High-Order Stochastic Adaptive Controller Design with Application to Mechanical System

Abstract

The main purpose of this paper is to apply stochastic adaptive controller design to mechanical system. Firstly, by a series of coordinate transformations, the mechanical system can be transformed to a class of special high-order stochastic nonlinear system, based on which, a more general mathematical model is considered, and the smooth state-feedback controller is designed. At last, the simulation for the mechanical system is given to show the effectiveness of the design scheme.

1. Introduction

In recent years, the study for deterministic high-order nonlinear systems has achieved remarkable development, see, for example, [1–3] and references herein. Inspired by these interesting and important results, it is natural to generalize their results to the following stochastic high-order nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input:𝑑𝑧=𝑓₀𝑧,𝑥₁𝑑𝑡+𝑔^𝑇₀𝑧,𝑥₁𝑑𝜔,𝑑𝑥_𝑖=𝑑_𝑖𝑥_𝑖,𝑡𝑥^𝑝𝑖_𝑖+1+𝑓_𝑖𝑧,𝑥_𝑖𝑑𝑡+𝑔^𝑇_𝑖𝑧,𝑥_𝑖𝑑𝜔,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑑_𝑛𝑥_𝑛,𝑡𝑢^𝑝𝑛+𝑓_𝑛𝑧,𝑥_𝑛𝑑𝑡+𝑔^𝑇_𝑛𝑧,𝑥_𝑛𝑑𝜔, (1.1) where (𝑧^𝑇,𝑥₁,…,𝑥_𝑛)^𝑇∈ℝ^𝑚+𝑛, and 𝑢∈ℝ are the measurable state and the input of system, respectively, 𝑥_𝑖=(𝑥₁,…,𝑥_𝑖)^𝑇, 𝑖=1,…,𝑛, 𝑧=(𝑧₁,…,𝑧_𝑚)^𝑇∈ℝ^𝑚 is referred to as the state of the stochastic inverse dynamics, 𝜔 is an 𝑟-dimensional standard Wiener process defined on a probability space (Ω,ℱ,𝑃) with Ω being a sample space, ℱ being a 𝜎-algebra, and 𝑃 being a probability measure, 𝑝_𝑖≥1, 𝑖=1,…,𝑛 are odd integers, and the functions 𝑓_𝑖(⋅) and 𝑔_𝑖(⋅), 𝑖=0,1,…,𝑛 are assumed to be smooth, vanishing at the origin (𝑧^𝑇,𝑥^𝑇_𝑛)=(0_1×𝑚,0_1×𝑛).

For (1.1) with 𝑑_𝑖(⋅)=1, Xie and Tian in [4] considered the state-feedback stabilization problem for the first time. After considering the stabilization of high-order stochastic nonlinear systems, [5] further addressed the problem of state-feedback inverse optimal stabilization in probability, that is, the designed stabilizing backstepping controller is also optimal with respect to meaningful cost functionals. When 𝑑_𝑖(⋅)≠1, [6] designed an adaptive state-feedback controller for a class of stochastic nonlinear uncertain systems with 0<𝜆_𝑖≤𝑑_𝑖(⋅)≤𝜇_𝑖≤𝜇, and [7] designed a smooth adaptive state-feedback controller for high-order stochastic systems with 𝜆_𝑖(𝑥_𝑖)≤𝑑_𝑖(⋅)≤𝜇_𝑖(𝑥_𝑖,𝜃) by using the parameter separation lemma and some flexible algebraic techniques. Recently, more excellent results [8–28] were achieved by Xie and his group.

However, all these theoretical results mentioned above are demonstrated only by some numerical simulation examples. Since many practical application systems in aerospace industry, industrial process control, and so forth, can be described by (or transformed to) stochastic high-order nonlinear systems, so it is very necessary to apply the control schemes to these systems. Based on this reason, we consider a practical example of mechanical movement in this paper. By a series of coordinate transformations, the mechanical system can be transformed to a high-order stochastic nonlinear system, based on which, we consider a more general mathematical model and design a smooth state-feedback control law. At last, the simulation for the mechanical system is given to show the effectiveness of the design scheme.

This paper is organized as follows. Section 2 gives a practical example. Section 3 provides preliminary knowledge and presents problem statement. Controller design and stability analysis are given in Section 4. The simulation for the practical example is provided to demonstrate the control scheme in Section 5. Section 6 gives some concluding remarks.

2. A Practical Example

Let us consider the following mechanical system which consists of two masses 𝑚₁ and 𝑚₂ on a horizontal smooth surface as shown in Figure 1. The mass 𝑚₁ is interconnected to the wall by a linear spring and to the mass 𝑚₂ by a nonlinear spring which has cubic force-deformation relation. Let 𝑥 be the displacement of mass 𝑚₁ and 𝑦 the displacement of mass 𝑚₂ such that at 𝑥=0 and 𝑦=0, that is, the springs are unstretched. A control force 𝑢 acts on 𝑚₁.

Figure 1 

A mechanical system.

Where the units of 𝑚₁, 𝑥, and 𝑢 are “kg”, “m”, and “N”, respectively, and 𝑦₁=𝑥−𝑦. The equations of motion for the system are described by𝑘𝑦̈=¹𝑚₂𝑥−𝑦³,𝑘𝑥̈=−𝑚₁𝑘𝑥−¹𝑚₁𝑥−𝑦³+𝑢𝑚₁, (2.1) where 𝑘 and 𝑘₁ are the spring coefficients, and their units are “N/m” and “N/m³”, respectively.

Introducing the smooth change of coordinates𝑥₁=𝑦,𝑥₂=𝑥̇₁𝑥=𝑦,₃̇=𝑥−𝑦³𝑘₁𝑚₁,𝑥₄=𝑥̇₃, (2.2) one gets𝑦=𝑥₁,𝑦̇=𝑥₂,𝑥𝑥=³₃𝑘₁/𝑚₁+𝑦,𝑥₄=₄𝑥₃𝑘₁/𝑚₁+𝑥₂. (2.3)

The linear spring constant 𝑘 has a specific nominal value 𝑘₀=1.5 which is considered uncertain, and 𝑘∈[0.75,2.25]. Let Δ(𝑡)=𝑘(𝑡)−𝑘₀. For all 𝑡≥0, Δ(𝑡) is the Gaussian white noise process with 𝐸Δ(𝑡)=0 and 𝐸Δ²(𝑡)=𝜎². We can choose the value of parameter 𝜎 such that 𝑘(𝑡) obeys the bound 0.75≤𝑘≤2.25 with a sufficiently high probability. This model of spring rate variations leads to an uncertain stochastic system. By (2.2), one chooses the smooth state-feedback control𝑢=𝑚₁₃𝑣𝑘₁/𝑚₁+𝑚₁+𝑚₂𝑚₂𝑚₁𝑥³₃, (2.4) which together with the property of Δ(𝑡) leads to𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=𝑚₁𝑚₂𝑥³₃𝑑𝑡,𝑑𝑥₃=𝑥₄𝑑𝑡,𝑑𝑥₄=𝑣𝑑𝑡+𝑘₀𝑓(𝑥)𝑑𝑡+𝜎𝑓(𝑥)𝑑𝜔,𝑦=𝑥₁, (2.5) where 𝑓(𝑥)=−𝑥₃/𝑚₁−³(𝑘₁/𝑚₁)(𝑥₁/𝑚₁), and 𝜔 is standard Wiener process.

This stochastic high-order nonlinear systems can be generalized to a more general system which will be given in the following section.

3. Preliminary Knowledge and Problem Statement

3.1. Preliminary Knowledge

In this section, we will introduce the concept of input-to-state practical stability (ISpS) in probability.

Consider the following stochastic nonlinear system𝑑𝑥=𝑓(𝑥,𝑢)𝑑𝑡+𝑔^𝑇(𝑥,𝑢)𝑑𝜔,𝑥(0)=𝑥₀∈ℝ^𝑛, (3.1) where 𝑥∈ℝ^𝑛, 𝑢∈ℝ^𝑚 are the state and the input of system, respectively. The Borel measurable functions 𝑓∶ℝ^𝑛+𝑚→ℝ^𝑛 and 𝑔∶ℝ^𝑛+𝑚→ℝ^𝑛×𝑟 are locally Lipschitz in 𝑥, and 𝜔∈ℝ^𝑟 is an 𝑟-dimensional independent standard Wiener process defined on the complete probability space (Ω,ℱ,𝑃).

The following definitions and lemmas will be used throughout the paper.

Definition 3.1 (see [29]). For any given 𝑉(𝑥)∈𝒞², associated with stochastic system (3.1), the differential operator ℒ is defined as follows:(𝑥)=(𝑥)ℒ𝑉𝜕𝑉𝑓()+1𝜕𝑥𝑥,𝑢2𝑔()𝜕Tr𝑥,𝑢²𝑉(𝑥)𝜕𝑥²𝑔^𝑇(𝑥,𝑢). (3.2)

Definition 3.2 (see [30]). The stochastic system (3.1) is input-to-state practically stable (ISpS) in probability if for any 𝜀>0, there exist a class 𝒦ℒ-function 𝛽(⋅), a class 𝒦_∞-function 𝛾(⋅), and a constant 𝑑₀ such that 𝑃|𝑥(𝑡)|<𝛽𝑥₀,𝑡+𝛾𝑢_𝑡+𝑑₀≥1−𝜀,𝑥₀∈ℝ^𝑛⧵{0}. (3.3)

Lemma 3.3 (see [30]). For system (3.1), if there exist a 𝒞² function 𝑉(𝑥), class 𝒦_∞ functions 𝛼₁, 𝛼₂, 𝜒, a class 𝒦 function 𝛼, and a constant 𝑑 such that 𝛼₁(|𝑥|)≤𝑉(𝑥)≤𝛼₂𝑥,(𝑥)𝑥𝑢+(||)ℒ𝑉≤−𝛼(||)+𝜒(||)𝑑, (3.4)(3.5) then(1)There exists an almost surely unique solution on [0,∞);(2)The system (3.1) is ISpS in probability.

Lemma 3.4 (see [6]). Let 𝑥 and 𝑦 be real variables. Then, for any positive integers 𝑚, 𝑛 and any nonnegative smooth function 𝑏(⋅), the following inequality holds:𝑥^𝑚𝑦^𝑛≤𝑚𝑏(⋅𝑥|𝑚+𝑛)|^𝑚+𝑛+𝑛𝑏(⋅)𝑚+𝑛^−𝑚/𝑛𝑦^𝑚+𝑛. (3.6)

Lemma 3.5 (see [2]). For real variables 𝑥≥0, 𝑦>0, and real number 𝑚≥1, the following inequality holds: 𝑥𝑥≤𝑦+𝑚^𝑚𝑚−1𝑦^𝑚−1. (3.7)

3.2. Problem Statement

From (2.5), we introduce a more general class of stochastic nonlinear systems as follows:𝑑𝑥_𝑖=𝑑_𝑖(𝑥)𝑥^𝑝𝑖_𝑖+1𝑑𝑡+𝑓_𝑖𝑥_𝑖+1𝑑𝑡+𝑔_𝑖𝑥_𝑖^𝑇𝑑𝜔,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑑_𝑛(𝑥)𝑢^𝑝𝑛𝑑𝑡+𝑓_𝑛𝑥_𝑛𝑑𝑡+𝑔_𝑛𝑥_𝑛^𝑇𝑑𝜔,𝑦=𝑥₁, (3.8) where 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈ℝ^𝑛, 𝑢,𝑦∈ℝ are the state, the input, and the measurable output of system, respectively, 𝑥_𝑖=(𝑥₁,…,𝑥_𝑖)^𝑇, 𝑝_𝑖, 𝑖=1,…,𝑛, are positive odd integers, 𝑓_𝑖(⋅)∶ℝ^𝑖+1→ℝ and 𝑔_𝑖(⋅)∶ℝ^𝑖→ℝ×ℝ^𝑟 are smooth functions with 𝑓_𝑖(0)=0 and 𝑔_𝑖(0)=0,𝑑_𝑖(𝑥) is unknown control coefficient with known sign, and 𝜔 is an 𝑟-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,𝑃).

The following assumptions are made on system (3.8).

A1: for each 𝑑_𝑖(𝑥), there exist unknown constant 𝜃^′>0 and known nonnegative smooth functions 𝑏_𝑖(𝑥_𝑖) and 𝑏_𝑖(𝑥_𝑖+1) such that0≤𝑏_𝑖𝑥_𝑖≤𝑑_𝑖(𝑥)≤𝜃^′𝑏_𝑖𝑥_𝑖+1. (3.9)

A2: for functions 𝑓_𝑖(⋅), 𝑔_𝑖(⋅), 𝑖=1,2,…,𝑛, there exist known nonnegative smooth functions 𝜑_𝑖𝑗(𝑥_𝑖) and 𝜓_𝑖(𝑥_𝑖) such that𝑓_𝑖𝑥_𝑖+1≤^𝑝𝑖−1_𝑗=0𝑥_𝑖+1^𝑗𝜑_𝑖𝑗𝑥_𝑖,𝑔_𝑖𝑥_𝑖≤𝑥₁^{(𝑝𝑖+1)/2}+⋯+𝑥_𝑖^{(𝑝𝑖+1)/2}𝜓_𝑖𝑥_𝑖. (3.10)

A3: the reference signal 𝑦_𝑟 and its derivative 𝑦_𝑟̇ are bounded.

The objective of this paper is to design an adaptive controller such that the closed-loop system is ISpS in probability and the tracking error 𝜉₁=𝑦−𝑦_𝑟 can be regulated to a neighborhood of the origin with radius as small as possible.

4. Controller Design and Stability Analysis

With the aid of Lemmas 3.3–3.5, we are ready to present the main results of this paper. In this section, we show that under A1–A3, it is possible to construct a globally stabilizing, state-feedback smooth controller for system (3.8). Introduce the odd positive integer 𝑝=max_{𝑖=1,…,𝑛}{𝑝_𝑖}, and the following coordinate change𝜉₁=𝑥₁−𝑦_𝑟,𝜉_𝑖=𝑥_𝑖−𝑥^∗_𝑖𝑥_𝑖−1,𝑦_𝑟,𝜃,𝑖=2,…,𝑛, (4.1) where 𝑥^∗_𝑖(𝑥_𝑖−1,𝑦_𝑟,𝜃), 𝑖=2,…,𝑛, are virtual smooth controllers to be designed later, 𝜃∶=max{𝜃^′,𝜃^{′(𝑝+3)/(𝑝−𝑝𝑖+3)}}, and 𝜃 denotes the estimate of 𝜃. Then, according to Ito differentiation rule, one has𝑑𝜉₁=𝑑₁𝑥^𝑝1₂𝑑𝑡+𝑓₁𝑑𝑡+𝑔^𝑇₁𝑑𝜔−𝑦_𝑟̇𝑑𝑡,𝑑𝜉_𝑖=𝑑_𝑖𝑥^𝑝𝑖_𝑖+1𝑑𝑡+𝑓_𝑖𝑑𝑡−^𝑖−1_𝑘=1𝜕𝑥^∗_𝑖𝜕𝑥_𝑘𝑥^𝑝𝑘_𝑘+1+𝑓_𝑘1𝑑𝑡−2^𝑖−1_𝑗,𝑘=1𝜕²𝑥^∗_𝑖𝜕𝑥_𝑗𝜕𝑥_𝑘𝑔^𝑇_𝑗𝑔_𝑘−𝑑𝑡𝜕𝑥^∗_𝑖𝜕𝜃𝜃̇𝑑𝑡−𝜕𝑥^∗_𝑖𝜕𝑦_𝑟𝑦_𝑟̇𝑑𝑡+𝑔^𝑇_𝑖−^𝑖−1_𝑘=1𝜕𝑥^∗_𝑖𝜕𝑥_𝑘𝑔^𝑇_𝑘𝑑𝜔,𝑖=2,…,𝑛−1,𝑑𝜉_𝑛=𝑑_𝑛𝑢^𝑝𝑛𝑑𝑡+𝑓_𝑛𝑑𝑡−^𝑛−1_𝑘=1𝜕𝑥^∗_𝑛𝜕𝑥_𝑘𝑥^𝑝𝑘_𝑘+1+𝑓_𝑘1𝑑𝑡−2^𝑛−1_𝑗,𝑘=1𝜕²𝑥^∗_𝑛𝜕𝑥_𝑗𝜕𝑥_𝑘𝑔^𝑇_𝑗𝑔_𝑘−𝑑𝑡𝜕𝑥^∗_𝑛𝜕𝜃𝜃̇𝑑𝑡−𝜕𝑥^∗_𝑛𝜕𝑦_𝑟𝑦_𝑟̇𝑑𝑡+𝑔^𝑇_𝑛−^𝑛−1_𝑘=1𝜕𝑥^∗_𝑛𝜕𝑥_𝑘𝑔^𝑇_𝑘𝑑𝜔. (4.2) Let 𝐺^𝑇_𝑖=𝑔^𝑇_𝑖−∑^𝑖−1_𝑘=1(𝜕𝑥^∗_𝑖/𝜕𝑥_𝑘)𝑔^𝑇_𝑘, 𝑖=2,…,𝑛. Next, we design the controller step by step by backstepping.

Step 1. Consider the 1st Lyapunov candidate function 𝑉₁𝜉₁,𝜃=1𝑝−𝑝₁𝜉+4^{𝑝−𝑝11+4}+12𝜃², (4.3) where 𝜃=𝜃−𝜃 is the parameter estimation error. In view of (3.2), (4.1), and (4.2), one has ℒ𝑉₁𝜉₁,𝜃=𝜉^𝑝−𝑝1₁⁺³𝑑₁(𝑥)𝑥^𝑝1₂+𝑓₁𝑥₂−𝑦_𝑟̇+12Tr𝑔₁𝑥₁𝑝−𝑝₁+3𝜉^𝑝−𝑝1₁⁺²𝑔^𝑇₁𝑥₁−𝜃𝜃̇≤𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^𝑝1₂+𝜉₁^{𝑝−𝑝1+3}𝑓₁𝑥₂−𝑦_𝑟̇+12𝑝−𝑝₁+3𝜉^𝑝−𝑝1₁⁺²𝑔₁𝑥₁²−𝜃𝜃̇. (4.4) By Lemma 3.4 and A2, there exist nonnegative smooth functions 𝜑₁(𝑥₁) and 𝜓₁(𝑥₁) such that 𝑓₁𝑥₂≤^𝑝1−1_𝑗=0𝑥₂^𝑗𝜑_1𝑗𝑥₁=^𝑝1−1_𝑗=0𝑥₂^𝑗𝜑^{1/(𝑝1−𝑗)}_1𝑗𝑥₁^𝑝1−𝑗≤^𝑝1−1_𝑗=0𝑗𝑝₁1𝑏2𝑗¹𝑥₁𝑥₂^𝑝1+𝑝₁−𝑗𝑝₁2𝑗𝑏₁𝑥₁^{𝑗/(𝑝1−𝑗)}𝜑^{𝑝1/(𝑝1−𝑗)}_1𝑗𝑥₁≤𝑏₁𝑥₁2𝑥₂^𝑝1+𝜑₁𝑥₁,𝑔₁𝑥₁≤𝜉₁^(𝑝1+1)/2𝜓₁𝑥₁, (4.5) which together with the boundedness of 𝑦_𝑟̇ imply that 𝑓₁−𝑦_𝑟̇≤𝑏₁𝑥₁2𝑥₂^𝑝1+𝜑^′₁𝑥₁,𝑦_𝑟, (4.6) where 𝜑^′₁(𝑥₁,𝑦_𝑟) is a nonnegative smooth function, 𝜓₁(𝑥₁)=𝜓₁(𝑥₁). Then, for any real number 𝛿₁>0, choosing 𝑎=|𝜉^𝑝−𝑝1₁⁺³|𝜑^′₁(𝑥₁,𝑦_𝑟), 𝑏=𝛿₁, 𝑚=(𝑝+3)/(𝑝−𝑝₁+3), by Lemma 3.5, there is a smooth function 𝜙₁₁(𝑥₁,𝑦_𝑟) such that 𝜉₁^{𝑝−𝑝1+3}𝑓₁−𝑦_𝑟̇≤𝜉₁^{𝑝−𝑝1+3}𝑏₁𝑥₁2𝑥₂^𝑝1+𝜑^′₁𝑥₁,𝑦_𝑟≤𝜉₁^{𝑝−𝑝1+3}𝑏₁𝑥₁2𝑥₂^𝑝1+𝛿₁+𝑝−𝑝₁+3𝜉₁^{𝑝−𝑝1+3}𝜑^′₁𝑥₁,𝑦_𝑟𝑝+3^{(𝑝+3)/(𝑝−𝑝1+3)}×𝑝₁𝛿₁𝑝−𝑝₁+3^{𝑝1/(𝑝−𝑝1+3)}=𝜉₁^{𝑝−𝑝1+3}𝑏₁𝑥₁2𝑥₂^𝑝1+𝜉₁^𝑝+3𝜙₁₁𝑥₁,𝑦_𝑟+𝛿₁, (4.7) where 𝜙₁₁(𝑥₁,𝑦_𝑟)=((𝑝−𝑝₁+3)𝜑^′₁(𝑥₁,𝑦_𝑟)/(𝑝+3))^{(𝑝+3)/(𝑝−𝑝1+3)}(𝑝₁/𝛿₁(𝑝−𝑝₁+3))^{𝑝1/(𝑝−𝑝1+3)}. Substituting (4.5) and (4.7) into (4.4), and adding and subtracting (𝑏₁(𝑥₁)/2)𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂ on the right-hand side of (4.4), we have ℒ𝑉₁≤𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^𝑝1₂+𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^𝑝1₂+𝜉₁^𝑝+3𝜙₁₁𝑥₁,𝑦_𝑟+𝛿₁+𝑝−𝑝₁+32𝜉^𝑝−𝑝1₁⁺²𝜉^𝑝1₁⁺¹𝜓²₁𝑥₁−𝜃𝜃̇=𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^𝑝1₂+𝑏₁𝑥₁2𝜉₁^{𝑝−𝑝1+3}𝑥₂^𝑝1+𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂−𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂+𝜉₁^𝑝+3𝜙₁₁𝑥₁,𝑦_𝑟+𝜉₁^𝑝+3𝜙₁₂𝑥₁+𝛿₁−𝜃𝜃̇, (4.8) where 𝜙₁₂(𝑥₁)=((𝑝−𝑝₁+3)/2)𝜉^𝑝−𝑝1₁𝜓²₁(𝑥₁). Suppose the virtual smooth controller 𝑥^∗₂=−𝜉₁𝛽₁(𝑥₁,𝑦_𝑟,𝜃) with 𝛽₁(𝑥₁,𝑦_𝑟,𝜃)>0, which together with A1 lead to 0≤−𝑏₁𝑥₁𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂≤−𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂,−𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂+𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂≤−𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂,−𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂≤−𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂−𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂. (4.9) Substituting (4.9) into (4.8), one can obtain ℒ𝑉₁≤𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^𝑝1₂+𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^𝑝1₂+𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂−𝑑₁(𝑥)𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂−𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂+𝛿₁−𝜃𝜃̇−𝑐₁𝜃𝜉₁^𝑝+3+𝑐₁𝜃𝜉₁^𝑝+3+𝜉₁^𝑝+3𝜙₁₁𝑥₁,𝑦_𝑟+𝜉₁^𝑝+3𝜙₁₂𝑥₁≤−𝑐₁𝜃𝜉₁^𝑝+3+𝜃𝑏₁𝑥₂+𝑏₁𝑥₁2𝜉₁^{𝑝−𝑝1+3}𝑥^𝑝1₂−𝑥^∗𝑝1₂+𝑏₁𝑥₁2𝜉^𝑝−𝑝1₁⁺³𝑥^∗𝑝1₂+𝑐₁𝜃𝜉₁^𝑝+3+𝜉₁^𝑝+3𝜙₁₁𝑥₁,𝑦_𝑟+𝜉₁^𝑝+3𝜙₁₂𝑥₁+𝛿₁+𝜃𝜏₁−𝜃̇, (4.10) where 𝜏₁=𝑐₁𝜉₁^𝑝+3 is a nonnegative smooth function. Choose 𝑥^∗₂ as follows: 𝑥^∗₂𝑥₁,𝑦_𝑟,𝜃=−𝜉₁𝛽₁𝑥₁,𝑦_𝑟,𝜃,𝛽₁𝑥₁,𝑦_𝑟,𝜃=2𝑏₁𝑥₁𝑐₁+𝜙₁₁𝑥₁,𝑦_𝑟+𝜙₁₂𝑥₁+𝑐₁1+𝜃^21/𝑝1, (4.11) where 𝛽₁(𝑥₁,𝑦_𝑟,𝜃)≥0 is a smooth function. Then, ℒ𝑉₁≤−𝑐₁𝜉₁^𝑝+3−𝑐₁𝜃𝜉₁^𝑝+3+𝜃𝑏₁𝑥₂+𝑏₁𝑥₁2𝜉₁^{𝑝−𝑝1+3}𝑥^𝑝1₂−𝑥^∗𝑝1₂+𝛿₁+𝜃𝜏₁−𝜃̇. (4.12)

Step i. 2≤𝑖≤𝑛: Assume that at Step 𝑖−1, there exists a smooth state-feedback virtual control 𝑥^∗_𝑖𝑥_𝑖−1,𝑦_𝑟,𝜃=−𝛽_𝑖−1𝑥_𝑖−1,𝑦_𝑟,𝜃𝜉_𝑖−1, (4.13) such that ℒ𝑉_𝑖−1≤−^𝑖−1_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜉_𝑗^𝑝+3−^𝑖−1_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜃𝜉_𝑗^𝑝+3+𝜃𝑏_𝑖−1+𝑏_𝑖−12𝜉_𝑖−1^{𝑝−𝑝𝑖−1+3}𝑥^𝑝𝑖−1_𝑖−𝑥^{∗𝑝𝑖−1}_𝑖+^𝑖−1_𝑗=1𝛿_𝑗+𝜃+^𝑖−1_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃𝜏_𝑖−1−𝜃̇, (4.14) where 𝛽_𝑖−1>0 is a smooth function, and 𝑉_𝑖−1=(1/4)∑^𝑖−1_𝑘=1𝜉^{𝑝−𝑝𝑘}_𝑘⁺⁴+(1/2)𝜃². We will prove that (4.14) still holds for Step 𝑖.
Define the 𝑖th Lyapunov candidate function 𝑉_𝑖=𝑉_𝑖−1+14𝜉^{𝑝−𝑝𝑖}_𝑖⁺⁴. (4.15) From (4.2) and (4.14), it follows that ℒ𝑉_𝑖≤−^𝑖−1_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜉_𝑗^𝑝+3−^𝑖−1_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜃𝜉_𝑗^𝑝+3+𝜃𝑏_𝑖−1+𝑏_𝑖−12𝜉_𝑖−1^{𝑝−𝑝𝑖−1+3}𝑥^𝑝𝑖−1_𝑖−𝑥^{∗𝑝𝑖−1}_𝑖+^𝑖−1_𝑗=1𝛿_𝑗+𝜃+^𝑖−1_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃𝜏_𝑖−1−𝜃̇+𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑑_𝑖(𝑥)𝑥^𝑝𝑖_𝑖+1+𝜉^{𝑝−𝑝𝑖}_𝑖⁺³×𝑓_𝑖−^𝑖−1_𝑘=1𝜕𝑥^∗_𝑖𝜕𝑥_𝑘𝑑_𝑘(𝑥)𝑥^𝑝𝑘_𝑘+𝑓_𝑘−12^𝑖−1_𝑗,𝑘=1𝜕²𝑥^∗_𝑖𝜕𝑥_𝑗𝜕𝑥_𝑘𝑔^𝑇_𝑗𝑔_𝑘−𝜕𝑥^∗_𝑖𝜕𝑦_𝑟𝑦_𝑟̇−𝜕𝑥^∗_𝑖𝜕𝜃𝜃̇+12Tr𝐺_𝑖𝑝−𝑝_𝑖+3𝜉^{𝑝−𝑝𝑖}_𝑖⁺²𝐺^𝑇_𝑖. (4.16) By A2 and Lemma 3.4, there is a smooth nonnegative function 𝜑_𝑖(𝑥_𝑖) such that 𝑓_𝑖𝑥_𝑖+1≤^𝑝𝑖−1_𝑗=0𝑥_𝑖+1^𝑗𝜑_𝑖𝑗𝑥_𝑖≤𝑏_𝑖𝑥_𝑖𝑥^𝑝𝑖_𝑖+12+𝜑_𝑖𝑥_𝑖, (4.17) then, 𝑓_𝑖−^𝑖−1_𝑘=1𝜕𝑥^∗_𝑖𝜕𝑥_𝑘𝑓_𝑘−12^𝑖−1_𝑗,𝑘=1𝜕²𝑥^∗_𝑖𝜕𝑥_𝑗𝜕𝑥_𝑘𝑔^𝑇_𝑗𝑔_𝑘−𝜕𝑥^∗_𝑖𝜕𝑦_𝑟𝑦_𝑟̇≤𝑏_𝑖𝑥_𝑖𝑥^𝑝𝑖_𝑖+12+𝜑_𝑖𝑥_𝑖,𝑦_𝑟,𝜃, (4.18) where 𝜑_𝑖(𝑥_𝑖,𝑦_𝑟,𝜃) is a smooth function. By A2, (4.1) and (4.13), there exists a nonnegative smooth function 𝜓^′_𝑖(𝑥_𝑖,𝑦_𝑟,𝜃) such that 𝐺_𝑖𝑥_𝑖≤𝜉₁^{(𝑝𝑖+1)/2}+⋯+𝜉_𝑖^{(𝑝𝑖+1)/2}𝜓^′_𝑖𝑥_𝑖,𝑦_𝑟,𝜃. (4.19) By (4.13), we have 𝜃𝑏_𝑖−1𝑥_𝑖+𝑏_𝑖−1𝑥_𝑖−12𝜉_𝑖−1^{𝑝−𝑝𝑖−1+3}𝑥^𝑝𝑖−1_𝑖−𝑥^{∗𝑝𝑖−1}_𝑖=𝜃𝑏_𝑖−1𝑥_𝑖+𝑏_𝑖−1𝑥_𝑖−12^𝑝𝑖−1_𝑘=1𝐶^𝑘_𝑝𝑖−1𝜉_𝑖^𝑘𝜉₁^{𝑝−𝑘+3}𝛽^𝑝𝑖−1₁^−𝑘≤^𝑖−1_𝑘=1𝑐_𝑖𝑘1𝜉_𝑘^𝑝+3+^𝑖−1_𝑘=1𝑐_𝑖𝑘𝜃𝜉_𝑘^𝑝+3+𝜑_𝑖1𝑥_𝑖,𝑦_𝑟,𝜃𝜉_𝑖^𝑝+3+𝜃𝜑_𝑖2𝜉_𝑖^𝑝+3, (4.20) where 𝜑_𝑖1(𝑥_𝑖,𝑦_𝑟,𝜃) and 𝜑_𝑖2(𝑥_𝑖,𝑦_𝑟,𝜃) are two smooth functions. From A1, (4.1), and (4.13), it follows that −𝜉^{𝑝−𝑝𝑖}_𝑖^+3𝑖−1_𝑘=1𝜕𝑥^∗_𝑖𝜕𝑥_𝑘𝑑_𝑘(𝑥)𝑥^𝑝𝑘_𝑘≤𝜃^′𝜉_𝑖^{𝑝−𝑝𝑖+3𝑖−1}_𝑘=1𝑏_𝑘−1𝑥_𝑘𝜕𝑥^∗_𝑖𝜕𝑥_𝑘𝜉_𝑘+𝑥^∗_𝑘^𝑝𝑘≤𝜃^{′(𝑝+3)/(𝑝−𝑝𝑖+3)}𝜉_𝑖^𝑝+3𝜑_𝑖3𝑥_𝑖,𝑦_𝑟,𝜃+𝛿_𝑖1≤𝜃𝜉_𝑖^𝑝+3𝜑_𝑖3𝑥_𝑖,𝑦_𝑟,𝜃+𝛿_𝑖1,𝜉_𝑖^{𝑝−𝑝𝑖+3}𝑓_𝑖−^𝑖−1_𝑘=1𝜕𝑥^∗_𝑖𝜕𝑥_𝑘𝑓_𝑘−12^𝑖−1_𝑗,𝑘=1𝜕²𝑥^∗_𝑖𝜕𝑥_𝑗𝜕𝑥_𝑘𝑔^𝑇_𝑗𝑔_𝑘−𝜕𝑥^∗_𝑖𝜕𝑦_𝑟𝑦_𝑟̇≤𝜉_𝑖^{𝑝−𝑝𝑖+3}𝑏_𝑖𝑥_𝑖𝑥_𝑖+1^𝑝𝑖2+𝜑^′_𝑖𝑥_𝑖,𝑦_𝑟,𝜃≤𝑏_𝑖𝑥_𝑖2𝜉_𝑖^{𝑝−𝑝𝑖+3}𝑥_𝑖+1^𝑝𝑖+𝜑_𝑖4𝑥_𝑖,𝑦_𝑟,𝜃𝜉_𝑖^𝑝+3+𝛿_𝑖2, (4.21)(4.22) where 𝜑_𝑖3(𝑥_𝑖,𝑦_𝑟,𝜃) and 𝜑_𝑖4(𝑥_𝑖,𝑦_𝑟,𝜃) are two smooth functions. From (4.19), one can obtain 12Tr𝐺_𝑖𝑝−𝑝_𝑖+3𝜉^{𝑝−𝑝𝑖}_𝑖⁺²𝐺^𝑇_𝑖≤𝑝−𝑝_𝑖+32𝜉^{𝑝−𝑝𝑖}₂⁺²𝜉₁^{(𝑝𝑖+1)/2}+⋯+𝜉_𝑖^{(𝑝𝑖+1)/22}𝜓_𝑖^′2𝑥_𝑖,𝑦_𝑟,𝜃≤^𝑖−1_𝑘=1𝑐_𝑖𝑘2𝜉_𝑘^𝑝+3+𝜑_𝑖5𝑥_𝑖,𝑦_𝑟,𝜃𝜉_𝑖^𝑝+3, (4.23) where 𝜑_𝑖5(𝑥_𝑖,𝑦_𝑟,𝜃) is a smooth nonnegative function. Substituting (4.20)–(4.23) into (4.16), one gets ℒ𝑉_𝑖≤−^𝑖−1_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜉_𝑗^𝑝+3−^𝑖_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜃𝜉_𝑗^𝑝+3+^𝑖−1_𝑗=1𝑐^′_𝑖𝑗𝜉_𝑗^𝑝+3+^𝑖−1_𝑗=1𝛿_𝑗+𝜃^𝑖−1_𝑗=1𝑐_𝑖𝑗𝜉_𝑗^𝑝+3+ℎ^′_𝑖1𝜉_𝑖^𝑝+3+𝜃ℎ_𝑖2𝜉_𝑖^𝑝+3−𝑐_𝑖𝜃𝜉_𝑖^𝑝+3+𝑐_𝑖𝜃𝜉_𝑖^𝑝+3+𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑑_𝑖(𝑥)𝑥^𝑝𝑖_𝑖+1+𝑏_𝑖𝑥_𝑖2𝜉_𝑖^{𝑝−𝑝𝑖+3}𝑥_𝑖+1^𝑝𝑖+𝑏_𝑖𝑥_𝑖2𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖₃−𝑏_𝑖𝑥_𝑖2𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖₃+𝜃+^𝑖−1_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃𝜏_𝑖−1−𝜃̇−𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝜕𝑥^∗_𝑖𝜕𝜃𝜃̇, (4.24) where 𝑐^′_𝑖𝑗=𝑐_𝑖𝑗1+𝑐_𝑖𝑗2ℎ,𝑗=1,…,𝑖−1,_′𝑖1=𝜑_𝑖1+𝜑_𝑖4+𝜑_𝑖5,ℎ_𝑖2=𝜑_𝑖2+𝜑_𝑖3. (4.25) Suppose the virtual smooth controller 𝑥^∗_𝑖+1=−𝜉_𝑖𝛽_𝑖(𝑥_𝑖,𝑦_𝑟,𝜃) with 𝛽_𝑖(𝑥_𝑖,𝑦_𝑟,𝜃)>0, which together with A2 render −𝑏_𝑖𝑥_𝑖2𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖_𝑖+1≤−𝑑_𝑖(𝑥)𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖_𝑖+1−𝑏_𝑖𝑥_𝑖2𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖_𝑖+1. (4.26) Substituting (4.26) into (4.24) leads to ℒ𝑉_𝑖≤−^𝑖−1_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜉_𝑗^𝑝+3−^𝑖_𝑗=1𝑐_𝑗−^𝑖_𝑘=𝑗+1𝑐_𝑘𝑗𝜃𝜉_𝑗^𝑝+3+^𝑖−1_𝑗=1𝑐^′_𝑖𝑗𝜉_𝑗^𝑝+3+ℎ^′_𝑖1𝜉_𝑖^𝑝+3+𝜃+𝜃ℎ_𝑖2𝜉_𝑖^𝑝+3+𝑐_𝑖𝜃+𝜃𝜉_𝑖^𝑝+3+𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑑_𝑖(𝑥)𝑥^𝑝𝑖_𝑖+1+𝑏_𝑖𝑥_𝑖2𝜉_𝑖^{𝑝−𝑝𝑖+3}𝑥_𝑖+1^𝑝𝑖+𝑏_𝑖𝑥_𝑖2𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖₃−𝑑_𝑖(𝑥)𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝2_𝑖+1−𝑏_𝑖𝑥_𝑖2𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖_𝑖+1+𝜃+^𝑖−1_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃𝜏_𝑖−1−𝜃̇−𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝜕𝑥^∗_𝑖𝜕𝜃𝜃̇+^𝑖−1_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃ℎ_𝑖2+𝑐_𝑖𝜉_𝑖^𝑝+3−^𝑖−1_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃ℎ_𝑖2+𝑐_𝑖𝜉_𝑖^𝑝+3+𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝜕𝑥^∗_𝑖𝜕𝜃𝜏_𝑖−𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝜕𝑥^∗_𝑖𝜕𝜃𝜏_𝑖, (4.27) where 𝜏_𝑖=𝜏_𝑖−1+(ℎ_𝑖2+𝑐_𝑖)𝜉_𝑖^𝑝+3. For (4.27), we have −^𝑖−1_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃ℎ_𝑖2+𝑐_𝑖𝜉_𝑖^𝑝+3≤𝜑_𝑖6𝑥_𝑖,𝑦_𝑟,𝜃𝜉_𝑖^𝑝+3,−𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝜕𝑥^∗_𝑖𝜕𝜃𝜏_𝑖≤∑^𝑖−1_𝑘=1𝑐_𝑖𝑘3𝜉_𝑘^𝑝+3+𝜑_𝑖7𝑥_𝑖,𝑦_𝑟,𝜃𝜉_𝑖^𝑝+3, (4.28) where 𝑐_𝑖𝑘3 is a design parameter, 𝜑_𝑖6(𝑥_𝑖,𝑦_𝑟,𝜃) and 𝜑_𝑖7(𝑥_𝑖,𝑦_𝑟,𝜃) are the smooth functions. Let 𝑐_𝑖𝑗=𝑐^′_𝑖𝑗+𝑐_𝑖𝑗3, ℎ_𝑖1=ℎ^′_𝑖1+𝜑_𝑖6+𝜑_𝑖7. (4.27) becomes ℒ𝑉_𝑖≤−^𝑖−1_𝑗=1𝑐_𝑗−^𝑖−1_𝑘=𝑗+1𝑐_𝑘𝑗𝜉_𝑗^𝑝+3−^𝑖_𝑗=1𝑐_𝑗−^𝑖_𝑘=𝑗+1𝑐_𝑘𝑗𝜃𝜉_𝑗^𝑝+3+^𝑖−1_𝑗=1𝑐_𝑖𝑗𝜉_𝑗^𝑝+3+ℎ^′_𝑖1𝜉_𝑖^𝑝+3+𝜃ℎ_𝑖2𝜉_𝑖^𝑝+3+𝜃𝑏_𝑖𝑥_𝑖+1+𝑏_𝑖𝑥_𝑖2𝜉_𝑖^{𝑝−𝑝𝑖+3}𝑥^𝑝𝑖_𝑖+1−𝑥^∗𝑝𝑖_𝑖+1+𝑏_𝑖𝑥_𝑖2𝜉^{𝑝−𝑝𝑖}_𝑖⁺³𝑥^∗𝑝𝑖_𝑖+1+𝜃+^𝑖_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃𝜏_𝑖−𝜃̇+^𝑖_𝑗=1𝛿_𝑗≤−^𝑖_𝑗=1𝑐_𝑗−^𝑖_𝑘=𝑗+1𝑐_𝑘𝑗𝜉_𝑗^𝑝+3−^𝑖_𝑗=1𝑐_𝑗−^𝑖_𝑘=𝑗+1𝑐_𝑘𝑗𝜃𝜉_𝑗^𝑝+3+𝜃𝑏_𝑖𝑥_𝑖+1+𝑏_𝑖𝑥_𝑖2𝜉_𝑖^{𝑝−𝑝𝑖+3}𝑥^𝑝𝑖_𝑖+1−𝑥^∗𝑝𝑖_𝑖+1+^𝑖_𝑗=1𝛿_𝑗+𝜃+^𝑖_𝑘=2𝜉^{𝑝−𝑝𝑘}_𝑘⁺³𝜕𝑥^∗_𝑘𝜕𝜃𝜏_𝑖−𝜃̇, (4.29) by choosing 𝑥^∗_𝑖+1𝑥_𝑖,𝑦_𝑟,𝜃=−𝜉_𝑖𝛽_𝑖𝑥_𝑖,𝑦_𝑟,𝜃,𝛽_𝑖𝑥_𝑖,𝑦_𝑟,𝜃=2𝑏_𝑖𝑥_𝑖𝑐_𝑖+ℎ_𝑖1+(ℎ_𝑖2+𝑐_𝑖)1+𝜃^21/𝑝𝑖, (4.30) where 𝛽_𝑖(𝑥_𝑖,𝑦_𝑟,𝜃)≥0 is a smooth function.
Finally, when 𝑖=𝑛,𝑥_𝑛+1=𝑥^∗_𝑛+1=𝑢 is the actual control. By choosing the actual control law and the adaptive law: 𝑢𝑥_𝑛,𝑦_𝑟,𝜃=−𝛽_𝑛𝑥_𝑛,𝑦_𝑟,𝜃𝜉_𝑛,𝜃̇=𝜏_𝑛=^𝑛_𝑘=1𝐻_𝑘2𝜉_𝑘^𝑝+3, (4.31) where 𝛽_𝑛≥0 and 𝐻₁₂,…,𝐻_𝑛2 are smooth functions, one gets ℒ𝑉_𝑛≤−^𝑛_𝑗=1𝑐_𝑗−^𝑛_𝑘=𝑗+1𝑐_𝑘𝑗𝜉_𝑗^𝑝+3−^𝑛_𝑗=1𝑐_𝑗−^𝑛_𝑘=𝑗+1𝑐_𝑘𝑗𝜃𝜉_𝑗^𝑝+3+^𝑛_𝑗=1𝛿_𝑗. (4.32)

Theorem 4.1. If A1–A3 hold for the high-order stochastic nonlinear system (3.8), under the smooth adaptive state-feedback controller (4.32), the closed-loop system is ISpS in probability, and the tracking error 𝜉₁=𝑦−𝑦_𝑟 can be regulated to a neighborhood of the origin in probability with radius as small as possible (Figure 2).

(a)

(b)

Figure 2 

Gives the response of the closed-loop system, from which, the effectiveness of the controller is demonstrated.

Proof. For 𝑉_𝑛=∑^𝑛_𝑖=1(1/4)𝜉^{𝑝−𝑝𝑖}_𝑖⁺⁴+(1/2)𝜃², it is obvious that 𝑉_𝑛 satisfies (3.4). Choosing all the design parameters 𝑐_𝑗 and 𝑐_𝑗 to satisfy 𝑐_𝑗>^𝑛_𝑘=𝑗+1𝑐_𝑘𝑗,𝑐_𝑗>^𝑛_𝑘=𝑗+1𝑐_𝑘𝑗,𝑗=1,…,𝑛, (4.33) such that (3.5) holds, and then using Lemma 3.3, one can prove Theorem 4.1.

5. Simulation

Now, we apply the control scheme to the mechanical system (2.5). Let 𝜉₁=𝑥₁−𝑦_𝑟 be the tracking error, where 𝑦_𝑟=sin𝑡 is a bounded smooth reference signal. For (2.5), 𝑑_𝑖(⋅)=1, and 𝑝=max{1,3}=3.

Choose 𝑉₁(𝜉₁)=(1/(𝑝−𝑝₁+4))𝜉^𝑝−𝑝1₁⁺⁴=𝜉⁶₁/6. Then,ℒ𝑉₁𝜉₁=𝜉⁵₁𝑥₂−𝑦_𝑟̇. (5.1) The smooth virtual controller can be chosen as 𝑥^∗₂(𝑥₁,𝑦_𝑟)=−𝑐₁𝜉₁+𝑦_𝑟̇, which rendersℒ𝑉₁𝜉₁=−𝑐₁𝜉⁶₁+𝜉⁵₁𝑥₂−𝑥^∗₂. (5.2)

Next, defining 𝑉₂(𝜉₁,𝜉₂)=𝑉₁+(1/(𝑝−𝑝₂+4))𝜉^𝑝−𝑝2₂⁺⁴=𝜉⁶₁/6+𝜉⁴₂/4, a direct calculation givesℒ𝑉₂=−𝑐₁𝜉⁶₁+𝜉⁵₁𝜉₂+𝜉³₂𝑥³₃−𝜕𝑥^∗₂𝜕𝑥₁𝑥₂−𝜕𝑥^∗₂𝜕𝑦_𝑟𝑦_𝑟̇=−𝑐₁𝜉⁶₁+𝜉⁵₁𝜉₂+𝜉³₂𝑥³₃−ℎ₂, (5.3) where 𝜉₂=𝑥₂−𝑥^∗₂. By Lemma 3.5, choosing 𝑚=3/2, one can obtain that for any constant 𝛿₂>0,𝜉⁴₂ℎ₂≤𝛿₂+2𝜉⁴₂ℎ₂3^3/212𝛿₂^1/2≤𝛿₂+𝜉⁶₂𝜑₂𝑥₂. (5.4) Then, by (5.4) and (5.5), it is easy to see thatℒ𝑉𝜉₁,𝜉₂=−𝑐₁−𝑐₂₁𝜉⁶₁−𝑐₂𝜉⁶₂+𝜉³₂𝑥³₃−𝑥₃^∗3+𝛿₂, (5.5) by choosing 𝑥^∗₃(𝑥₁,𝑥₂,𝑦_𝑟)=−𝜉₂(𝑐₂+𝑑₂+𝜑₂)^1/3.

Defining 𝜉₃=𝑥₃−𝑥^∗₃ and the Lyapunov function 𝑉₃(𝜉₁,𝜉₂,𝜉₃)=𝑉₂(𝜉₁,𝜉₂)+(1/6)𝜉⁶₃, one getsℒ𝑉₃≤−𝑐₁−𝑐₂₁𝜉⁶₁−𝑐₂𝜉⁶₂+𝜉³₂𝑥³₃−𝑥₃^∗3+𝛿₂+𝜉⁵₃𝑥₄−𝜕𝑥^∗₃𝜕𝑥₁𝑥₂−𝜕𝑥^∗₃𝜕𝑥₂𝑥³₃−𝜕𝑥^∗₃𝜕𝑦_𝑟𝑦_𝑟̇≤−𝑐₁−𝑐₂₁𝜉⁶₁−𝑐₂𝜉⁶₂+𝛿₃₁+𝜉⁶₃𝜑₃₁+𝛿₃₂+𝜉⁶₃𝜑₃₂+𝜉⁵₃𝑥₄−𝑥^∗₄+𝜉⁵₃𝑥^∗₄=−𝑐₁−𝑐₂₁𝜉⁶₁−𝑐₂𝜉⁶₂−𝑐₃𝜉⁶₃+𝜉⁵₃𝑥₄−𝑥^∗₄+𝛿₂+𝛿₃, (5.6) by choosing 𝑥^∗₄(𝑥₁,𝑥₂,𝑥₃,𝑦_𝑟)=−𝜉₃(𝑐₃+𝜑₃₁+𝜑₃₂). At last, choosing 𝜉₄=𝑥₄−𝑥^∗₄,𝑉₄(𝜉₁,𝜉₂,𝜉₃,𝜉₄)=𝑉₃(𝜉₁,𝜉₂,𝜉₃)+(1/6)𝜉⁶₄, a direct calculation gives𝑉₄≤−𝑐₁−𝑐₂₁𝜉⁶₁−𝑐₂𝜉⁶₂−𝑐₃𝜉⁶₃+𝜉⁵₃𝑥₄−𝑥^∗₄+𝛿₂+𝛿₃+𝜉⁵₄𝑣+𝑘₀𝑓−𝜕𝑥^∗₄𝜕𝑥₁𝑥₂−𝜕𝑥^∗₄𝜕𝑥₂𝑥³₃−𝜕𝑥^∗₄𝜕𝑥₃𝑥₄−𝜕𝑥^∗₄𝜕𝑦_𝑟𝑦_𝑟̇+5𝜉⁴₄𝜎²𝑓²≤−𝑐₁−𝑐₂₁𝜉⁶₁−𝑐₂𝜉⁶₂−𝑐₃𝜉⁶₃−𝑐₄𝜉⁶₄+𝛿₂+𝛿₃+𝛿₄₁+𝜉⁶𝜑₄₁+𝛿₄₂+𝜉⁶𝜑₄₂+𝜉⁵₄𝑣=−𝑐₁−𝑐₂₁𝜉⁶₁−𝑐₂𝜉⁶₂−𝑐₃𝜉⁶₃−𝑐₄𝜉⁶₄+𝛿₂+𝛿₃+𝛿₄, (5.7) by choosing𝑣=−𝜉₄𝑐₄+𝜑₄₁+𝜑₄₂. (5.8)

Choose the design parameters 𝜎=0.125,𝛿₂=0.01,𝛿₃=0.01, and 𝛿₄=0.01. Moreover, to satisfy (5.3), we choose 𝑐₁=1>𝑐₂₁=5/6,𝑐₂=1.5,𝑐₃=0.5 and 𝑐₄=0.5. Choose the initial values 𝑥₁(0)=0.45,𝑥₂(0)=0.5,𝑥₃(0)=0.5,and𝑥₄(0)=0.5.

6. Concluding Remarks

In this paper, a mechanical system is firstly introduced. Then, by a series of coordinate transformations, the mechanical system can be transformed to a class of high-order stochastic nonlinear system, based on which, a more general mathematical model is considered and the smooth state-feedback controller is designed which guarantees that the tracking error 𝜉₁=𝑦−𝑦_𝑟 can be regulated to a neighborhood of the origin in probability with radius as small as possible. At last, the simulation is given to show the effectiveness of the design scheme.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no.61004003) and Research Supporting Foundation of Young and Middle-Aged Scientists of Shandong Province (Grant no. BS2009DX015).