Stabilization of a Class of Stochastic Nonlinear Systems

Abstract

This paper addresses two control schemes for stochastic nonlinear systems. Firstly, an adaptive controller is designed for a class of motion equations. Then, a robust finite-time control scheme is proposed to stabilize a class of nonlinear stochastic systems. The stability of the closed-loop systems is established based on stochastic Lyapunov stability theorems. Links between these two methods are given. The efficiency of the control schemes is evaluated using numerical simulations.

1. Introduction

There has been conspicuous attention toward extending popular nonlinear control design in deterministic setting to stochastic framework. In particular, the integrator backstepping was generalized for stochastic nonlinear system in [1]. The author in [2] gave an extension of output feedback backstepping design for stochastic systems. Battiloti [3] investigated stabilization for a class of stochastic nonlinear systems in upper triangular. A stochastic version of nonlinear small gain was given in [4]. Using the small gain condition, the authors provided an adaptive backstepping controller. Other attempts toward this end have been reported in [5–10] and references therein.

Asymptotic stabilization is an important issue in many engineering applications. But for very demanding applications, finite-time stabilization offers an effective alternative, which yields, in some sense, fast response, high tracking precision, and disturbance-rejection properties [11]. Finite-time stability [12, 13] allows solving the finite-time stabilization problem. Finite-time stabilization method was introduced by Bhat and Bernstein [12] and then it has been developed by many other researchers (see, e.g., [13, 14]). As mentioned above, over the last few decades, considerable research works have been devoted to analysis and design of nonlinear stochastic systems. Recently, Chen and Jiao [15] have extended finite-time stability of deterministic systems to stochastic framework using the Itô differential equation.

In this paper, we provide two nonlinear control designs with applications to a guidance system in stochastic setting. The former gives an adaptive control law for the guidance system. The latter stabilizes a class of stochastic system in finite-time; as a special case, it is applied to the guidance system. Two numerical simulations illustrate the effectiveness of the proposed control schemes. Moreover, links between these two methods are given in Section 4.

The rest of this paper is organized as follows. In Section 2, notions for stochastic nonlinear systems are reviewed. In Section 3, a Lyapunov-based adaptive stochastic control is presented for the guidance system; then the result is verified with numerical simulations. In Section 4, a finite-time stochastic control is investigated. Using this control method, a robust finite-time guidance law is derived. In Section 5, concluding remarks are placed.

2. Preliminaries

List of Properties(i)𝑅₊ is the nonnegative real numbers.(ii)𝐶^𝑛 denotes the set of 𝑛-times differentiable functions 𝑅^𝑛 to 𝑅₊.(iii)|𝐱| stands for Euclidean norm.(iv)A function 𝛼∶𝑅₊→𝑅₊ is of class-𝒦 (𝛼∈𝒦) if it is continuous, zero at zero and strictly increasing. Furthermore, it is of class-𝒦_∞(𝛼∈𝒦_∞) if |𝑠|→∞ as 𝑠→∞.(v)A function 𝛽∶𝑅₊×𝑅₊→𝑅₊ is of class-𝒦ℒ if for each fixed 𝑡≥0, 𝛽(⋅,𝑡)∈𝒦 and for each fixed 𝑠≥0, 𝛽(𝑠,⋅) is strictly decreasing and 𝛽(𝑠,𝑡)→0 as 𝑡→∞.(vi)𝐸[⋅] denotes the expectation of stochastic variable 𝑥.

Consider the following stochastic nonlinear system: 𝑑𝐱=(𝐟(𝐱)+𝐠(𝐱)𝑢)𝑑𝑡+𝐡(𝐱)𝑑𝐰, (1) where 𝐱=[𝑥₁,…,𝑥_𝑛]^𝑇∈𝑅^𝑛 is the state of the system which is assumed to be available for measurement, 𝑢∈𝑅 is the control input, 𝑡∈𝑅₊ is time, 𝐟∈𝑅^𝑛,𝐠∈𝑅^𝑛, and 𝐡∈𝑅^𝑛×𝑟 are continuous functions, and 𝐰 is an 𝑟-dimensional standard Brownian motion. Let 𝑥(𝑡,𝑥₀) denote the solution to the system (1) starting from the initial value 𝑥₀. The system (1) can be thought of as a perturbation from the deterministic system 𝐱̇=𝐟(𝐱)+𝐠(𝐱)𝑢 by an additive white noise.

Let 𝑉∶𝑅^𝑛→𝑅⁺ with the property that 𝑉∈𝐶². The differential operator 𝐿 is defined by 𝐿𝑉∶=𝜕𝑉(𝐟(𝐱)(𝐱)𝑢)+1𝜕𝐱+𝐠2Tr𝐡^𝑇𝜕²𝑉𝜕𝐱²𝐡, (2) where Tr{⋅} denotes the matrix trace. We borrow some notions on stability of stochastic system from [4, 15].

Definition 1 (see [4]). The system (1) is said to be input-to-state practically stable (ISpS) in probability if for any 𝜀>0, there exist some 𝛽∈𝒦ℒ, 𝛾∈𝒦, and 𝛿>0 such that the following hold: 𝑃|𝑥|≤𝛽𝑥₀,𝑡+𝛾(|𝑢|)+𝛿≥1−𝜀,∀𝑡≥0,𝑥₀∈𝑅^𝑛⧵{0}. (3)

Definition 2 (see [15]). For system (1), define 𝑇(𝑥₀,𝑤)={𝑇≥0∶𝑥(𝑡,𝑥₀)≡0∀𝑡>𝑇}, which is called the stochastic settling time function.

Definition 3 (see [15]). For stochastic system (1), the origin 𝐱=0 is said to be globally stochastically finite-time stable, if for each 𝑥₀∈𝑅^𝑛, the following conditions hold:(i)stochastic settling time function 𝑇(𝑥₀,𝑤) exists with probability one;(ii)if 𝑇(𝑥₀,𝑤) exists, then 𝐸[𝑇(𝑥₀,𝑤)]<+∞;(iii)the origin is stable.

Theorem 4 (see [4]). For system (1), there exist 𝑉∶𝑅^𝑛→𝑅₊ function with the property that 𝑉∈𝐶², functions 𝛼,𝛼,𝛼∈𝒦_∞ and 𝛾∈𝒦, and 𝛿>0 such that the following hold: 𝛼(𝐱)≤𝑉(𝐱)≤𝛼(𝐱),𝐿𝑉≤−𝛼(|𝐱|)+𝛾(|𝐱|)+𝛿; (4) then system (1) is ISpS in probability sense. Such a function 𝑉 is called an ISpS Lyapunov function for (1).

Theorem 5 (see [15]). Consider system (1) with 𝑢=0. If there exist a positive definite, twice continuously differentiable, and radially unbounded Lyapunov function 𝑉∶𝑅^𝑛→𝑅₊ and real numbers 𝛽>0 and 0<𝛼<1, such that 𝐿𝑉(𝐱)≤−𝛽[𝑉(𝐱)]^𝛼, (5) then the origin of system (1) is globally stochastically finite-time stable.

Lemma 6 (Young’s inequality). For any (𝑎,𝑏)∈𝑅×𝑅, for all (𝑝,𝑞)∈𝑅₊×𝑅₊ with (1/𝑝)+(1/𝑞)=1, and for each 𝜀>0, the following holds: 𝜀𝑎𝑏≤^𝑝𝑝|𝑎|^𝑝+1𝜀^𝑞𝑞|𝑏|^𝑞. (6)

3. Adaptive Control Design for a Guidance System

In this section, we design an adaptive control law for the guidance system below under the presence of noise. The geometry of planar interception is shown in Figure 1. The two-dimensional motion equation of the planar interception is [16] 𝑟̇=𝑉_𝑇cos𝑞−𝜑_𝑇−𝑉_𝑀cos𝑞−𝜑_𝑀,𝑟𝑞̇=−𝑉_𝑇sin𝑞−𝜑_𝑇+𝑉_𝑀sin𝑞−𝜑_𝑀, (7) where 𝑟 is the relative range, 𝑞 is LOS angle, 𝜑_𝑀 and 𝜑_𝑇 are flight-path angle for missile and target, and 𝑉_𝑀 and 𝑉_𝑇 are missile velocity and target velocity, respectively. Differentiating (7) with respect to time yields [16]: 𝑟̈=𝑟𝑞̇²̈2𝑟̇+𝑤−𝑢,𝑞=−𝑟̇1𝑞+𝑟1𝑤−𝑟𝑢, (8) where 𝑢 and 𝑤 are missile acceleration normal to line of sight and target acceleration normal to line of sight, respectively. Let 𝑢 be the control input. Also, define 𝑥(𝑡)∶=𝑞(̇𝑡) for all 𝑡≥0. So we get ̇2𝑟̇𝑥=−𝑟1𝑥+𝑟1𝑤−𝑟𝑢. (9)

Figure 1 

Planar interception geometry. 𝑀 and 𝑇 denote the missile and the target, respectively.

Assumption 7. Suppose that 𝑟(̇𝑡) and 𝑤(𝑡) are stochastic processes defined by 𝑟(̇𝑡)=𝑟̇(𝑡)(𝑡),𝑤(𝑡)=+𝜁𝑤(𝑡)+𝜁(𝑡), (10) where 𝑟(̇‌𝑡) and 𝑤(𝑡) are deterministic and 𝜁(𝑡) is white noise degrading measurements. Using Assumption 7, the system (9) can be represented as −2̇𝑑𝑥=𝑟𝑟1𝑥+𝑟1𝑤−𝑟𝑢1𝑑𝑡−𝑟(2𝑥−1)𝑑𝑤. (11)

Theorem 8. Consider the stochastic system (11). The following guidance law 𝑢=𝑐𝑥+𝑤−𝑁̇𝜃𝑥𝑟𝑥+𝑟𝑥+6𝑟−6𝑟,𝑁>2 (12) with the update law ̇𝜃=𝑘𝜎𝜃^∘−𝜃+𝑥⁴𝑟²𝑐,𝜎,𝑘>0,𝜃^∘∈𝑅 (13) guarantees the boundedness of 𝐱(𝑡) and 𝜃(𝑡) and the convergence of 𝐱(𝑡) to an arbitrary small neighborhood of the origin.

Proof. Define the following Lyapunov function: 𝑉𝜃=1𝑥,4𝑥⁴+1𝜃2𝑘², (14) where 𝜃=𝜃−𝜃. According to the Itô differential rule [17], we have 𝐿𝑉=𝜕𝑉−2̇𝜕𝑥𝑟𝑟1𝑥+𝑟1𝑤−𝑟𝑢+12𝜕²𝑉𝜕𝑥²2𝑥−1𝑟²+𝜕𝑉𝜕𝜃̇𝜃. (15)
Adding and subtracting 𝜎𝜃(𝜃^∘−𝜃) to the right-hand sight of (15) give 𝐿𝑉=𝑥³−2𝑟̇𝑟1𝑥+𝑟1𝑤−𝑟𝑢+32𝑥𝑟²4𝑥²−1−4𝑥+1𝑘𝜃̇𝜃±𝜎𝜃𝜃^∘−𝜃. (16)
This can be simplified as 𝑥𝐿𝑉=³𝑟−2𝑟̇𝑥+𝑤−𝑢+6𝑥𝑟−6𝑟+32𝑥²𝑟²−1𝑘𝜃̇𝜃±𝜎𝜃𝜃^∘−𝜃. (17)
Rewrite (17) as 𝑥𝐿𝑉=³𝑟−2𝑟𝑥̇+𝑤−𝑢+6𝑥𝑟−6𝑟+32𝑥²𝑟²−𝜃𝑘̇𝜃−𝜎𝑘𝜃^∘−𝜃−𝜎𝜃𝜃^∘−𝜃, (18) so we get 𝑥𝐿𝑉=³𝑟−2𝑟𝑥̇+𝑤−𝑢+6𝑥𝑟−6𝑟+𝜃𝑥𝑟−𝜃𝑥⁴𝑟²+32𝑥²𝑟²−𝜃𝑘̇𝜃−𝜎𝑘𝜃^∘−𝜃−𝜎𝜃𝜃^∘−𝜃. (19) Recalling the fact that 𝜃=𝜃−𝜃, (19) can be written as 𝑥𝐿𝑉=³𝑟−2𝑟𝑥̇+𝑤−𝑢+6𝑥𝑟−6𝑟+𝜃𝑥𝑟+𝑥𝜃−𝜃⁴𝑟²+32𝑥²𝑟²−𝜃𝑘̇𝜃−𝜎𝑘𝜃^∘−𝜃−𝜎𝜃𝜃^∘−𝜃. (20)
Therefore, we have 𝑥𝐿𝑉=³𝑟−2𝑟𝑥̇+𝑤−𝑢+6𝑥𝑟−6𝑟+𝜃𝑥𝑟𝑥−𝜃⁴𝑟²+32𝑥²𝑟²−𝜃𝑘̇𝜃−𝜎𝑘𝜃^∘−𝜃𝑥−𝑘⁴𝑟²−𝜎𝜃𝜃^∘−𝜃. (21)
It follows with the following update law ̇𝜃=𝑘𝜎𝜃^∘−𝜃+𝑥⁴𝑟² (22)
that 𝑥𝐿𝑉=³𝑟−2𝑟𝑥̇+𝑤−𝑢+6𝑥𝑟−6𝑟+𝜃𝑥𝑟𝑥−𝜃⁴𝑟²+32𝑥²𝑟²−𝜎𝜃𝜃^∘−𝜃. (23)
The last term in (23) can be written as 2𝜃𝜃^∘−𝜃=𝜃²+𝜃^∘−𝜃²−𝜃^∘−𝜃². (24)
Substituting (24) into (21) gives 𝑥𝐿𝑉=³𝑟−2𝑟𝑥̇+𝑤−𝑢+6𝑥𝑟−6𝑟+𝜃𝑥𝑟−𝜎2𝜃²+𝜃^∘−𝜃²−𝜃^∘−𝜃²𝑥−𝜃⁴𝑟²+32𝑥²𝑟². (25)
Pick the following control law: 𝑢=𝑐𝑥+𝑤−𝑁̇𝜃𝑥𝑟𝑥+𝑟𝑥+6𝑟−6𝑟𝑁>2. (26)
So we obtain 𝑥𝐿𝑉=−𝑐⁴𝑟+𝑟𝑥̇⁴𝑟()𝑥𝑁−2−𝜃⁴𝑟²+32𝑥²𝑟²−𝜎2𝜃²+𝜃^∘−𝜃²−𝜃^∘−𝜃². (27) For 𝑁>2, the second term on the right-hand side of (27) is negative (i.e., (𝑟̇𝑥⁴/𝑟)(𝑁−2)<0), thus 𝑥𝐿𝑉≤−𝑐⁴𝑟𝑥−𝜃⁴𝑟²+32𝑥²𝑟²−𝜎2𝜃²+𝜃^∘−𝜃²−𝜃^∘−𝜃². (28)
A simplified but conservative version of (28) is 𝑥𝐿𝑉≤−𝑐⁴𝑟𝑥−𝜃⁴𝑟²+32𝑥²𝑟²−𝜎2𝜃²−𝜃^∘−𝜃². (29)
The last term in (29) on the right-hand side of (29) is negative (i.e., −𝜎/2(𝜃^∘−𝜃)²<0), so 𝑥𝐿𝑉≤−𝑐⁴𝑟𝑥−𝜃⁴𝑟²+32𝑥²𝑟²−𝜎2𝜃²+𝜎2𝜃^∘−𝜃². (30)
The second and third terms on the right-hand side of (30) can be written as 𝑥−𝜃⁴𝑟²+32𝑥²𝑟²𝜃=−𝑟²𝑥⁴−3𝑥2𝜃²𝜃=−𝑟²𝑥⁴−3𝑥2𝜃²+34𝜃²−34𝜃²𝜃=−𝑟²𝑥²−34𝜃²+916𝜃𝑟². (31)
By substituting (31) into (30), we get 𝑥𝐿𝑉≤−𝑐⁴𝑟−𝜃𝑟²𝑥²−34𝜃²+916𝜃𝑟²−𝜎2𝜃²+𝜎2𝜃^∘−𝜃². (32)
So 𝑥𝐿𝑉≤−𝑐⁴𝑟−𝜎2𝜃²+916𝜃𝑟²+𝜎2𝜃^∘−𝜃²𝜃>0. (33)
Let 𝜀=(9/16𝜃𝑟²)+(𝜎/2)(𝜃^∘−𝜃)². Substituting (14) into the right-hand side of above inequality gives 𝐿𝑉≤−𝑐₀𝑉+𝜀, (34) where 𝑐₀=min{𝑘𝜎,4𝑐/𝑟}. Inequality (34) implies that 𝑉 is an ISpS Lyapunov function. Therefore, it ensures the boundedness of 𝐱(𝑡) and 𝜃(𝑡) and convergence of 𝐱(t) to an arbitrary small neighborhood of the origin.

Corollary 9. Under the assumption that 𝑤(𝑡) is a pure deterministic variable. The following guidance law 𝑢=𝑐𝑥+𝑤−𝑁𝑟𝑥̇+𝜃𝑥𝑟,𝑁>2 (35) with an update law 𝜃̇=𝑘𝑥⁴𝑟²𝑘>0 (36) guarantees the boundedness of 𝐱(𝑡) and 𝜃(𝑡) and the convergence of 𝐱(𝑡) to the origin.

Proof. The proof is a simple conclusion of Theorem 8.

Three-Dimensional Case. With the same arguments, the result can be extended to the three-dimensional case. Consider the following stochastic system: 𝑑𝑥₁=−2𝑟̇𝑟𝑥₁+1𝑟𝑤₁−1𝑟𝑢₁1𝑑𝑡−𝑟2𝑥₁−1𝑑𝑤₁,𝑑𝑥₂=−2𝑟̇𝑟𝑥₂+1𝑟𝑤₂−1𝑟𝑢₂1𝑑𝑡−𝑟2𝑥₂−1𝑑𝑤₂, (37) where 𝑤₁=𝑎_𝑇𝜃 and 𝑤₂=𝑎_𝑇𝜙. The following guidance laws 𝑢₁=𝑐₁𝑥+𝑤₁−𝑁𝑟𝑥̇₁+𝜃₁𝑥₁𝑟𝑥+6¹𝑟−6𝑟𝑢,𝑁>2,₂=𝑐₂𝑥+𝑤₂−𝑁𝑟𝑥̇₂+𝜃₂𝑥₂𝑟𝑥+6²𝑟−6𝑟,𝑁>2 (38) with the update laws ̇𝜃₁=𝑘₁𝜎₁𝜃^∘₁−𝜃₁+𝑥⁴₁𝑟²,𝑐₁,𝜎₁,𝑘₁>0,𝜃^∘₁̇𝜃∈𝑅,₂=𝑘₂𝜎₂𝜃^∘₂−𝜃₂+𝑥⁴₂𝑟²,𝑐₂,𝜎₂,𝑘₂>0,𝜃^∘₂∈𝑅 (39) guarantee the boundedness of 𝐱(𝑡) and the convergence of 𝐱(𝑡) to an arbitrary small neighborhood of the origin.

3.1. Numerical Simulation

For the closed loop system, the mentioned stochastic differential equations governed by this guidance system can be simplified as (𝑡)=−()̇𝑑𝑥𝑁+2𝑟(𝑡)𝑟(𝑡)𝑥(𝑡)−𝑐+𝜃(𝑡)𝑟(𝑡)𝑥(𝑡)+6𝑟²(𝑡)𝑥(𝑡)−6𝑟²(𝑡)+𝑑𝑡−2𝑟(𝑡)𝑥(𝑡)+1𝑟(𝑡)𝑑𝑤(𝑡), (40) where ̇𝜃(𝑡) and r(𝑡) are simultaneously solved by the following: ̇𝜃(𝑡)=𝑘𝜎𝜃^∘−𝜃(𝑡)+𝑥⁴(𝑡)𝑟(𝑡)𝑐,𝜎,𝑘>0𝜃^∘𝑟(̈𝑡)=𝑟(𝑡)𝑥(𝑡)̇∈𝑅,−𝑐−𝑁𝑟(𝑡)−𝜃(𝑡)+6𝑟(𝑡)(𝑡)(𝑡)−6×𝑥+𝜁𝑟(𝑡). (41)

To solve these equations numerically, we follow the methods in [18, 19]. The guidance law and update law parameters are chosen as 𝑁=2,𝑐=500,𝜎=2,𝐾=5, and 𝜃⁰=0.2. Initial conditions of system (11) and adaptive guidance law (13) are selected as 𝑟(̇0)=200, 𝑟(0)=1,𝜃(0)=0, and 𝑥(0)=0.1. Sampling time is set to the value ℎ=0.005. Applying our proposed method, the line of sight rate 𝑥(𝑡) is depicted in Figure 2.

Figure 2 

The line of sight rate 𝑥(𝑡).

The update parameter 𝜃(𝑡) is shown in Figure 3. As we expect, after some transient time (0.4 second), the update parameter stays close to the value of 0.2.

Figure 3 

The adaptation parameter 𝜃(𝑡).

The control input 𝑢(𝑡) is shown in Figure 4.

Figure 4 

The applied control effort.

4. Finite-Time Control Law for Stochastic systems

In the previous section, the variable 𝑞̇ is only controlled (see (9)). One expects to get better performance of the guidance system if both the LOS angle 𝑞 and the angular velocity 𝑞̇ are controlled. In this case, we get the following state-space equations: 𝑥̇₁=𝑥₂,𝑥̇₂2𝑟̇=−𝑟𝑥₂+1𝑟1𝑤−𝑟𝑢. (42) Using Assumption 7, the system (42) can be represented as 𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=−2𝑟̇𝑟𝑥₂+1𝑟1𝑤−𝑟𝑢1𝑑𝑡−𝑟2𝑥₂−1𝑑𝑤. (43) So we aim to develop a guidance law to stabilize the origin of the system (43). On the other hand, the notion of finite-time stability is very important in guidance problems since the guidance equations are valid as long as the intercept point is met in finite-time [16, 20, 21]. The adaptive guidance law proposed in the previous section does not provide the finite-time convergence of the guidance system although it gives the effective robustness. These motivate us to give a robust finite-time guidance law in stochastic setting.

Consider the following stochastic nonlinear system: 𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=𝑓(𝐱,𝑡)+𝑢𝑑𝑡+𝑔(𝐱,𝑡)𝑑𝑤, (44) where 𝑤∈𝑅 and 𝐱=[𝑥₁,𝑥₂]^𝑇∈𝑅² is assumed to be available for measurement. A guidance system can be modeled in this form. We emphasize that the guidance system (43) is a special case of the system (44).

Assumption 10. For any 𝐱∈𝑅², the nonlinear part of (42) can be bounded by 𝑓(𝐱,𝑡)≤𝛾𝑥₁,𝑥₂𝑥₁+𝑥₂, (45) where 𝛾(𝑥₁,𝑥₂)≥0 is a known C¹ function.
It should be noted that the function 𝑓(⋅,⋅) is unknown, in general. We only need that the upper bound (45) is given.

Theorem 11. Consider the stochastic system (42) with Assumption 10. The following control law 𝑢=−𝛽𝑥₁,𝑥₂𝑥₂^5/3+0.15𝑥₁^1/5−73𝜛𝑥₁,𝑥₂+𝛿⁴4𝑥₂^8/3𝜉₂^1/5 (46) guarantees the boundedness of 𝐱(𝑡) and the convergence of 𝐱(𝑡) to the origin in finite-time.

Proof. Define the following Lyapunov function: 𝑉(𝐱)=12𝑥²₁+^𝑥2_𝑥∗2𝑠^5/3−𝑥^∗5/3₂^7/5𝑑𝑠. (47)Fact 1. Recall that for a given dynamical system 𝑑𝐱=𝑓(𝐱,𝑡)𝑑𝑡+𝑔(𝐱,𝑡)𝑑𝐰(𝑡). (48)
According to the Itô differential rule 𝐿𝑉(𝐱) is (𝐱)=(𝐱)𝐿𝑉𝜕𝑉𝑓()+1𝜕𝐱𝐱,𝑡2𝑇𝑟𝑔^𝑇()𝜕𝐱,𝑡²𝑉(𝐱)𝜕𝐱²𝑔(𝐱,𝑡). (49) So 𝐿𝑉(𝐱) for the system (42) is (𝐱)=(𝐱)𝐿𝑉𝜕𝑉𝜕𝑥₁𝑥₂+𝜕𝑉(𝐱)𝜕𝑥₂𝑓()+1𝐱,𝑡+𝑢2𝜕²𝑉(𝐱)𝜕𝑥²₂𝑔²(𝐱,𝑡). (50) The partial derivative of 𝑉 respect to 𝑥₁ is 𝜕𝑉(𝐱)𝜕𝑥₁=𝑥₁+𝜕𝜕𝑥₁^𝑥2_𝑥^∗₂𝑠^5/3−𝑥^∗5/3₂^7/5𝑑𝑠. (51)
Fact 2. From the elementary calculus, we get 𝑑𝑑𝐱_𝑎^𝑔(𝐱)𝑓()=𝐱,𝑧𝑑𝑧_{𝑎𝑔(𝐱)}𝜕𝑓(𝐱,𝑧)(𝐱)𝜕𝐱𝑑𝑧+𝜕𝑔𝑓(𝜕𝑥𝐱,𝑧)_{𝑧=𝑔(𝐱)}. (52) Using Fact 2, (49) is rewritten as 𝜕𝑉(𝐱)𝜕𝑥₁=𝑥₁+^𝑥2_𝑥∗2𝜕𝜕𝑥₁𝑠^5/3−𝑥^∗5/3₂^7/5−𝑑𝑠𝜕𝑥^∗₂𝜕𝑥₁𝑠^5/3−𝑥^∗5/3₂^7/5_{𝑠=𝑥∗2}. (53) It follows with the fact that (𝜕𝑥^∗₂/𝜕𝑥₁)(𝑠^5/3−𝑥^∗5/3₂)^7/5|_{𝑠=𝑥∗2}=0 that 𝜕𝑉(𝐱)𝜕𝑥₁=𝑥₁+^𝑥2_𝑥∗275𝑠^5/3−𝑥^∗5/3₂^2/5×−53𝑥^∗2/3₂𝜕𝑥^∗₂𝜕𝑥₁𝑑𝑠. (54) Recall that 𝑥^∗2/3₂𝜕𝑥^∗₂𝜕𝑥₁=−2𝑥₁^3/52/3−65𝑥₁^−2/56=−5(4)^1/3; (55) we obtain 𝜕𝑉(𝐱)𝜕𝑥₁=𝑥₁+145(4)^1/3𝑥2_𝑥∗2𝑠^5/3−𝑥^∗5/3₂^2/5𝑑𝑠. (56) The partial derivative of 𝑉(𝐱) with respect to 𝑥₂ is 𝜕𝑉(𝐱)𝜕𝑥₂=𝑥₂^5/3−𝑥^∗5/3₂^7/5=𝜉₂^7/5 (57) and the second derivative with respect to 𝑥₂ is 𝜕²𝑉(𝐱)𝜕𝑥²₂=73𝜉₂^2/5𝑥₂^2/3. (58) So we have 𝐿𝑉(𝐱)=𝑥₁+145(4)^1/3𝑥2_𝑥∗2𝑠^5/3−𝑥^∗5/3₂^2/5𝑑𝑠𝑥₂+𝑥₂^7/5𝑓(𝐱)+7+𝑢6𝑥₂^2/5𝑥₂^2/3𝑔²(𝐱). (59)
Define the following variables: 𝜉₁∶=𝑥₁,𝜉₂∶=𝑥₂^5/3−𝑥^∗5/3₂,𝑥^∗₂∶=−2𝑥₁^3/5. (60)
Therefore, one yields: 𝐿𝑉=𝜉₁−75^𝑥2_𝑥^∗₂𝑠^5/3−𝑥^∗5/3₂^2/5d𝑠𝑥₂+𝜉₂^7/5𝑓(𝐱)+7+𝑢3𝑔(𝐱)²𝑥₂^2/3𝑥₂^5/3−𝑥^∗5/3₂^2/5. (61)
Let 𝑢∶=𝑢_𝑑+𝑢_𝑠; the deterministic term is 𝑢_𝑑=−𝛽𝑥₁,𝑥₂𝑥₂^5/3+0.15𝑥₁^1/5, (62) where 𝛽(𝑥₁,𝑥₂)>0 is a 𝐶¹ function. Particularly, 𝑢_𝑑 and 𝑢_𝑠 provide the deterministic and stochastic parts of (44), respectively. Substituting (62) into the right-hand side of (61) gives 𝐿𝑉≤−𝜉₁^8/5+𝜉₂^8/5+𝜉₂^7/5𝑢_𝑠+73𝑔(𝐱)²𝑥₂^2/3𝜉₂^2/5. (63)
Assume that 𝑔(𝐱,𝑡)²≤𝜛𝑥₁,𝑥₂𝜉₁^6/5+𝜉₂^6/5, (64) where 𝜛(𝑥₁,𝑥₂)>0 is a 𝐶¹ function. From the aforementioned assumption and Young’s inequality, we get 𝐿𝑉≤−𝜉₁^8/5+𝜉₂^8/5+𝜉₂^7/5𝑢_𝑠+73𝜛𝑥₁,𝑥₂𝜉₁^6/5+𝜉₂^6/5𝑥₂^2/3𝜉₂^2/5≤−𝜉₁^8/5+𝜉₂^8/5+1(4/3)𝛿^4/3𝜉₁^8/5+𝜉₂^7/5𝑢_𝑠+73𝜛𝑥₁,𝑥₂+𝛿⁴4𝑥₂^8/3𝜉₂^1/5, (65) where 𝛿∈(0,1). Let 𝑢_𝑠 be 𝑢_𝑠7=−3𝜛𝑥₁,𝑥₂+𝛿⁴4𝑥₂^8/3𝜉₂^1/5. (66) With the similar arguments as above, we have 𝐿𝑉≤−(1−𝛿)𝜉₁^8/5+𝜉₂^8/5. (67)
It follows from Proposition 2 and Lemma 2.3 in [13] that 𝐿𝑉≤−(1−𝛿)𝜉₁^8/5+𝜉₂^8/5≤−𝜌𝑉^4/5𝜌∈(0,0.5). (68)
This shows that the origin of the system is globally stochasticly finite-time stable.

4.1. Numerical Simulation

In this section, the performance of the proposed control schemes is evaluated via a numerical example. Consider the following dynamical system: 𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=𝑢𝑑𝑡+𝑥₁𝑑𝑤, (69) where 𝑤≈𝑁(0,0.05). Let 𝛽(𝑥₁,𝑥₂)=0.4 and 𝜎=0.8. Initial condition is selected as 𝑥₁(0)=1 and 𝑥₂(0)=−1. A lower bound for 𝜛(𝑥₁,𝑥₂) is 𝜛𝑥₁,𝑥₂≥𝑔(𝐱)²𝜉₁^6/5+𝜉₂^6/5. (70)

Therefore, we can choose 𝜛(𝑥₁,𝑥₂) as 𝜛𝑥₁,𝑥₂=𝑏𝑔(𝐱)²𝜉₁^6/5+𝜉₂^6/5, (71) where 𝑏=1.1. The control law obtained from the results in Theorem 11 is 𝑢=−𝛽𝑥₁,𝑥₂𝑥₂^5/3+0.15𝑥₁^1/5−73𝜛𝑥₁,𝑥₂+𝛿⁴4𝑥₂^8/3𝜉₂^1/5, (72) where 𝜉₁=𝑥₁,𝜉₂=𝑥₂^5/3−𝑥^∗5/3₂,𝑥^∗₂=−2𝑥₁^3/5. (73)

Numerical simulations are implemented in MATLAB with the step size 0.001. Figures 5 and 6 illustrate the effectiveness of simulation results using the control law (70). The state trajectories are shown in Figure 5. The control input is depicted in Figure 6. Obviously, the control signal (70) is not smooth although it gives a finite-time convergence.

Figure 5 

The state trajectories 𝑥(𝑡) due to the control scheme.

Figure 6 

Input control 𝑢(𝑡) due to the control scheme.

5. Conclusions

Two guidance laws were proposed in stochastic setting. First, an adaptive guidance was presented to achieve target interception under measurement noise. The effectiveness of this result was evaluated using numerical simulations. Although this guidance law provided the effective robustness, it did not guarantee the finite-time convergence for the guidance system which is preferably required. Next, a robust control method was proposed to stabilize both the LOS angle and its angular velocity at the origin in finite-time under measurement noise. Simulation results demonstrated that the second guidance approach provided the finite-time convergence, but the signal input was not as smooth as the one in the first method. So this showed that balance between these approaches was needed.

Conflict of Interests

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