# Stabilization of a Class of Stochastic Nonlinear Systems

Authors: Valiollah Ghaffari; Hamid Reza Karimi; Navid Noroozi; S. Vahid Naghavi
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Section4.The rest of this paper is organized as follows. In Section2, 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  Rn to  R+.
(iii)
|
x
| stands for Euclidean norm.
(iv)
A functionα: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 functionβ: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:(1)dx=(f(x)+g(x)u)dt+h(x)dw,
where x=[x1,…,xn]T∈Rn 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∈Rn,g∈Rn, and h∈Rn×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 x˙=f(x)+g(x)u by an additive white noise.LetV:Rn→R+ with the property that V∈C2. The differential operator L is defined by
(2)LV:=∂V∂x(f(x)+g(x)u)+12Tr{hT∂2V∂x2h},
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:
(3)P{|x|≤β(|x0|,t)+γ(|u|)+δ}≥1-ε,∀t≥0,x0∈Rn∖{0}.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∈Rn, the following conditions hold:(i)
stochastic settling time functionT(x0,w) exists with probability one;(ii)
ifT(x0,w) exists, then E[T(x0,w)]<+∞;(iii)
the origin is stable.Theorem 4 (see [4]).
For system (1), there exist  V:Rn→R+ function with the property that V∈C2, functions α,α_,α-∈𝒦∞ and γ∈𝒦, and δ>0 such that the following hold:
(4)α_(x)≤V(x)≤α-(x),LV≤-α(|x|)+γ(|x|)+δ;
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:Rn→R+ and real numbers β>0 and 0<α<1, such that
(5)LV(x)≤-β[V(x)]α,
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:
(6)ab≤εpp|a|p+1εqq|b|q.


## 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 Figure1. The two-dimensional motion equation of the planar interception is [16]
(7)r˙=VTcos(q-φT)-VMcos(q-φM),rq˙=-VTsin(q-φT)+VMsin(q-φM),
where r is the relative range, q is LOS angle, φM and φT are flight-path angle for missile and target, and  VM and VT are missile velocity and target velocity, respectively. Differentiating (7) with respect to time yields [16]:
(8)r¨=rq˙2+w-u,q¨=-2r˙rq˙+1rw-1ru,
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
(9)x˙=-2r˙rx+1rw-1ru.Figure 1
Planar interception geometry.M and T denote the missile and the target, respectively.Assumption 7.
Suppose thatr˙(t) and w(t) are stochastic processes defined by
(10)r˙(t)=r-˙(t)+ζ(t),w(t)=w-(t)+ζ(t),
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
(11)dx=(-2r-˙rx+1rw--1ru)dt-1r(2x-1)dw.Theorem 8.
Consider the stochastic system (11). The following guidance law
(12)u=cx+w--Nr-˙x+θ^xr+6xr-6r,N>2
with the update law
(13)θ^˙=k[σ(θ°-θ^)+x4r2]c,σ,k>0,θ°∈R
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:(14)V(x,θ^)=14x4+12kθ~2,
where θ~=θ-θ^. According to the Itô differential rule [17], we have
(15)LV=∂V∂x(-2r-˙rx+1rw--1ru)+12∂2V∂x2(2x-1r)2+∂V∂θ^θ^˙.
Adding and subtractingσθ~(θ°-θ^) to the right-hand sight of (15) give
(16)LV=x3(-2r-˙rx+1rw--1ru)+32(xr)2(4x2-4x+1)-1kθ~θ^˙±σθ~(θ°-θ^).
This can be simplified as(17)LV=x3r(-2r-˙x+w--u+6xr-6r)+32x2r2-1kθ~θ^˙±σθ~(θ°-θ^).
Rewrite (17) as
(18)LV=x3r(-2r-˙x+w--u+6xr-6r)+32x2r2-θ~k(θ^˙-σk(θ°-θ^))-σθ~(θ°-θ^),
so we get
(19)LV=x3r(-2r-˙x+w--u+6xr-6r+θ^xr)-θ^x4r2+32x2r2-θ~k(θ^˙-σk(θ°-θ^))-σθ~(θ°-θ^).
Recalling the fact that  θ~=θ-θ^, (19) can be written as
(20)LV=x3r(-2r-˙x+w--u+6xr-6r+θ^xr)+(θ~-θ)x4r2+32x2r2-θ~k(θ^˙-σk(θ°-θ^))-σθ~(θ°-θ^).
Therefore, we have(21)LV=x3r(-2r-˙x+w--u+6xr-6r+θ^xr)-θx4r2+32x2r2-θ~k(θ^˙-σk(θ°-θ^)-kx4r2)-σθ~(θ°-θ^).
It follows with the following update law(22)θ^˙=k[σ(θ°-θ^)+x4r2]
that(23)LV=x3r(-2r-˙x+w--u+6xr-6r+θ^xr)-θx4r2+32x2r2-σθ~(θ°-θ^).
The last term in (23) can be written as
(24)2θ~(θ°-θ^)=θ~2+(θ°-θ^)2-(θ°-θ)2.
Substituting (24) into (21) gives
(25)LV=x3r(-2r-˙x+w--u+6xr-6r+θ^xr)-σ2(θ~2+(θ°-θ^)2-(θ°-θ)2)-θx4r2+32x2r2.
Pick the following control law:(26)u=cx+w--Nr-˙x+θ^xr+6xr-6rN>2.
So we obtain(27)LV=-cx4r+r-˙x4r(N-2)-θx4r2+32x2r2-σ2(θ~2+(θ°-θ^)2-(θ°-θ)2).
For N>2, the second term on the right-hand side of (27) is negative (i.e., (r-˙x4/r)(N-2)<0), thus
(28)LV≤-cx4r-θx4r2+32x2r2-σ2(θ~2+(θ°-θ^)2-(θ°-θ)2).
A simplified but conservative version of (28) is
(29)LV≤-cx4r-θx4r2+32x2r2-σ2(θ~2-(θ°-θ)2).
The last term in (29) on the right-hand side of (29) is negative (i.e., -σ/2(θ°-θ^)2<0), so
(30)LV≤-cx4r-θx4r2+32x2r2-σ2θ~2+σ2(θ°-θ)2.
The second and third terms on the right-hand side of (30) can be written as
(31)-θx4r2+32x2r2=-θr2(x4-32θx2)=-θr2(x4-32θx2+(34θ)2-(34θ)2)=-θr2(x2-34θ)2+916θr2.
By substituting (31) into (30), we get
(32)LV≤-cx4r-θr2(x2-34θ)2+916θr2-σ2θ~2+σ2(θ°-θ)2.
So(33)LV≤-cx4r-σ2θ~2+916θr2+σ2(θ°-θ)2θ>0.
Letε=(9/16θr2)+(σ/2)(θ°-θ)2. Substituting (14) into the right-hand side of above inequality gives
(34)LV≤-c0V+ε,
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 thatw(t) is a pure deterministic variable. The following guidance law
(35)u=cx+w-Nr-˙x+θ^xr,N>2
with an update law
(36)θ˙=kx4r2k>0
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 Theorem8.Three-Dimensional Case. With the same arguments, the result can be extended to the three-dimensional case. Consider the following stochastic system:
(37)dx1=(-2r-˙rx1+1rw-1-1ru1)dt-1r(2x1-1)dw1,dx2=(-2r-˙rx2+1rw-2-1ru2)dt-1r(2x2-1)dw2,
where w1=aTθ and w2=aTϕ. The following guidance laws
(38)u1=c1x+w-1-Nr-˙x1+θ^1x1r+6x1r-6r,N>2,u2=c2x+w-2-Nr-˙x2+θ^2x2r+6x2r-6r,N>2
with the update laws
(39)θ^˙1=k1[σ1(θ1°-θ^1)+x14r2],c1,σ1,k1>0,θ1°∈R,θ^˙2=k2[σ2(θ2°-θ^2)+x24r2],c2,σ2,k2>0,θ2°∈R
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(40)dx(t)=(-(N+2)r-˙(t)r(t)x(t)-c+θ^(t)r(t)x(t)+6r2(t)x(t)-6r2(t)-c+θ^(t)r(t))dt+(-2r(t)x(t)+1r(t))dw(t),
where θ^˙(t) and r(t) are simultaneously solved by the following:
(41)θ^˙(t)=k[σ(θ°-θ^(t))+x4(t)r(t)]c,σ,k>0θ°∈R,r¨(t)=(r(t)x(t)-c-Nr-˙(t)-θ^(t)+6r(t))×x(t)+ζ(t)-6r(t).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 ratex(t).The update parameterθ^(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 parameterθ^(t).The control inputu(t)  is shown in Figure 4.Figure 4
The applied control effort.


## 3.1. Numerical Simulation

For the closed loop system, the mentioned stochastic differential equations governed by this guidance system can be simplified as(40)dx(t)=(-(N+2)r-˙(t)r(t)x(t)-c+θ^(t)r(t)x(t)+6r2(t)x(t)-6r2(t)-c+θ^(t)r(t))dt+(-2r(t)x(t)+1r(t))dw(t),
where θ^˙(t) and r(t) are simultaneously solved by the following:
(41)θ^˙(t)=k[σ(θ°-θ^(t))+x4(t)r(t)]c,σ,k>0θ°∈R,r¨(t)=(r(t)x(t)-c-Nr-˙(t)-θ^(t)+6r(t))×x(t)+ζ(t)-6r(t).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 ratex(t).The update parameterθ^(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 parameterθ^(t).The control inputu(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 variableq˙ 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 q˙  are controlled. In this case, we get the following state-space equations:
(42)x˙1=x2,x˙2=-2r˙rx2+1rw-1ru.
Using Assumption 7, the system (42) can be represented as
(43)dx1=x2dt,dx2=(-2r-˙rx2+1rw--1ru)dt-1r(2x2-1)dw.
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:(44)dx1=x2dt,dx2=(f(x,t)+u)dt+g(x,t)dw,
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 anyx∈R2, the nonlinear part of (42) can be bounded by
(45)|f(x,t)|≤γ(x1,x2)(|x1|+|x2|),
where γ(x1,x2)≥0 is a known C1 function.
It should be noted that the functionf(·,·) 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
(46)u=-β(x1,x2)(x25/3+0.15x1)1/5-[73ϖ(x1,x2)+δ44x28/3]ξ21/5
guarantees the boundedness of x(t) and the convergence of  x(t) to the origin in finite-time.Proof.
Define the following Lyapunov function:(47)V(x)=12x12+∫x2*x2(s5/3-x2*5/3)7/5ds.Fact1. Recall that for a given dynamical system
(48)dx=f(x,t)dt+g(x,t)dw(t).
According to the Itô differential ruleLV(x) is
(49)LV(x)=∂V(x)∂xf(x,t)+12Tr(gT(x,t)∂2V(x)∂x2g(x,t)).
So LV(x) for the system (42) is
(50)LV(x)=∂V(x)∂x1x2+∂V(x)∂x2(f(x,t)+u)+12∂2V(x)∂x22g2(x,t).
The partial derivative of  V respect to x1 is
(51)∂V(x)∂x1=x1+∂∂x1∫x2*x2(s5/3-x2*5/3)7/5ds.
Fact
2. From the elementary calculus, we get
(52)ddx(∫ag(x)f(x,z)dz)=∫ag(x)∂f(x,z)∂xdz+∂g(x)∂xf(x,z)|z=g(x).
Using Fact 2, (49) is rewritten as
(53)∂V(x)∂x1=x1+∫x2*x2∂∂x1(s5/3-x2*5/3)7/5ds-∂x2*∂x1(s5/3-x2*5/3)7/5|s=x2*.
It follows with the fact that  (∂x2*/∂x1)(s5/3-x2*5/3)7/5|s=x2*=0  that
(54)∂V(x)∂x1=x1+∫x2*x275(s5/3-x2*5/3)2/5×(-53x2*2/3)∂x2*∂x1ds.
Recall that
(55)x2*2/3∂x2*∂x1=(-2x13/5)2/3(-65x1-2/5)=-65(4)1/3;
we obtain
(56)∂V(x)∂x1=x1+145(4)1/3∫x2*x2(s5/3-x2*5/3)2/5ds.
The partial derivative of  V(x) with respect to x2 is
(57)∂V(x)∂x2=(x25/3-x2*5/3)7/5=ξ27/5
and the second derivative with respect to x2 is
(58)∂2V(x)∂x22=73ξ22/5x22/3.
So we have
(59)LV(x)=(x1+145(4)1/3∫x2*x2(s5/3-x2*5/3)2/5ds)x2+x27/5(f(x)+u)+76x22/5x22/3g2(x).
Define the following variables:(60)ξ1:=x1,ξ2:=x25/3-x2*5/3,x2*:=-2x13/5.
Therefore, one yields:(61)LV=(ξ1-75∫x2*x2(s5/3-x2*5/3)2/5ds)x2+ξ27/5(f(x)+u)+73g(x)2x22/3(x25/3-x2*5/3)2/5.
Letu:=ud+us; the deterministic term is
(62)ud=-β(x1,x2)(x25/3+0.15x1)1/5,
where β(x1,x2)>0 is a C1 function. Particularly, ud and us provide the deterministic and stochastic parts of (44), respectively. Substituting (62) into the right-hand side of (61) gives
(63)LV≤-(ξ18/5+ξ28/5)+ξ27/5us+73g(x)2x22/3ξ22/5.
Assume that(64)|g(x,t)|2≤ϖ(x1,x2)(|ξ1|6/5+|ξ2|6/5),
where ϖ(x1,x2)>0 is a C1 function. From the aforementioned assumption and Young’s inequality, we get
(65)LV≤-(ξ18/5+ξ28/5)+ξ27/5us+73ϖ(x1,x2)(|ξ1|6/5+|ξ2|6/5)x22/3ξ22/5≤-(ξ18/5+ξ28/5)+1(4/3)δ4/3ξ18/5+ξ27/5(us+[73ϖ(x1,x2)+δ44x28/3]ξ21/5),
where  δ∈(0,1). Let  us be
(66)us=-[73ϖ(x1,x2)+δ44x28/3]ξ21/5.
With the similar arguments as above, we have
(67)LV≤-(1-δ)(ξ18/5+ξ28/5).
It follows from Proposition 2 and Lemma 2.3 in [13] that
(68)LV≤-(1-δ)(ξ18/5+ξ28/5)≤-ρV4/5ρ∈(0,0.5).
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:(69)dx1=x2dt,dx2=udt+x1dw,
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
(70)ϖ(x1,x2)≥|g(x)|2|ξ1|6/5+|ξ2|6/5.Therefore, we can chooseϖ(x1,x2) as
(71)ϖ(x1,x2)=b|g(x)|2|ξ1|6/5+|ξ2|6/5,
where  b=1.1.  The control law obtained from the results in Theorem 11 is
(72)u=-β(x1,x2)(x25/3+0.15x1)1/5-[73ϖ(x1,x2)+δ44x28/3]ξ21/5,
where
(73)ξ1=x1,ξ2=x25/3-x2*5/3,x2*=-2x13/5.Numerical simulations are implemented in MATLAB with the step size 0.001. Figures5 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 trajectoriesx(t) due to the control scheme.Figure 6
Input controlu(t) due to the control scheme.


## 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:(69)dx1=x2dt,dx2=udt+x1dw,
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
(70)ϖ(x1,x2)≥|g(x)|2|ξ1|6/5+|ξ2|6/5.Therefore, we can chooseϖ(x1,x2) as
(71)ϖ(x1,x2)=b|g(x)|2|ξ1|6/5+|ξ2|6/5,
where  b=1.1.  The control law obtained from the results in Theorem 11 is
(72)u=-β(x1,x2)(x25/3+0.15x1)1/5-[73ϖ(x1,x2)+δ44x28/3]ξ21/5,
where
(73)ξ1=x1,ξ2=x25/3-x2*5/3,x2*=-2x13/5.Numerical simulations are implemented in MATLAB with the step size 0.001. Figures5 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 trajectoriesx(t) due to the control scheme.Figure 6
Input controlu(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.



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