k )$. This completes the proof. □

Similar to the method used in Theorem 2, the following theorem gives the method for solving the upper bound.

Theorem 5

After deriving the controller gain matrix K from (26), if there exist matrices $Q_{11} > 0$, $Q_{22} > 0$ and W, a positive scalar θ, satisfying

$$\begin{aligned}& \begin{pmatrix} W & Q_{21} \\ Q_{21}^{T} & Q_{22} \end{pmatrix} > 0,\qquad W < \theta Q_{11}, \qquad \psi _{0} < 0,\qquad \psi _{1} < - \theta \psi _{0}, \end{aligned}$$

(31)

then, the SMDs (17) is ISS with respect to observer error $e ( k )$ for any $\varepsilon \in ( 0,\bar{\varepsilon} ]$, where $\bar{\varepsilon} = \theta ^{ - 1}$.

4 Numerical examples

In this section, we present two examples to verify the effectiveness of the obtained results.

Example 1

Consider the following uncertain DTSPS with external disturbance.

$$\begin{aligned}& A = \begin{pmatrix} - 0.5 & 0.6 \\ - 0.6 & - 0.5 \end{pmatrix},\qquad B_{u} = \begin{pmatrix} 0 \\ 1 \end{pmatrix},\qquad B_{w} = \begin{pmatrix} 0.2 \\ 0.1 \end{pmatrix}, \\& M = N = \begin{pmatrix} - 0.02 & - 0.06 \\ - 0.02 & 0.04 \end{pmatrix}, \\& C = \begin{pmatrix} 0.05 & 0.01 \end{pmatrix},\qquad E_{0} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\qquad E_{\varepsilon} = \begin{pmatrix} \varepsilon & 0 \\ 0 & 1 \end{pmatrix}. \end{aligned}$$

We might as well make $F=\left(\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right)$, the following results can be derived via solving the LMI (9).

$$ P = \begin{pmatrix} 0.3413 & 0 \\ 0 & 0.2188 \end{pmatrix},\qquad Y = \begin{pmatrix} - 6.0374 \\ 1.8511 \end{pmatrix},\qquad \gamma _{\min} = 0.0017. $$

The observed gain matrix $L = P^{ - T}Y = ( - 17.6892 \ 8.4593 )^{T}$. According to Theorem 2, the upper bound of perturbed parameter $\bar{\varepsilon} = 1.7758$ can be obtained by solving the LMI (15).

Furthermore, according to Theorem 4, we obtain the following results.

$$\begin{aligned}& X = \begin{pmatrix} 1.7502 & 0 \\ 0.7544 & 1.0996 \end{pmatrix},\qquad G = \begin{pmatrix} 0.2484 & - 1.2169 \end{pmatrix}, \\& K = GX^{ - 1} = \begin{pmatrix} 0.6189 & - 1.1067 \end{pmatrix}. \end{aligned}$$

Furthermore, we get the upper bound $\tilde{\varepsilon} = 4.0535$ by solving GEVP (31). Moreover, when the small parameter ε is assumed to be 0.01, given the initial conditions $x ( 0 ) = ( 0.5 \ {-} 15 )^{T}$, $\hat{x} ( 0 ) = ( 1\ {-} 1 )^{T}$, the external disturbance $w ( k ) =1 / ( 1 + t^{3} )$, and the nonlinear disturbance $\varphi ( u ( k ) ) = u \cdot \cos u$. Then, the simulation results are presented by the following graphs. Figure 1 shows the responses of the closed-loop system. The responses of the observer error system are depicted in Fig. 2. From these simulations, it is clear that the proposed observer-based sliding mode control strategy efficiently reduces the impacts of nonlinearities and ensures the closed-loop system’s asymptotic stability. Moreover, the response of the sliding mode variable $s(k)$ is given in Fig. 3, which shows that the sliding motion in the sliding surface $s(x{\hat{\ }}(t))=0$ can be attained in a finite time.

Figure 1

States of closed-loop system

Figure 2

States of observer error system

Figure 3

State of the sliding mode variable

Example 2

Consider the DTSPS with the following coefficient matrices.

$$\begin{aligned}& A = \begin{pmatrix} - 0.2090 & - 0.6744 & - 0.9126 & 0.1024 \\ 1.4737 & 0.0150 & 0.0203 & 0.0023 \\ - 0.0015 & 0 & 0.9886 & 0.0130 \\ 0.0005 & 0 & - 0.0625 & 0.9932 \end{pmatrix},\qquad B = \begin{pmatrix} 0.1432 \\ - 16.9432 \\ - 0.0022 \\ - 0.1139 \end{pmatrix},\\& C = \begin{pmatrix} - 0.0500 & 0.0010 & 0.0010 & - 0.002 \\ 0.0010 & - 0.0010 & 0.0100 & 0.0050 \end{pmatrix},\qquad D = \begin{pmatrix} 0.1000 & 0 & 0 & 1.0000 \end{pmatrix}^{T},\\& M = \begin{pmatrix} 0.2 & 0.2 & 0.2 & 0.2 \end{pmatrix}^{T},\qquad N = \begin{pmatrix} 1 & 1 & 1 & 1 \end{pmatrix}. \end{aligned}$$

Then, let $F = 0.5$, the following results can be obtained

$$\begin{aligned}& Y = \begin{pmatrix} - 1.1789 & - 0.1840 \\ 0.0191 & - 0.0408 \\ 0.0637 & 0.4720 \\ - 0.0228 & 0.2108 \end{pmatrix}, \qquad P = \begin{pmatrix} 0.0603 & - 0.0010 & 0 & 0 \\ - 0.0010 & 0.0003 & 0 & 0 \\ 0 & 0 & 0.0110 & 0.0000 \\ 0 & 0 & 0.0000 & 0.0026 \end{pmatrix},\\& L = P^{ - T}Y = \begin{pmatrix} - 19.6045 & - 5.8739 \\ - 2.9772 & - 167.4233 \\ 5.8031 & 42.4154 \\ - 8.9673 & 81.3245 \end{pmatrix}, \end{aligned}$$

the minimum of disturbance attenuation level is given by $r_{\min} = 0.0461$. According to Theorem 2, the upper bound of perturbed parameter $\bar{\varepsilon} = 1.4909$ is obtained. Solving LMI (25) in a same manner, the following results are presented

$$\begin{aligned}& G = \begin{pmatrix} 29.8446 \\ - 4.1426 \\ 0.2307 \\ 2.4064 \end{pmatrix},\qquad X = \begin{pmatrix} 59.8528 & 33.4401 & 0 & 0 \\ 33.4401 & 59.6633 & 0 & 0 \\ 46.2038 & 630.2194 & 8.8697 & 2.7035 \\ 630.2194 & 222.9048 & 2.7035 & 51.4828 \end{pmatrix},\\& K = GX^{ - 1} = \begin{pmatrix} 0.3049 & - 0.5389 & 0.0120 & 0.0461 \end{pmatrix}. \end{aligned}$$

In addition, according to Theorem 5, the upper bound $\tilde{\varepsilon} = 0.0023$ is solved by GEVP.

Next, we choose $\varepsilon = 0.01$, given the initial conditions $x ( 0 ) = ( - 0.05 \ 0.75 \ {-} 1.5 \ 0.12 )^{T}$, $\hat{x} ( 0 ) = ( {-} 0.3 \ {-} 0.3 \ {-} 1.9 \ 0.5 )^{T}$, the disturbance $w ( k ) = 1 / ( 1 + k^{2} )$, and the nonlinear disturbance $\varphi ( u ( k ) ) = \cos u^{2}$. The responses of the closed-loop system and the observer error system are plotted in Figs. 4–5, from which the presented sliding mode control scheme guarantees the asymptotic stability of the closed-loop system. The State of the sliding mode variable $s(k)$ is given in Fig. 6, which demonstrates that the system trajectory is pushed onto the predetermined sliding mode surface in a finite time.

Figure 4

States of closed-loop system

Figure 5

States of observer error system

Figure 6

State of the sliding mode variable

5 Conclusion

The H∞ based on the observer sliding mode control for uncertain DTSPSs has been considered. A suitable SMC law has been designed to make the trajectories of the system tend to the sliding mode surface in a finite time. Based on the Lyapunov method, the observer error system is ensured to be ISS with the given H∞ performance index γ. Meanwhile, the upper bound of the small parameter and the minimum H∞ performance index have been obtained in a workable way. Beyond that, the sufficient condition for ISS of the SMDs with regard to the observer error has also been derived in the state estimation space. Finally, two numerical examples have demonstrated the effectively of the provided methods. One of the further research topics is to extend the present results to more general cases, for example, the case that the system is purely nonlinear, the case when the effect of the time delay is considered.