Variable Structure Disturbance Rejection Control for Nonlinear Uncertain Systems with State and Control Delays via Optimal Sliding Mode Surface Approach

Abstract

The paper considers the problem of variable structure control for nonlinear systems with uncertainty and time delays under persistent disturbance by using the optimal sliding mode surface approach. Through functional transformation, the original time-delay system is transformed into a delay-free one. The approximating sequence method is applied to solve the nonlinear optimal sliding mode surface problem which is reduced to a linear two-point boundary value problem of approximating sequences. The optimal sliding mode surface is obtained from the convergent solutions by solving a Riccati equation, a Sylvester equation, and the state and adjoint vector differential equations of approximating sequences. Then, the variable structure disturbance rejection control is presented by adopting an exponential trending law, where the state and control memory terms are designed to compensate the state and control delays, a feedforward control term is designed to reject the disturbance, and an adjoint compensator is designed to compensate the effects generated by the nonlinearity and the uncertainty. Furthermore, an observer is constructed to make the feedforward term physically realizable, and thus the dynamical observer-based dynamical variable structure disturbance rejection control law is produced. Finally, simulations are demonstrated to verify the effectiveness of the presented controller and the simplicity of the proposed approach.

1. Introduction

Various approaches have been proposed to solve disturbance rejection problems, such as 𝐻_∞ control [1], adaptive control [2], internal model control [3], variable structure control (VSC) [4], and optimal control [5–7]. Optimal disturbance rejection appears in a variety of applications, vehicle engine [5], damped systems [6], and spacecraft attitude control [7], for instance. However, in reality, the factor of nonlinearity, uncertainty, or time delay exhibits much influence on practical system [8–19], especially on today’s large information communication system. The factor of nonlinearity, uncertainty, or time delay leads the system and computation to be more complex and difficult to calculate. In addition, it brings discrepancy between the real parameter and the precise one. Therefore, it is necessary to take account of nonlinearity, uncertainty, or time delays in system modeling. In recent years, these issues have been given many attention; for example, see [1, 8–30]. Moreover, there are series techniques appeared to design controllers for such systems. Seeing that it is not an easy work to solve one of these problems, say nothing of concentrating on all of these issues. This study explores a disturbance rejection control design for a nonlinear uncertain time-delay system using the variable structure control approach, for the reason of VSC’s insensitivity to a wide class of uncertainties or disturbances [4, 8, 12, 13, 16, 30–34]. In previous VSC investigations, some reports gave the solutions for uncertain or nonlinear systems, for example, [8, 30, 34]; some gave for the time delay systems [12, 13]; but few appeared to focus on systems of both nonlinear uncertain and time-delay. This paper is different from these bodies of literature. It presents a relative simple VSC solution to a nonlinear uncertain and time-delay system.

It is based on the ideas of the approximating sequence method [25–27] and the finite spectrum assignment approach [20–22]. The contributions of this study are as follows: firstly, the functional transformation method is applied, which transforms the time-delay system to a delay-free one and reduces the original problem from an infinite-dimension space to a finite-dimension one; subsequently, the optimal sliding mode surface (OSMS) is designed by using approximating sequence method, which simplifies the nonlinear OSMS design problem to a linear two-point boundary value (TPBV) one; furthermore, the corresponding compensators are designed in the variable structure disturbance rejection control (VSDC) so that the nonlinearity, uncertainty, and the time-delay effects are entirely compensated consequentially; moreover, the disturbance effect is reduced by the feedforward compensation term; additionally, to realize its physical implementation, a reduced-order observer is constructed to reconstruct the disturbance state vectors, and thus the observer-based dynamical VSDC is produced; finally, the designed VSDC is employed to a quarter-car suspension model which possesses the nonlinear, uncertain, and time-delay properties, and by comparing the system responses with the open-loop system (OLS), the effectiveness and the simplicity of the designed control are demonstrated.

This paper is organized as follows. After an introduction, in Section 2, system description is done. The OSMS and the corresponding VSDC in finite-time horizon are designed in Section 3. In Section 4, the infinite-time horizon ones are given. In Section 5, numerical simulations are illustrated. Concluding remarks are given in Section 6.

2. System Descriptions

Consider the uncertain nonlinear system with state and control delays: 𝑥̇(𝑡)=𝐴₀𝑥(𝑡)+𝐴₁𝑥(𝑡−𝜎)+𝐵₀𝑢(𝑡)+𝐵₁𝑢(𝑡−𝜏)+𝐷𝑣(𝑡)+𝑑()()𝑥(𝑡)(𝑡)[]𝑢(𝑡)[),𝑥,𝑡+𝑓𝑥,𝑡=𝜙,𝑡∈−𝜎,0=0,𝑡∈−𝜏,0 (1) where 𝑥(𝑡)∈ℝ^𝑛, 𝑢(𝑡)∈ℝ^𝑚 are state and control vector, respectively; 𝐴₀,𝐴₁∈ℝ^𝑛×𝑛, 𝐵₀,𝐵₁∈ℝ^𝑛×𝑚, and 𝐷∈ℝ^𝑛×𝑞 are constant matrices; 𝜏,𝜎∈ℝ⁺ are constant control and state delays, respectively; and 𝜙(𝑡)∈𝐂([−𝜎,0],ℝ^𝑛) is the initial state vector. 𝑑(𝑥,𝑡) represents a bounded matching uncertainty produced by the system perturbation or internal disturbance. 𝑓(𝑥,𝑡)∈𝐂¹(ℝ^𝑛×ℝ,ℝ^𝑛) is the nonlinearity with 𝑓(0,𝑡)=0 satisfying Lipschitz condition. In this study, control 𝑢(𝑡) is assumed unlimited. Meanwhile, the following assumption is needed for the derivation.

Assumption 1. It is assumed that system (1) is spectral controllable.
In system (1), 𝑣(𝑡)∈ℝ^𝑞 is the external disturbance input whose dynamical characteristic is known and can be described by the following exosystem: 𝑤̇(𝑡)(𝑡)𝑣(𝑡)(𝑡),=𝐺𝑤=𝐹𝑤 (2) where 𝑤(𝑡)∈ℝ^𝑝 is the disturbance state vector and  𝐺∈ℝ^𝑝×𝑝 and 𝐹∈ℝ^𝑞×𝑝 are constant matrices. The pair (𝐺,𝐹) is observable completely. The initial condition 𝑤(0) can be unknown. Exosystem (2) can describe most general disturbances, such as step signal with unknown amplitude, sinusoidal signal with known frequency and unknown amplitude and phase [28], or random signal [29]. Due to the dynamical characteristics of the external disturbances, exosystem (2) may be Lyapunov stable or asymptotically stable; that is, Re𝜇_𝑗(𝐺)=0 or Re𝜇_𝑗(𝐺)<0(𝑗=1,2,…,𝑝), respectively, where 𝜇_𝑗(⋅) denote the eigenvalues of a matrix.

In what follows, we will reduce the controller design problem of original system (1) to that of a delay-free one referring to the method proposed in [20] which applied this method to a linear time-delay system. Now, we develop it to the nonlinear time-delay system (1).

Define the following functional transformation: 𝑦(𝑡)=𝑥(𝑡)+^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ−𝜎)}𝐴₁𝑥(ℎ)+𝑑ℎ_{𝑡𝑡−𝜏}𝑒^{𝐴(𝑡−ℎ−𝜏)}𝐵₁𝑢(ℎ)𝑑ℎ, (3) where 𝑦 is absolutely continuous and therefore differentiable almost everywhere and 𝐴 is an 𝑛×𝑛 matrix to be defined. Employing differentiation under the integral, (3) in conjunction with (1) which gives 𝑦̇(𝑡)=𝐴𝑦(𝑡)+𝐵₀+𝑒^−𝐴𝜏𝐵₁𝑢(𝑡)+𝐷𝑣(𝑡)+𝑑()()−𝑥,𝑡+𝑓𝑥,𝑡𝐴−𝐴₀−𝑒^−𝐴𝜎𝐴₁𝑥(𝑡). (4) From (4), it is observed that the range of (3) defines an ordinary differential system given by𝑦̇(𝑡)=𝐴𝑦(𝑡)+𝐵𝑢(𝑡)+𝐷𝑣(𝑡)+𝑑()()𝑦(0)𝑥,𝑡+𝑓𝑥,𝑡=𝜙₀=𝜙(0)+⁰_−𝜎𝑒^{−𝐴(ℎ+𝜎)}𝐴₁𝑥(ℎ)𝑥(𝑡)(𝑡)−𝑑ℎ=𝑦^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−𝛿)}𝐴₁𝑥(𝛿)−𝑑𝛿_{𝑡𝑡−𝜏}𝑒^{𝐴(𝑡−𝛿)}𝐵₁𝑢(𝛿)𝑑𝛿. (5a)(5b)If the following definitions are adopted: 𝐴=𝐴₀+𝐴₁,𝐵=𝐵₀+𝐵₁,𝐴₁=𝑒^−𝐴𝜎𝐴₁,𝐵₁=𝑒^−𝐴𝜏𝐵₁, (6) in which 𝐴 is referred to as the characteristic matrix equation and the method of its solution can be found in the researches of Fiagbedzi and Pearson [20, 21] and Zheng, Cheng, and Gao [22], and so forth, system (1) is assumed spectral controllable, which implies that system (5a) and (5b) is completely controllable [20]. Actually, here, 𝑓(𝑦,𝑡)≜𝑓(𝑥(𝑦),𝑡); but we choose the denotation 𝑓(𝑥,𝑡) instead of 𝑓(𝑦,𝑡) in order to simplify the derivation in the rest of the sections. Hence, time-delay system (1) is transformed into the delay-free system (5a) and (5b).

Notice that system (5a) and (5b) is coupled with 𝑦 and 𝑥(𝑦). Indeed, the term 𝑓(𝑥,𝑡) is nonlinear. From the previous investigations, it can be seen that by the functional transformation method [20–22], 𝑦 is admissible to system (5a) and (5b) if and only if 𝑥 is admissible to system (1). Moreover, the controller stabilizing (1) also stabilizes (5a) and (5b). Based on this idea, in what follows, we will develop this approach to solve the VSDC problem of system (1) through the equivalent delay-free system (5a) and (5b).

First, system (5a) and (5b) will be converted into a regular form. Assume that the uncertainty is matched; that is, there exists rank𝐵𝑑=rank𝐵, where 𝐵𝑑 is of full column rank [34], so that there exists a nonsingular matrix Ξ∈ℝ^𝑛×𝑛 such that Ξ𝐷0𝐷=,Ξ0𝐵𝐵=,Ξ𝑑()=0𝑑()𝑥,𝑡𝑥,𝑡, (7) where 𝐷∈ℝ^{(𝑛−𝑛1)×𝑞} and 𝐵∈ℝ^𝑛1×𝑚 is nonsingular. Then, denote the following nonsingular transformations: 𝑧(𝑡)=𝑧₁(𝑡)𝑧₂(𝑡)(𝑡),𝑀≜Ξ𝑦𝑀=¹¹𝑀₁₂𝑀₂₁𝑀₂₂≜Ξ𝐴Ξ⁻¹,𝑓()=𝑓𝑥,𝑡¹()𝑓𝑥,𝑡₂()(),𝑥,𝑡≜Ξ𝑓𝑥,𝑡𝐵≜Ξ𝐵,𝐷≜Ξ𝑑()𝐷,𝑥,𝑡≜Ξ𝑑(𝑥,𝑡), (8) where 𝑧₁(𝑡)∈ℝ^𝑛−𝑛1,𝑧₂(𝑡)∈ℝ^𝑛1. Via above transformations, system (5a) and (5b) can be converted into the regular form of𝑧̇₁(𝑡)=𝑀₁₁𝑧₁(𝑡)+𝑀₁₂𝑧₂(𝑡)+𝐷𝑣(𝑡)+𝑓₁()𝑧̇𝑥,𝑡₂(𝑡)=𝑀₂₁𝑧₁(𝑡)+𝑀₂₂𝑧₂(𝑡)+𝐵𝑢(𝑡)+𝑓₂(𝑥,𝑡)+𝑑(𝑥,𝑡), (9a)(9b)

where 𝑓_𝑖(𝑥,𝑡),𝑖=1,2 is the nonlinear and Lipschitz functions with 𝑓_𝑖(0,𝑡)=0. The pair (𝐴,𝐵) is controllable and guarantees (𝑀₁₁,𝑀₁₂) controllable. And 𝑑(𝑥,𝑡)=[𝑑₁(𝑥,𝑡),𝑑₂(𝑥,𝑡),…,𝑑_𝑛1(𝑥,𝑡)]^𝑇 is the uncertain function, where there exists a vector function 𝜌(𝑥,𝑡)=[𝜌₁(𝑥,𝑡),𝜌₂(𝑥,𝑡),…,𝜌_𝑛1(𝑥,𝑡)]^𝑇∶ℝ^𝑛×ℝ⁺→ℝ^𝑛1, subject to 𝑑(𝑥,𝑡)≤𝜌(𝑥,𝑡), (10) in which 𝜌_𝑖(𝑥,𝑡)(𝑖=1,2,…,𝑛₁) are scalar functions and the components of 𝑑(𝑥,𝑡) are less than or equivalent to the relevant components of 𝜌(𝑥,𝑡), that is, 𝑑_𝑖(𝑥,𝑡)≤𝜌_𝑖(𝑥,𝑡),∀𝑖=1,2,…,𝑛₁. (11) This is supposed a known to be condition to the controller designers [34].

Thereby, we will demonstrate the control law design process of finite-time horizon and infinite-time horizon in Sections 3 and 4, respectively.

3. Design of VSDC in Finite-Time Horizon

This section is to outline the sufficient and necessary condition for the optimality of an optimal sliding mode surface 𝑠^∗(𝑡) subject to the nonlinear dynamical constraint (9a) and (9b). In what follows, we will give the design procedure in two steps: OSMS design and VSDC design.

3.1. Design of OSMS in Finite-Time Horizon

To design a sliding mode surface 𝑠(𝑧)=𝐶₁𝑧₁(𝑡)+𝐶₂𝑧₂(𝑡) for system (9a) and (9b), treat 𝑧₂ as a virtual control of system (9a). And respecting the finite-time performance index 𝐽(⋅)=12𝑧^𝑇₁𝑡_𝑓𝑄_𝑓𝑧₁𝑡_𝑓+12^𝑡𝑓₀𝑧^𝑇₁(𝑡)𝑄𝑧₁(𝑡)+𝑧^𝑇₂(𝑡)𝑅𝑧₂(𝑡)𝑑𝑡, (12) where 𝑄_𝑓∈ℝ^{(𝑛−𝑛1)×(𝑛−𝑛1)} and 𝑄=𝐶^𝑇𝐶∈ℝ^{(𝑛−𝑛1)×(𝑛−𝑛1)} are positive semidefinite matrices, 𝑅∈ℝ^𝑛1×𝑛1 is a positive definite matrix, and (𝐴,𝐶) is assumed observable; the OSMS will be obtained by solving the optimal control problem of system (9a) with performance index (12).

3.1.1. Global OSMS of Nonlinear Systems

Consider the optimal control problem (9a) subject to (12), treating 𝑓₁(𝑥,𝑡) as an excitation term despite its relationship with 𝑦. Thus, the main result of Theorem 2 is achieved.

Theorem 2. Consider the optimal sliding mode surface design problem described by system (1) under disturbance (2) with quadratic performance index (12). The optimal sliding mode surface is existent and unique which is given by 𝑠^∗=𝑀^𝑇₁₂𝑃₁(𝑡)𝑅×Ξ𝑥^∗(𝑡)+^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥^∗(ℎ)𝑑ℎ+^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢^∗(ℎ)𝑑ℎ+𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)+𝑔(𝑡), (13) where 𝑃₁(𝑡) is the unique positive definite solution of Riccati matrix differential equation: −𝑃̇₁(𝑡)=𝑀^𝑇₁₁𝑃₁(𝑡)+𝑃₁(𝑡)𝑀₁₁−𝑃₁(𝑡)𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑃+𝑄₁𝑡_𝑓=𝑄_𝑓,𝑡∈0,𝑡_𝑓. (14)𝑃₂(𝑡) is the unique solution of Sylvester matrix differential equation: −𝑃̇₂(𝑡)=𝑀₁₁−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)^𝑇𝑃₂(𝑡)+𝑃₂(𝑡)𝐺+𝑃₁(𝑡)𝑃𝐷𝐹₂𝑡_𝑓=0, (15) and 𝑔(𝑡) is the unique solution of the adjoint equation: 𝑔̇(𝑡)=𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)−𝑀₁₁^𝑇𝑔(𝑡)−𝑃₁(𝑡)𝑓₁(𝑥(𝑡))𝑔𝑡,𝑡_𝑓=0. (16) The optimal state 𝑥^∗(𝑡) is the solution of the closed-loop system: 𝑧̇^∗₁(𝑡)=𝑀₁₁−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑧^∗₁(𝑡)+𝐷𝐹−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑤(𝑡)+𝑓₁𝑥^∗,𝑡−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑔(𝑡)𝜙₁(0)=𝜙₁𝑧^∗₂(𝑡)=−𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑧^∗₁(𝑡)+𝑃₂(𝑡)𝑤(𝑡)(𝑡𝑥+𝑔)_∗(𝑡)=Ξ⁻¹𝑧^∗(𝑡)−^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−𝛿)}𝐴₁𝑥^∗(𝛿)𝑑𝛿−^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−𝛿)}𝐵₁𝑢^∗(𝛿)𝑑𝛿. (17)

Proof. In analogy to classical linear quadratic regulator optimal control theory from minimum principle, the Hamiltonian for the linear-quadratic control problem (9a) with respect to (10) and (11) becomes 𝐻𝑧₁(𝑡),𝑧₂(𝑡)(𝑡)=1,𝜆,𝑡2𝑧^𝑇₁(𝑡)𝑄𝑧₁(𝑡)+𝑧^𝑇₂(𝑡)𝑅𝑧₂(𝑡)+𝜆^𝑇(𝑡)𝑀₁₁𝑧₁(𝑡)+𝑀₁₂𝑧₂(𝑡)+𝐷𝑣(𝑡)+𝑓₁(𝑥,𝑡), (18) which satisfies the canonical equations: 𝑧̇₁(𝑡)=𝜕𝐻𝜕𝜆(𝑡)=𝑀^𝑇₁₁𝑧₁(𝑡)−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝜆(𝑡)+𝐷𝑣(𝑡)+𝑓₁(),𝜆̇(𝑡)𝑥,𝑡=−𝜕𝐻𝜕𝑧₁(𝑡)=−𝑄𝑧₁(𝑡)−𝑀^𝑇₁₁𝜆(𝑡), (19)(20) with the transversality condition: 𝜆𝑡_𝑓=𝑄_𝑓𝑧₁𝑡_𝑓 (21) and the control equation: 𝜕𝐻𝜕𝑧₂(𝑡)=𝑅𝑧₂(𝑡)+𝑀^𝑇₁₁𝜆(𝑡)=0 (22) giving 𝑧^∗₂(𝑡)=−𝑅⁻¹𝑀^𝑇₁₂𝜆(𝑡) (23) along an optimal trajectory. Then, the canonical equations (19) and (20) result in the coupled nonlinear TPBV problem: 𝑧̇₁(𝑡)𝜆̇(𝑡)=𝑀₁₁−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂−𝑄−𝑀^𝑇₁₁𝑧₁(𝑡)𝜆(𝑡)+𝐷0𝑣(𝑡)+𝑓₁()0𝑧𝑥,𝑡₁(0)𝜆𝑡_𝑓=𝜙₁0. (24) The virtual optimal control law is (23). The optimal sliding mode surface is defined by 𝑠^∗(𝑧)=𝑅𝑧^∗₂(𝑡)+𝑀^𝑇₁₂𝜆(𝑡)=0. (25) For 𝑠=[𝑠₁,𝑠₂,…,𝑠_𝑛1]∈ℝ^𝑛1. The objective is to design 𝑧^∗₂(𝑡) forcing system trajectories reach sliding mode surface (25).
From (21), it shows that the costate 𝜆 has linear relationship with state 𝑧₁(𝑡). Thus, denote the costate vector 𝜆(𝑡)=𝑃₁(𝑡)𝑧₁(𝑡)+𝑃₂(𝑡)𝑤(𝑡)+𝑔(𝑡), (26) where unknown matrices 𝑃₁(𝑡),𝑃₂(𝑡), and continuous function 𝑔(𝑡) are to be determined. Differentiating (26) and substituting the first equation of (24) and (26) into the result follow the following equation: 𝜆̇(𝑡)=𝑃̇₁(𝑡)+𝑃₁(𝑡)𝑀₁₁−𝑃₁(𝑡)𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑧₁(𝑡)+𝑃̇₂(𝑡)+𝑃₁(𝑡)𝐷𝐹+𝑃₂(𝑡)𝐺−𝑃₁(𝑡)𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)−𝑃₁(𝑡)𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑔(𝑡)+𝑃₁(𝑡)𝑓₁(𝑥,𝑡)+𝑔̇(𝑡). (27) Moreover, putting (26) into the second equation of (24) yields the following equation: 𝜆̇(𝑡)=−𝑄+𝑀^𝑇₁₁𝑃₁(𝑡)𝑧₁(𝑡)−𝑀^𝑇₁₁𝑃₂(𝑡)𝑤(𝑡)−𝑀^𝑇₁₁𝑔(𝑡). (28) Equations (27) and (28) are equivalent directly and give the Riccati differential equation (14), Sylvester differential equation (15), and the adjoint differential equation (16). Then, substituting (26) into the optimal sliding mode surface (25) yields 𝑠^∗=𝑅𝑧^∗₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝑧₁(𝑡)+𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)+𝑀^𝑇₁₂𝑔(𝑡)=0. (29) From the denotation in (8), the OSMS (29) becomes 𝑠^∗=𝑀^𝑇₁₂𝑃₁(𝑡)𝑅Ξ𝑦(𝑡)+𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)+𝑔(𝑡). (30) Replacing 𝑦(𝑡) of (30) by that of (3) yields the optimal sliding mode surface (13).
From (23) and (26), it follows the optimal virtual control 𝑧^∗₂(𝑡)=−𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑧₁(𝑡)+𝑃₂(𝑡)𝑤(𝑡)+𝑔(𝑡). (31) Substituting (31) into (9a) with (5b) and (8) obtains the closed-loop system (17).
Matrix differential equations (14)–(16) satisfy the conditions of existence and uniqueness which implies that the OSMS (13) is existent and unique. The proof is completed.

3.1.2. Approximations of Sequences of State and Costate Equations

However, noting that (16) and (17) are coupled nonlinear differential equations, they are complex and seldom have analysis solutions. So, the sequence approximation method is adopted to obtain the solutions. Differential equations (16) and (17) can be replaced by sequences of linear time-varying (LTV) approximations [27]: 𝑔̇^(𝑘)(𝑡)=𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)−𝑀₁₁^𝑇𝑔^(𝑘)(𝑡)−𝑃₁(𝑡)𝑓₁𝑥^(𝑘−1)(𝑡)𝑔,𝑡_(𝑘)𝑡_𝑓𝑧̇=0,𝑘=1,2,…,_1(𝑘)(𝑡)=𝑀₁₁−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑧₁^(𝑘)+𝐷𝐹−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑤(𝑡)+𝑓₁𝑥^(𝑘−1)(𝑡),𝑡−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑔^(𝑘)(𝑡)𝑧₁^(𝑘)(0)=𝜙₁𝑧₂^(𝑘)(𝑡)=−𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑧₁^(𝑘)(𝑡)+𝑃₂(𝑡)𝑤(𝑡)+𝑔^(𝑘)(𝑡𝑥_(𝑘))(𝑡)=Ξ⁻¹𝑧^(𝑘)(𝑡)−^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−𝛿)}𝐴₁𝑥^(𝑘)(𝛿)−𝑑𝛿_{𝑡𝑡−𝜏}𝑒^{𝐴(𝑡−𝛿)}𝐵₁𝑢^(𝑘)(𝛿)𝑑𝛿. (32)(33) Notice that approximation sequences (32) and (33) are represented as inhomogeneous linear differential equations, which have the following solutions given by the variation of constant formula: 𝑔⁽⁰⁾(𝑡)𝑔=0_(𝑘)(𝑡)=^∞_𝑡Φ^𝑇(𝑟−𝑡)𝑃₁(𝑡)𝑓₁𝑥^(𝑘−1)(𝑟)𝑔,𝑟𝑑𝑟_(𝑘)𝑡_𝑓𝑧=0,_1(𝑘)(𝑡)=Φ(𝑡)𝑧₁^(𝑘)(0)+^𝑡₀Φ(𝑡−𝑟)𝐷𝐹−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑟)+𝑓₁𝑥^(𝑘−1)(𝑟),𝑟−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑔^(𝑘)(𝑟𝑧)𝑑𝑟_1(𝑘)(0)=𝜙₁𝑧₂^(𝑘)(𝑡)=−𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑧₁^(𝑘)(𝑡)+𝑃₂(𝑡)𝑤(𝑡)+𝑔^(𝑘)(𝑡𝑥_(𝑘))(𝑡)=Ξ⁻¹𝑧^(𝑘)(𝑡)−^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−𝛿)}𝐴₁𝑥^(𝑘)(𝛿)𝑑𝛿−^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−𝛿)}𝐵₁𝑢^(𝑘)(𝛿)𝑑𝛿 (34)(35) in which Φ(𝑡) denotes the transition matrix generated by [𝑀₁₁−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)], that is, Φ(𝑡)=exp𝑀₁₁−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡)𝑡. (36)

In what follows, with the purpose of proving sequences {𝑔^(𝑘)} and {𝑧₁^(𝑘)}, {𝑥^(𝑘)} converge to the solution of (16) and (17), and some preliminaries should be carried out.

Firstly, we should arrange the equations in compact forms. Noting that (32) and (33) are equivalent to (34) and (35), respectively, one formula will be chosen for proving. We select the latter giving the proof. In order to simplify the derivation, denote (16) and (17) in a compact form: 𝑧̇₁(𝑡)=𝐾₁𝑧₁(𝑡)+ℎ(𝑡,𝑥)𝑧₁(0)=𝜙₁𝑥(𝑡)=Ξ⁻¹𝑧(𝑡)−^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−𝛿)}𝐴₁𝑥(𝛿)−𝑑𝛿_{𝑡𝑡−𝜏}𝑒^{𝐴(𝑡−𝛿)}𝐵₁𝑢(𝛿)𝑑𝛿 (37) and combine (32) and (33) in another compact form: 𝑧₁⁽⁰⁾(𝑡)=Ψ(𝑡)𝑧₁^(𝑘)(0),𝑧₁^(𝑘)(𝑡)=Ψ(𝑡)𝑧₁^(𝑘)(0)+^𝑡₀Ψ(𝑡−𝛿)ℎ𝛿,𝑥^(𝑘−1)(𝛿)𝑑𝛿𝑧₁^(𝑘)(0)=𝜙₁𝑥^(𝑘)(𝑡)=Ξ⁻¹𝑧^(𝑘)(𝑡)−^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−𝛿)}𝐴₁𝑥^(𝑘)(𝛿)−𝑑𝛿_{𝑡𝑡−𝜏}𝑒^{𝐴(𝑡−𝛿)}𝐵₁𝑢^(𝑘)(𝛿)𝑑𝛿, (38) where Ψ denotes the state transition matrix corresponding to matrix 𝐾₁ and the nonlinear function ℎ(⋅)∈𝐂¹([𝑡₀,𝑡_𝑓]×ℝ^𝑛,ℝ^𝑛) with ℎ(𝑡,0)=0 satisfies the uniformly boundedness condition and the Lipschitz conditions, respectively; that is, (A1)‖ℎ(𝑡,𝑥)‖≤𝛾,   for all  𝑥∈ℝ^𝑛, (A2) ‖ℎ(𝑡,𝑥₁)−ℎ(𝑡,𝑥₂)‖≤𝛽‖𝑥₁−𝑥₂‖,  for all  𝑥₁,𝑥₂∈ℝ^𝑛, where 𝛾 is some finite constant and 𝛽>0.

Secondly, we will prove approximating sequences (38) convergent to the solution of (37). As a result, Lemma 3 is available.

Lemma 3. Let ℎ(𝑡,𝑥) satisfy (A1) and (A2). Then, the limit of the solution of the approximating sequences (38) on 𝐂([𝑡₀,𝑡_𝑓];ℝ^𝑛) converges to the unique solution of (37) on [𝑡₀,𝑡_𝑓].

Proof. Let sup‖Ψ(𝑡−𝛿)‖=𝜁,supℎ𝑡,𝑥⁽⁰⁾(𝑡)=𝛾. (39) Note that ‖Ψ(𝑡₀−𝑡₀)‖=‖𝐼‖=1. Then, 𝜁≥1. From (38), it follows that 𝑧₁⁽¹⁾(𝑡)−𝑧₁⁽⁰⁾(𝑡)=^𝑡₀Ψ(𝑡−𝛿)ℎ𝛿,𝑥⁽⁰⁾(𝛿)𝑑𝛿 (40) With (A2), (5b), and (8), it gives ℎ𝑡,𝑥^(𝑘)(𝑡)−ℎ𝑡,𝑥^(𝑘−1)(𝑡)≤𝛽𝑥^(𝑘)(𝑡)−𝑥^(𝑘−1)(𝑡𝑧)≤𝛽₁^(𝑘)(𝑡)−𝑧₁^(𝑘−1)(𝑡), (41) where 𝛽,𝛽>0. The above inequality gives 𝑧₁⁽¹⁾(𝑡)−𝑧₁⁽⁰⁾(𝑡)=𝜁^𝑡₀ℎ𝑡,𝑥⁽⁰⁾(𝑡)𝑑𝛿≤𝜁𝛾𝑡. (42) Subsequently, 𝑧₁⁽²⁾(𝑡)−𝑧₁⁽¹⁾(𝑡)=𝜁𝛽^𝑡₀𝑧₁⁽²⁾(𝑡)−𝑧₁⁽¹⁾(𝑡1)𝑑𝛿≤2𝜁²𝛽𝛾𝑡². (43) By analogy, it has 𝑧₁^(𝑘+1)(𝑡)−𝑧₁^(𝑘)(𝑡)≤𝜁^𝑘+1𝛽^𝑘𝛾^𝑘𝑡^𝑘+1()!.𝑘+1 (44) According to trigonometry inequality, for any 𝑗, the following holds: 𝑧₁^(𝑘+1)(𝑡)−𝑧₁^(𝑘)(𝑡≤𝑧_1(𝑘+𝑗))(𝑡)−𝑧₁^{(𝑘+𝑗−1)}(𝑡+𝑧_{1(𝑘+𝑗−1)})(𝑡)−𝑧₁^{(𝑘+𝑗−2)}(𝑡𝑧)+⋯+₁^(𝑘+1)(𝑡)−𝑧₁^(𝑘)(𝑡≤_{𝑘+𝑗−1𝑖=𝑘})𝜁^𝑖+1𝛽^𝑖𝛾^𝑖𝑡^𝑖+1()!≤𝜁𝑖+1_𝑘+1𝛽^𝑘𝛾^𝑘𝑡^𝑘+1(𝑘+1)!^𝑗−1_𝑖=0𝜁^𝑖𝛽^𝑖𝛾^𝑖𝑡^𝑖≤𝜁𝑖!_𝑘+1𝛽^𝑘𝛾^𝑘𝑡^𝑘+1()!;𝑘+1exp𝜁𝛽𝛾𝑡 (45) that is, there exists lim_𝑘→∞𝑧₁^(𝑘+𝑗)(𝑡)−𝑧₁^(𝑘)(𝑡)=0. (46) Via the same way, there exists lim_𝑘→∞‖𝑧₂^(𝑘+𝑗)(𝑡)−𝑧₂^(𝑘)(𝑡)‖=0. Hence, lim_𝑘→∞‖𝑧^(𝑘+𝑗)(𝑡)−𝑧^(𝑘)(𝑡)‖=0. From (5b) and (8), it follows lim_𝑘→∞𝑥^(𝑘+𝑗)(𝑡)−𝑥^(𝑘)(𝑡)=0 (47) Therefore, {𝑧^(𝑘)(𝑡)} and {𝑥^(𝑘)(𝑡)} are sequences of Cauchy in Banach space 𝐂([𝑡₀,𝑡_𝑓];ℝ^𝑛), respectively. Therefore, 𝑧^(𝑘)(𝑡)→𝑧(𝑡) on 𝐂([𝑡₀,𝑡_𝑓];ℝ^𝑛) and 𝑥^(𝑘)(𝑡)→𝑥(𝑡) on 𝐂([𝑡₀,𝑡_𝑓];ℝ^𝑛). The proof of Lemma 4 is completed.

Thirdly, result of (38) and (37) described by Lemma 3 directly carries out the same result of (32)–(35) and (16)-(17) described by Lemma 4.

Lemma 4. Let 𝑓₁(𝑥,𝑡) be bounded and Lipschitz continuous in its arguments. Then, the limits of the solutions of the approximating sequences (32) and (33) (or (34) and (35)) on 𝐂([𝑡₀,𝑡_𝑓];ℝ^𝑛) globally converge to the unique solutions of (16) and (17), respectively, on [𝑡₀,𝑡_𝑓].

Last but not least, the main result of OSMS by approximation sequences is got as Theorem 5.

Theorem 5. Given the nonlinear system (1) and the cost functional (12), then the optimal sliding mode surface is given by the limit of the sequence 𝑠^∗(𝑘)(𝑡)=𝑀^𝑇₁₂𝑃₁(𝑡)𝑅×Ξ𝑥^(𝑘)(𝑡)+^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥^(𝑘)(ℎ)𝑑ℎ+^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢^(𝑘)(ℎ)𝑑ℎ+𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)+𝑔^(𝑘)(𝑡), (48) where 𝑃₁(𝑡)∈ℝ^𝑛×𝑛 is the unique positive definite solution of Riccati matrix differential equation (14), 𝑃₂(𝑡)∈ℝ^𝑛×𝑝 is the unique solution of the Sylvester matrix differential equation (15), and 𝑔^(𝑘)(𝑡) is given by the converged unique solution of the LTV differential equation sequence (32) or (34). The optimal state 𝑥^∗(𝑡) is the solution of the closed-loop system (33) or (35).

Remark 6. Actually, in practice, the limit of lim_𝑘→∞𝑔^(𝑘) can be obtained by replacing ∞ with a positive integer 𝑀. Relevantly, the following suboptimal sliding mode surface is achieved: 𝑠^(𝑀)(𝑡)=𝑀^𝑇₁₂𝑃₁(𝑡)𝑅×Ξ𝑥^(𝑀)(𝑡)+^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥^(𝑀)(ℎ)𝑑ℎ+^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢^(𝑀)(ℎ)𝑑ℎ+𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)+𝑔^(𝑀)(𝑡), (49) corresponding to the following performance index: 𝐽^(𝑀)=12𝑧₁^(𝑀)𝑇𝑡_𝑓𝑄_𝑓𝑧₁^(𝑀)𝑡_𝑓+12^𝑡𝑓₀𝑧₁^(𝑀)𝑇(𝑡)𝑄𝑧₁^(𝑀)(𝑡)+𝑧₂^(𝑀)𝑇(𝑡)𝑅𝑧₂^(𝑀)(𝑡)𝑑𝑡, (50) where 𝑀 is determined by a small enough error criterion 𝜃>0 when the following inequality holds: 𝐽^(𝑀−1)−𝐽^(𝑀)𝐽^(𝑀)<𝜃. (51)
Consequently, we give the algorithm of OSMS.

Algorithm 7. OSMS of system (1). 𝑆𝑡𝑒𝑝1. Solve 𝑃₁(𝑡) and 𝑃₂(𝑡) from (14) and (15). Give some positive real constant 𝜃>0. Set 𝑧₁⁽⁰⁾(𝑡)=𝑔⁽⁰⁾(𝑡)=𝐽⁽⁰⁾=0 and 𝑘=1. 𝑆𝑡𝑒𝑝 2. Obtain the 𝑘th adjoint vector 𝑔^(𝑘)(𝑡) from (34). 𝑆𝑡𝑒𝑝 3. Let 𝑀=𝑘. Calculate 𝑧₂^(𝑀) from (35) and 𝑠^(𝑀) from (49). 𝑆𝑡𝑒𝑝4. Calculate 𝐽^(𝑀) from (50). 𝑆𝑡𝑒𝑝5. When condition (51) is satisfied, stop and output 𝑠^(𝑀). 𝑆𝑡𝑒𝑝6. Calculate 𝑥^(𝑘) from (35).

3.2. Design of VSDC in Finite-Time Horizon

In this section, we will present the VSDC design of finite-time horizon based on the proposed OSMS in the last section.

3.2.1. VSDC Design

To begin with, a trending law should be chosen. As known to all, different variable structure control laws can be designed corresponding to different sliding mode reachable conditions. In this paper, we will adopt the exponential trending law introduced by Gao [30, 34] as follows: 𝑠̇=−𝑘𝑠−𝜀sign(𝑠),𝑘,𝜀>0, (52) where 𝑘=diag𝑘₁,𝑘₂,…,𝑘_𝑛1,𝜀=diag𝜀₁,𝜀₂,…,𝜀_𝑛1sign(𝑠)=sign𝑠₁,sign𝑠₂,…,sign𝑠_𝑛1^𝑇 (53) and 𝑘_𝑖,𝜀_𝑖 are parameters to be selected which determine system instant response and convergence speed of the sliding mode reaching, respectively. The related reachable condition should be satisfied: 𝑠_𝑖̇≥−𝑘_𝑖𝑠_𝑖−𝜀_𝑖sign𝑠_𝑖,𝑠_𝑖𝑠̇<0,_𝑖≤−𝑘_𝑖𝑠_𝑖−𝜀_𝑖sign𝑠_𝑖,𝑠_𝑖>0, (54) for 𝑖=1,2,…,𝑛₁, that is, 𝑠𝑠̇<0. For the reason of the reachable condition (54), variable structure control satisfying condition (54) will drive the state trajectories to reach the sliding mode surface in exponential velocity and then remain on it [30, 34]. Consequentially, in the basis of trending law (52), the VSDC is presented as Theorem 8.

Theorem 8. Consider system (1) under disturbance (2) with quadratic performance index (12). The sliding mode surface is given by (13). Then, the state trajectories can reach the sliding mode surface in finite time and remain on it by the variable structure control law:𝑢^∗(𝑡)=−(𝑅𝐵)⁻¹𝑅𝑀₂₁+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₁+𝑘𝑀^𝑇₁₂𝑃₁(𝑡)+𝑀^𝑇₁₂𝑃̇₁(𝑡)𝑅𝑀₂₂+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₂𝑥(𝑡)+𝑘𝑅×Ξ+∫^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥(ℎ)𝑑ℎ+∫^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢(ℎ)𝑑ℎ+𝑀^𝑇₁₂𝑔̇(𝑡)+𝑘𝑔(𝑡)+𝑘𝑀^𝑇₁₂𝑃₂(𝑡)+𝑀^𝑇₁₂𝑃₂(𝑡)𝐺+𝑀^𝑇₁₂𝑃̇₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝐷𝐹𝑤(𝑡)+𝑅𝑓₂(𝑥,𝑡)+𝜀+𝜌(𝑥,𝑡)𝑅sign(𝑠). (55)

Proof. Differentiating sliding mode surface (29) with respect to time and substituting system (9a) and (9b) into it give 𝑠̇(𝑡)=𝑅𝑀₂₁+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₁+𝑀^𝑇₁₂𝑃̇₁(𝑡)𝑧₁(𝑡)+𝑅𝑀₂₂+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₂𝑧₂(𝑡)(𝑡)+𝑀+𝑅𝐵𝑢_𝑇12𝑃₁(𝑡)𝐷𝐹+𝑀^𝑇₁₂𝑃₂(𝑡)𝐺₂+𝑀^𝑇₁₂𝑃̇₂(𝑡)𝑤(𝑡)+𝑅𝑓₂(𝑥,𝑡)+𝑀^𝑇₁₂𝑔̇(𝑡). (56) Substituting (26) into (25) and putting the result into (52) yield 𝑠̇=−𝑘𝑀^𝑇₁₂𝑃₁(𝑡)𝑧₁(𝑡)−𝑘𝑅𝑧₂(𝑡)−𝑘𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)−𝑘𝑀^𝑇₁₂𝑔(𝑡)−𝜀sign(𝑠). (57) Comparing (56) and (57), the variable structure control law is designed: 𝑢^∗(𝑡)=−(𝑅𝐵)⁻¹𝑅𝑀₂₁+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₁+𝑘𝑀^𝑇₁₂𝑃₁(𝑡)+𝑀^𝑇₁₂𝑃̇₁(𝑡)𝑧₁(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₂+𝑅𝑀₂₂+𝑘𝑅𝑧₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝐷𝐹+𝑀^𝑇₁₂𝑃₂(𝑡)𝐺+𝑘𝑀^𝑇₁₂𝑃₂(𝑡)+𝑀^𝑇₁₂𝑃̇₂(𝑡)𝑤(𝑡)+𝑀^𝑇₁₂𝑔̇(𝑡)+𝑘𝑔(𝑡)+𝑅𝑓₂()+()𝑅(𝑠.𝑥,𝑡𝜀+𝜌𝑥,𝑡sign) (58) Replacing 𝑧₁,𝑧₂ in (58) by 𝑥 of (8) and (3) yields the VSDC (55).
Besides, the reachable condition (54) is verified in what follows. Substituting the VSDC (58) into (56) gives 𝑠̇=−𝑘𝑠−𝜀sign(𝑠)+𝑅𝑑(𝑥,𝑡)−𝜌(𝑥,𝑡)sign(𝑠). (59) If 𝑠_𝑖>0, the following holds: 𝑅𝑑(𝑥,𝑡)−𝜌(𝑥,𝑡)sign(𝑠)=𝑟_𝑖𝑑_𝑖(𝑥,𝑡)−𝜌(𝑥,𝑡) (60) for all  𝑖=1,2,…,𝑛₁. Since 𝑑_𝑖(𝑥,𝑡)≤𝜌_𝑖(𝑥,𝑡) (61) and 𝑟_𝑖>0. Thus, (59) becomes 𝑠_𝑖̇≤−𝑘_𝑖𝑠_𝑖−𝜀_𝑖sign𝑠_𝑖<0. (62) Alternatively, if 𝑠_𝑖<0, employing the same results, 𝑠_𝑖̇≥−𝑘_𝑖𝑠_𝑖−𝜀_𝑖sign𝑠_𝑖>0. (63) Therefore, combining (62) and (63), the VSDC (55) which satisfies reachable condition 𝑠𝑠̇<0, can make the state trajectories reach the sliding mode surface (13) in finite time and remain on it [30, 34]. The proof is completed.

Remark 9. Actually, for the reason in Remark 6, the suboptimal sliding mode surface (SOSMS) (49) can be applied in practice. Thus, the relevant VSDC for SOSMS (49) can be gotten:𝑢^(𝑀)(𝑡)=−(𝑅𝐵)⁻¹𝑅𝑀₂₁+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₁+𝑘𝑀^𝑇₁₂𝑃₁(𝑡)+𝑀^𝑇₁₂𝑃̇₁(𝑡)𝑅𝑀₂₂+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₂𝑥+𝑘𝑅×Ξ^(𝑀)(𝑡)+∫^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥^(𝑀)(ℎ)𝑑ℎ+∫^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢^(𝑀)(ℎ)𝑑ℎ+𝑀^𝑇₁₂𝑔̇^(𝑀)(𝑡)+𝑘𝑔^(𝑀)(𝑡)+𝑀^𝑇₁₂𝑃₂(𝑡)𝐺+𝑘𝑀^𝑇₁₂𝑃₂(𝑡)+𝑀^𝑇₁₂𝑃̇₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝐷𝐹𝑤(𝑡)+𝑅𝑓₂𝑥^(𝑀−1),𝑡+𝜀+𝜌𝑥^(𝑀−1),𝑡𝑅sign𝑠^(𝑀). (64)

Remark 10. Summarily, on the ideal sliding mode surface, there exists 𝑠̇=0. And then, putting virtual control (31) into system (9a) yields the ideal sliding mode equations of system (1): 𝑧̇^∗₁(𝑡)=𝑀₁₁−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑧^∗₁(𝑡)+𝐷𝐹−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑃₁(𝑡𝑤(𝑡)())+𝑓𝑥,𝑡−𝑀₁₂𝑅⁻¹𝑀^𝑇₁₂𝑔(𝑡)𝜙₁(0)=𝜙₁𝑅𝑧^∗₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝑧^∗₁(𝑡)+𝑃₂(𝑡)𝑤(𝑡)+𝑔(𝑡)=0. (65) At this time, taking 𝑠=0 into VSDC (55) leads to the equivalent control 𝑢_𝑒(𝑡) of system (1):𝑢_𝑒(𝑡)=−(𝑅𝐵)⁻¹𝑅𝑀₂₁+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₁+𝑘𝑀^𝑇₁₂𝑃₁(𝑡)+𝑀^𝑇₁₂𝑃̇₁(𝑡)𝑅𝑀₂₂+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₂𝑥(𝑡)++𝑘𝑅×Ξ^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥(ℎ)𝑑ℎ+^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢(ℎ)𝑑ℎ+𝑀^𝑇₁₂𝑔̇(𝑡)(𝑡++𝑘𝑔)𝑘𝑀^𝑇₁₂𝑃₂(𝑡)+𝑀^𝑇₁₂𝑃₂(𝑡)𝐺+𝑀^𝑇₁₂𝑃̇₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝐷𝐹𝑤(𝑡)+𝑅𝑓₂(𝑥,𝑡). (66)

3.2.2. Closed-Loop Stability Analysis

Select the OSMS function 𝑠(𝑧) as a Lyapunov function candidate, that is, 𝑉(𝑧)=(1/2)𝑠²(𝑧). Then, the increment of it is 𝑉̇=𝑠𝑠̇. As known, the OSMS satisfies the reachable condition (54), that is, 𝑠𝑠̇<0. Thus, 𝑉̇ is negative definite, which implies lim_𝑡→∞𝑧(𝑡)=0. Moreover, from (3) and (8), it can be proved that 𝑥→0 as 𝑧→0 indicating the closed-loop system states asymptotically stable.

3.2.3. Physical Realization of VSDC

The VSDC 𝑢^∗(𝑡) in (55) contains disturbance state 𝑤(𝑡) which is physically unmeasurable. To solve this problem, one can construct a reduced-order observer for disturbance vector so as to reconstruct the disturbance variables [28, 29, 35].

Since rank𝐹=𝑞, there exists an arbitrary matrix 𝐻∈ℝ^{(𝑝−𝑞)×𝑝} such that Γ=[𝐹^𝑇𝐻^𝑇]^𝑇∈ℝ^𝑝×𝑝 is nonsingular. Note that Π=Γ⁻¹=[Π₁∶Π₂], where Π₁∈ℝ^𝑝×𝑞 and Π₂∈ℝ^{𝑝×(𝑝−𝑞)}. It is obvious that 𝐼_𝑝=ΓΠ=𝐹Π₁𝐹Π₂𝐻Π₁𝐻Π₂=𝐼_𝑞00𝐼_𝑝−𝑞, (67) where 𝐼_𝑛 denotes the 𝑛-order identity matrix. Consequently, the following result can be obtained.

Theorem 11. Consider system (1) under disturbance (2) with quadratic performance index (12). Respecting the sliding mode surface (13), the dynamical sliding mode control law is given𝜓̇(𝑡)=(𝐻−𝐿𝐹)𝐺Π₂𝜓(𝑡)+Π₁+Π₂𝐿𝑣(𝑡𝑢_𝑑)(𝑡)=−(𝑅𝐵)⁻¹𝑅𝑀₂₁+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₁+𝑘𝑀^𝑇₁₂𝑃₁(𝑡)+𝑀^𝑇₁₂𝑃̇₁(𝑡)𝑅𝑀₂₂+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₂𝑥(𝑡)++𝑘𝑅×Ξ^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥(ℎ)𝑑ℎ+^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢(ℎ)dℎ+𝑀^𝑇₁₂𝑔̇(𝑡)(𝑡++𝑘𝑔)𝑘𝑀^𝑇₁₂𝑃₂(𝑡)+𝑀^𝑇₁₂𝑃₂(𝑡)𝐺+𝑀^𝑇₁₂𝑃̇₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)×Π𝐷𝐹₂𝜂(𝑡)+Π₁+Π₂𝐿𝑣(𝑡)+𝑅𝑓₂(𝑥,𝑡)+𝜀+𝜌(𝑥,𝑡)𝑅sign(𝑠), (68)by which the state trajectories can reach the sliding mode surface in finite time and remain on it.

Proof. Defining the nonsingular transformation Γ𝑤(𝑡)≜𝑤(𝑡)=𝑤₁(𝑡)𝑤₂(𝑡), (69) it gets 𝑤̇‌₂(𝑡)=𝐻𝐺Π₂𝜁₂(𝑡)+𝐻𝐺Π₁𝑣(𝑡)𝑣̇(𝑡)=𝐹𝐺Π₂𝑤₂(𝑡)+𝐹𝐺Π₁𝑣(𝑡). (70) Introduce a new variable 𝜓(𝑡)=𝑤₂(𝑡)−𝐿𝑣(𝑡), (71) where 𝐿 is the gain matrix to be selected. Then, differentiating 𝜓(𝑡) in (71) with respect to time and substituting (70) into it yield the observation state equation: 𝜓̇(𝑡)=(𝐻−𝐿𝐹)𝐺Π₂𝜓(𝑡)+Π₁+Π₂𝐿𝑣(𝑡)𝑤₂(𝑡)=𝜓(𝑡)+𝐿𝑣(𝑡), (72) where 𝑤₂(𝑡) is the component state to be reconstructed. Note that the other component state can be measured by 𝑣(𝑡)=𝑤₁(𝑡). On the other hand, from (69), it follows that 𝑤(𝑡)=Π₁𝑣(𝑡)+Π₂𝑤₂(𝑡). (73) Thus, substituting the second equation in (72) into (73) yields 𝜓̇(𝑡)=(𝐻−𝐿𝐹)𝐺Π₂𝜓(𝑡)+Π₁+Π₂𝐿𝑣(𝑡𝑤(𝑡))=Π₂𝜓(𝑡)+Π₁+Π₂𝐿𝑣(𝑡). (74) Relevant to (74), the reduced-order observer can be written as 𝜓̇(𝑡)=(𝐻−𝐿𝐹)𝐺Π₂𝜓(𝑡)+Π₁+Π₂𝐿𝑣(𝑡𝑤(𝑡))=Π₂𝜓(𝑡)+Π₁+Π₂𝐿𝑣(𝑡), (75) where 𝜓(𝑡), 𝑤(𝑡) represent the observed values of 𝜓(𝑡), 𝑤(𝑡), respectively. Denote the related errors as 𝑒_𝜓(𝑡)=𝜓(𝑡)−𝜓(𝑡), 𝑒_𝑤(𝑡)=𝑤(𝑡)−𝑤(𝑡). Subtracting (74) from (75) results in the error equation 𝑒_𝜓̇(𝑡)=(𝐻−𝐿𝐹)𝐺Π₂𝑒_𝜓(𝑡)𝑒_𝑤(𝑡)=Π₂𝑒_𝜓(𝑡). (76)
Now, the problem is converted into finding the observer gain 𝐿 such that error system (76) is asymptotically stable. That the pair (𝐺,𝐹) is observable ensures the pair (𝐹𝐺Π₂,𝐻𝐺Π₂) observable; thus there exists the gain 𝐿 such that all eigenvalues of matrix (𝐻−𝐿𝐹)𝐺Π₂ can be assigned to the desired position in left-half plane, which guarantees error system (76) exponentially stable. It implies lim_𝑡→∞𝑒_𝑤(𝑡)=0, that is, 𝑤(𝑡)→𝑤(𝑡). Hence, rewriting VSDC (57) by replacing the disturbance state 𝑤 by the estimated state 𝑤 of (75):𝑢_𝑑(𝑡)=−(𝑅𝐵)⁻¹𝑅𝑀₂₁+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₁+𝑘𝑀^𝑇₁₂𝑃₁(𝑡)𝑅𝑀₂₂+𝑀^𝑇₁₂𝑃₁(𝑡)𝑀₁₂𝑥(𝑡)+𝑘𝑅×Ξ+∫^𝑡_𝑡−𝜎𝑒^{𝐴(𝑡−ℎ)}𝐴₁𝑥(ℎ)𝑑ℎ+∫^𝑡_𝑡−𝜏𝑒^{𝐴(𝑡−ℎ)}𝐵₁𝑢(ℎ)𝑑ℎ+𝑀^𝑇₁₂𝑔̇(𝑡)(𝑡+𝑀+𝑘𝑔)_𝑇12𝑃₁(𝑡)𝐷𝐹+𝑀^𝑇₁₂𝑃₂(𝑡)𝐺+𝑘𝑀^𝑇₁₂𝑃₂(𝑡)𝑤(𝑡)+𝑅𝑓₂(𝑥,𝑡)+𝜀+𝜌(𝑥,𝑡)𝑅sign(𝑠) (77)yields the dynamical VSDC (68). The proof is completed.

At this time, the VSDC design in finite-time horizon is completed. In the next section, we will discuss the corresponding results of infinite-time horizon.

4. VSDC Design in Infinite-Time Horizon

To design an OSMC in the infinite-time horizon, regarding the stability situations of exosystem (2), two different performance indexes in infinite-time horizon can be chosen. When the disturbance is asymptotically stable, that is, Re𝜇_𝑗(𝐺)<0, the following general one should be selected: 𝐽(⋅)=^∞₀𝑧^𝑇₁(𝑡)𝑄𝑧₁(𝑡)+𝑧^𝑇₂(𝑡)𝑅𝑧₂(𝑡)𝑑𝑡, (78) where 𝑄=𝐶^𝑇𝐶∈ℝ^{(𝑛−𝑛1)×(𝑛−𝑛1)} is a positive semidefinite matrix, 𝑅∈ℝ^𝑛1×𝑛1 is a positive definite matrix, and (𝐴,𝐶) is observable. On the other hand, when the disturbance is Lyapunov stable, that is, Re𝜇_𝑗(𝐺)=0, since 𝑧₂ in (78) concluding disturbance 𝑤 may make the general infinite-horizon performance index (78) not convergent, in this case, the following average one can be selected: 𝐽(⋅)=lim_𝑇→∞1𝑇^𝑇₀𝑧^𝑇₁(𝑡)𝑄𝑧₁(𝑡)+𝑧^𝑇₂(𝑡)𝑅𝑧₂(𝑡)𝑑𝑡. (79)

Then, our object is to design OSMS and VSDC with respect to performance index (78) and (79) subject to dynamic constraint (9a) and (9b).

In reality, when 𝑡_𝑓→∞, the time-variable matrices 𝑃₁(𝑡_𝑓) and 𝑃₂(𝑡_𝑓) of finite-time horizon approach to some constant matrices denoted by 𝑃₁ and 𝑃₂, respectively, that is, lim_{𝑡𝑓→∞}𝑃₁(𝑡_𝑓)=𝑃₁ and lim_{𝑡𝑓→∞}𝑃₂(𝑡_𝑓)=𝑃₂ [35]. Therefore, the results of the infinite-time horizon can be directly obtained from those of the finite-time horizon. Yet, for the sake of briefness, the corresponding results are omitted here.

5. Simulation Example

In this section, a 2DOF quarter-car model is applied to simulate the system responses under the designed VSDC comparing with those of OLS. The governing dynamic equations are 𝑚_𝑠+Δ𝑚_𝑠𝑥̈_𝑠(𝑡)+𝑏_𝑠𝑥̇_𝑠(𝑡)−𝑥̇_𝑢(𝑡+𝑘_1𝑠)+Δ𝑘_1𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡+𝑘_2𝑠)+Δ𝑘_2𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)³(𝑡)𝑚=𝑢_𝑢𝑥̈_𝑢(𝑡)−𝑏_𝑠𝑥̇_𝑠(𝑡)−𝑥̇_𝑢(𝑡)−𝑘_1𝑠+Δ𝑘_1𝑠×𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)−𝑘_2𝑠+Δ𝑘_2𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)³+𝑘_𝑡+Δ𝑘_𝑡𝑥_𝑢(𝑡)−𝑥_𝑟(𝑡)=−𝑢(𝑡), (80) where 𝑚_𝑠 is the sprung mass representing car chassis; 𝑚_𝑢 is the unsprung mass representing wheel assembly; 𝑘_1𝑠 is the linear stiffness coefficients; 𝑏_𝑠 is the damping of the uncontrolled suspension; 𝑘_𝑡 stands for the stiffness of tire respectively; 𝑥_𝑠,𝑥_𝑢 are displacements of sprung and unsprung masses, respectively; 𝑥_𝑟 is the road displacement input. The actuator force 𝑢 acts between sprung and unsprung masses. Let 𝑘_2𝑠 represent the spring nonlinearity and Δ𝑚_𝑠, and let  Δ𝑘_𝑖(𝑖=1𝑠,2𝑠,𝑡) indicate the uncertainty parameters of spring mass, stiffness of spring, and tire with known bounds. Summarizing equations (80), one can get the nonlinearity 𝑓_𝑖(𝑥)(𝑖=1,2) of the system, which was employed in some references, for example, [14, 15] 𝑓₁(𝑥)=𝑘_2𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)³,𝑓₂(𝑥)=−𝑘_2𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)³, (81) and the uncertainties Δ𝑓_𝑖(𝑥) of the system which are referred to [16–18]: Δ𝑓₁(𝑥)=Δ𝑚_𝑠𝑥̈_𝑠(𝑡)+Δ𝑘_1𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)+Δ𝑘_2𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)³Δ𝑓₂(𝑥)=−Δ𝑘_1𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)−Δ𝑘_2𝑠𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡)³+Δ𝑘_𝑡𝑥_𝑢(𝑡)−𝑥_𝑟(𝑡). (82) More characteristic details about these kinds of nonlinearity and uncertainties can be found in abovementioned [14–18]. By defining the set of state variable 𝑥₁(𝑡)=𝑥_𝑠(𝑡)−𝑥_𝑢(𝑡),𝑥₂(𝑡)=𝑥_𝑢(𝑡)−𝑥_𝑟(𝑡),𝑥₃(𝑡)=𝑥̇_𝑠(𝑡),𝑥₄(𝑡)=𝑥̇_𝑢(𝑡), (83) where 𝑥₁ is the suspension deflection, 𝑥₂ the tire deflection, 𝑥₃ the sprung mass velocity, and 𝑥₄ the unsprung mass velocity, the state vector is denoted by 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡)]^𝑇. Then, the vehicle suspension system is rewritten in state-space representation (1) with 𝐴₁=001−10001−𝑘_1𝑠𝑚_𝑠0−𝑏_𝑠𝑚_𝑠𝑏_𝑠𝑚_𝑠𝑘_1𝑠𝑚_𝑢−𝑘_𝑡𝑚_𝑢𝑏_𝑠𝑚_𝑢−𝑏_𝑠𝑚_𝑢,𝐵₁=001𝑚_𝑠−1𝑚_𝑢,000,𝑓(𝑥)=00𝐷=−1−𝑘_2𝑠𝑚_𝑠𝑥³₁𝑘_2𝑠𝑚_𝑢𝑥³₁,𝑑(𝑥)=00−Δ𝑘_1𝑠𝑚_𝑠𝑥₁−Δ𝑘_2𝑠𝑚_𝑠𝑥³₁−Δ𝑚_𝑠𝑚_𝑠𝑥̇₃Δ𝑘_1𝑠𝑚_𝑢𝑥₁+Δ𝑘_2𝑠𝑚_𝑢𝑥³₁−Δ𝑘_𝑡𝑚_𝑢𝑥₂. (84)

Adopting the parameter values listed in Table 1 [19], the associated matrices and vectors of (84) are as follows:𝐴₁=,𝑓(𝑥)=00001−10001−43.9410−1.1261.126376.053−890.0979.639−9.768−4.394𝑥³₁37.605𝑥³₁,𝐵₁=00000.0010.001−0.009−0.009,𝐴₀=𝐵₀=0. (85)

Table 1 

Parameters and values of quarter-car suspension.

Set the initial value 𝑥₀=[1,0.1,0.1,0.1]^𝑇 and time-delays 𝜏=0.01𝑠, 𝜎=0.02𝑠. Take Δ𝑚_𝑠=𝑚_𝑠𝑑_𝑚𝑠𝛿_𝑚𝑠, Δ𝑘_𝑖=𝑘_𝑖𝑑_𝑙𝛿_𝑙 for 𝑖=1𝑠,2𝑠,𝑡;  𝑙=𝑘_1𝑠,𝑘_2𝑠,𝑘_𝑡, where 𝑑_𝑙 indicates the percentage of variation allowed around its nominal value and 𝛿_𝑗(𝑗=𝑚_𝑠,𝑘_1𝑠,𝑘_2𝑠,𝑘_𝑡) determine the actual parameter derivation changing in interval [0,1] [18]. Referring to [16, 17], given the possible variations of uncertain parameters: 𝑑_𝑚𝑠=0.285, 𝑑_𝑘1𝑠=0.15, 𝑑_𝑘2𝑠=0.15, and 𝑑_𝑘𝑡=0.25, note they represent 28.5% uncertainty in 𝑚_𝑠, 15% in 𝑘_1𝑠 and 𝑘_2𝑠, and 25% in 𝑘_𝑡, which yields the uncertainty bound parameters: Δ𝑚_𝑠≤100 kg, Δ𝑘_1𝑠≤2175 N/m, Δ𝑘_2𝑠≤24000 N/m, Δ𝑘_𝑡≤47500 N/m. Consequentially, the bound vector function 𝜌 is produced: 𝜌(𝑥)=00−6.21𝑥₁−68.57𝑥³₁−0.29𝑥̇₃36.86𝑥₁+406.78𝑥³₁−805.08𝑥₂. (86)

Then, transform the system in the form of (9a) and (9b). To simulate the road profiles, two cases of sinusoidal signal and attenuated signal are employed.

Case 1. Consider sinusoidal disturbance described by exosystem (2) with 𝐺=00100001−10000−200, (87) which produces the eigenvalues of 𝐺 are ±1𝑗,  ±1.4142𝑗. Take the performance index (79) with 𝑄=𝑅=𝐼₂. First of all, with (14) and (15) the matrices 𝑃₁, 𝑃₂ are computed. Moreover, from (32) and (33), 𝑔^(𝑀) and 𝑥^(𝑀) are solved. Consequentially, setting 𝑘=1 and 𝜀=0.05 in trending law (52), the OSMS (49) is gotten as well as VSDC (77) is obtained. Finally, the suspension responses by VSDC and OLS are demonstrated in Figures 1 and 2.

(a)

(b)

(c)

(d)

Figure 1 

States trajectories of VSDC and OLS under sinusoidal disturbance.

(a)

(b)

(c)

Figure 2 

Control inputs and sliding mode surface under sinusoidal disturbance.

From Figure 1, it can be seen that the states are stabilized and become stable by using VSDC and the magnitudes of the suspension response are obviously reduced comparing with those of the OLS. The reason can be concluded from Figure 2, that by using VSDC, the states were controlled towards the OSMS; while being forced onto the OSMS, the equivalent control 𝑢_𝑒 drove them stabilized and hold good dynamical characteristics.

Case 2. Consider attenuated disturbance described by exosystem (2) with 𝐺=00100001−0.050−0.400−0.340−0.3, (88) which yields that the eigenvalues of 𝐺 are −0.2±0.1𝑗,  −0.15±0.5635𝑗. Take performance index (12) with 𝑄=𝑅=𝐼₂. Through the same way, from (14) and (15), it yields the values of matrix 𝑃₁, 𝑃₂. And from (32) and (33), 𝑔^(𝑀) and 𝑥^(𝑀) are solvable. Then, the OSMS (49) and VSDC (77) are obtained taking 𝑘=1 and 𝜀=0.05 in trending law (52). Hence, as displayed in Figures 3 and 4, the suspension responses by VSDC and OLS are illustrated.

(a)

(b)

(c)

(d)

Figure 3 

States trajectories of VSDC and OLS under attenuated disturbance.

(a)

(b)

(c)

Figure 4 

Control inputs and sliding mode surface under attenuated disturbance.

It is shown that the state variables are reduced by VSDC comparing with OLS, which are controlled asymptotically stable at about 30 sec. Clearly, the VSDC rejected the disturbance which is faced by the system and makes the state variable of OLS vibrate.

Summarily, comparing the simulation results between Figures 1 and 2 and Figures 3 and 4, we can see that, although the states are reduced by VSDC and lower than those of OLS when facing sinusoidal disturbance, it still vibrates a little; whereas when facing attenuated disturbance, they converge to zero gradually. The reason lies in the main different feature of these two kinds of signals. Notice that in VSDC (55), there exists the feedforward compensation term described as −(𝑅𝐵)⁻¹[𝑀^𝑇₁₂𝑃₂(𝑡)𝐺+𝑘𝑀^𝑇₁₂𝑃₂(𝑡)+𝑀^𝑇₁₂𝑃̇₂(𝑡)+𝑀^𝑇₁₂𝑃₁(𝑡)𝐷𝐹]𝑤(𝑡), which causes the closed-loop system to still be affected by disturbance 𝑤(𝑡). For this reason, facing sinusoidal disturbance, the VSDC is able to compensate the effect generated by disturbance so that the state amplitudes are reduced but is not able to eliminate it entirely; while when facing attenuated disturbance, the disturbance effect can be entirely eliminated and state variables approach zero. In addition, the sinusoidal and random disturbances can be entirely eliminated by using dynamic internal-model compensator, which can be referred to some bodies of literature, for example, [35] or our previous studies [28, 36].

6. Conclusions

In this paper, the variable structure disturbance rejection strategy for nonlinear systems with uncertainty and time delay is presented. The original system has been transformed into the delay-free one as well as the optimal problem has been reduced to that of lower dimensions. The designed OSMS and VSDC in both finite-time and infinite-time horizon are proposed, where the disturbance is reduced and the uncertainty and the nonlinearity are entirely compensated. Numerical simulations have demonstrated the effectiveness of the designed control law as well as the simplicity of the proposed approach. It is verified that this proposed approach is easy to implement and of reduced complexity.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the National Natural Science Foundations (61364012, 11361076), the China Scholarship Council Foundation (201208535084), the Natural Science Foundation of Yunnan Province (2011FZ169), the Key Program of the Scientific Research Foundation of the Education Department of Yunnan Province (2013Z034), and the Open Fund of Key Laboratory in Software Engineering of Yunnan Province (2011SE15). The authors would like to thank Editor Valery Y. Glizer and the Reviewers for their careful review and significant comments improving the presentation of this paper.