# Influence of Seeker Disturbance Rejection and Radome Error on the Lyapunov Stability of Guidance Systems

Authors: Kaiwei Chen; Qunli Xia; Xiao Du; Yuemin Yao
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This study analyzes the effects of disturbance rejection and radome error on the stability of guidance systems. First, disturbance rejection rate transfer function (DRRTF) models are established. Second, the Lyapunov stability of a proportional navigation guidance system is proposed. The passivity theorems are introduced for the analysis of the Lyapunov stability of the nonlinear time-varying system. Finally, the influence of the different DRRTF models on the stability of the guidance system is analyzed by mathematical simulations. The simulation results indicate that the stable boundary of the guidance system varies and its tolerance to disturbance differs significantly for various disturbance torque types and radome slope.


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## Body

## 1. Introduction

The stabilized platform of a gimbaled seeker isolates the body motion caused by missile maneuvering and vibration during flight. However, the isolation is incomplete due to wire pulling inside the seeker and friction among components. Consequently, missile disturbance coupling enters the seeker, thereby affecting the accuracy of the seeker output [1–3] and leading to the disturbance rejection effect. The seeker disturbance rejection effect then forms a parasitic loop within the missile guidance loop, that is, the disturbance rejection rate parasitic loop (DRRPL).Radomes are attached to the nose of radar homing missiles. The refraction of the electromagnetic wave by the radome causes an error in the line-of-sight (LOS) angle, that is, the radome refraction angle, which also creates a parasitic loop with the guidance homing loop. In certain situations, the disturbance rejection and radome error act to reduce the stability of the guidance system and cause an increase in miss distance [4–7].Several stability analysis approaches for guidance systems have been suggested in the literature.(1) The first and most commonly used approach is based on the “frozen time” assumption, transforming the guidance system into a time-invariant system and analyzing the stability by the Routh criterion [8–11]. However, when the distance between the missile and the target is short or the dynamics of the guidance system are relatively fast, the “frozen time” assumption is no longer valid. Moreover, applying the Routh criterion to analyze the stability generates a large error.(2) The second approach assumes that the terminal guidance time is infinite and uses the stability analysis method of time-varying systems on the basis of infinite time, such as the Popov and circle criteria [4, 12, 13]. These stability conditions are defined over an infinite-horizon time domain and thus are not suitable for the stability analysis of a homing guidance loop, which is only defined over a finite time interval.(3) The third approach analyzes stability by using the finite time stability theory of time-varying systems [14–17]. This approach is practical in real engagement.This study evaluates the Lyapunov stability over a finite time interval on the basis of the linearized model of the proportional navigation (PN) guidance system. The disturbance rejection rate is the transfer function which represents the incorrect angular rate caused by an attitude disturbance applied to the missile’s body [18]. Therefore, this study establishes the disturbance rejection rate transfer function (DRRTF) models under the joint action of the disturbance torque and the radome error. A PN guidance system model is established on the basis of DRRTF. The passivity theorems for analyzing the Lyapunov stability of nonlinear time-varying systems are convenient for engineering application because they do not need to build the state space models and construct Lyapunov functions [19–21]. Therefore, this study analyzes the Lyapunov stability of the PN guidance system on the basis of the passivity theorems.This paper is structured as follows. Section2 establishes the DRRTF and PN guidance system models, which consider the different disturbance torque types and radome error slope. Section 3 defines the uniform asymptotical stability of the PN guidance system and introduces the Lyapunov stability represented by passivity theorems. Section 4 analyzes the stability of the PN guidance system on the basis of passivity theorems. Section 5 investigates the influence of spring torque, damping torque, and radome error slope on the Lyapunov stability of the PN guidance system by simulations. Section 6 concludes this paper.


## 2. Problem Formulation

### 2.1. Modeling of the DRRTF

Figure1 shows the basic geometry in pitch plane. In Figure 1, qt represents the LOS angle, qs is the seeker rotation angle, ϑ denotes the missile attitude angle, φr refers to the gimbal angle, and ε stands for the seeker LOS error angle. From geometry, the detector error angle can be written as follows:(1)ε=qt-qsFigure 1
Basic geometry in pitch plane.Figure2 presents a typical control loop model of a gimbaled seeker.Figure 2
Control loop model of a gimbaled seeker.In Figure2, L is the inductance of armature winding, R denotes the total resistance, KT represents the torque constant, KE stands for the back electromotive force (EMF), Tem(t) refers to the electromagnetic torque, TD signifies the total load torque of the motor and load, J indicates the total motor rotational inertia, and q˙ symbolizes the LOS rate required for guidance.Figure2 shows that the model mainly includes the stabilized, tracking, disturbance torque, and back EMF loops. The disturbance torque loop is caused by wire pulling and friction among rotation joints, which happens in the seeker rotation process. In Figure 2, various types of disturbance torque are equivalent to the transfer function GD(s). The disturbance torque is the main factor leading to problems of the disturbance rejection rate of the gimbaled seeker, including the spring and damping torques. A real gimbaled seeker disturbance torque is a nonlinear model. A simplified model is adopted to analyze the influence of disturbance torque on seeker disturbance rejection rate and output accuracy. The expression is as follows [22]:(2)GDs=Kns+Kω,where Kn is the spring torque coefficient and Kω denotes the damping torque coefficient.The occurrence of the disturbance torque couples the missile and seeker motions, thereby causing DRRPL and reducing the accuracy of seeker control. The DRRTFR(s) is defined as follows:(3)Rs=Δq˙sϑ˙s,where Δq˙(s) is the additional rotation angular rate of the platform caused by the missile motion in relation to the inertial space. From the definition, the smaller the amplitude of R(s), the stronger the seeker’s isolation capability to the disturbance of the missile.In accordance with Figure2, assuming H(s)=1 and ignoring the effect of back EMF, the seeker DRRTF can be educed as(4)Rs=Δq˙sϑ˙s=sGDsLs+RsJsLs+R+GDsLs+R+G2sk2KT+G1sG2sk1k2KTWith the damping torqueGD(s)=Kω and spring torque GD(s)=Kn/s models substituted into (4), the DRRTF models under the effect of the damping and spring torques are shown in (5)Rvs=Δq˙vsϑ˙s=sKωLs+RsJsLs+R+KωLs+R+G2sk2KT+G1sG2sk1k2KT(6)Rss=Δq˙ssϑ˙s=KnLs+RsJsLs+R+G2sk2KT+KnLs+R+G1sG2sk1k2KT


### 2.2. Radome Parasitic Effects in a Stabilized Seeker

The existence of radome affects the seeker measurement on the real target position. Figure3 shows the basic missile–target geometry under the existence of the radome error.Figure 3
Basic geometry for radome analysis.In Figure3, Δq denotes the radome refraction angle, ε∗ is the error angle between the seeker centerline and apparent LOS, and q∗ represents the apparent LOS angle caused by the radome error.The refraction angleΔq varies with the seeker gimbal angle φr. For preliminary analysis, the refraction angle is assumed to be linearly proportional to the gimbal angle(7)Δq=Rdomφr,where Rdom is the radome slope. Rdom is generally regarded as a constant despite being variable during flight [23, 24].In accordance with the angular relationships in Figure3, q∗ can be expressed as follows:(8)q∗=qt+qs-ϑRdom.Considering a stable seeker tracking,ε is generally a small parameter. Thus, qs≈qt. Then, (8) can be rewritten as follows:(9)q∗≅qt+qt-ϑRdom=qt1+R-ϑRdom.Generally speaking,Rdom≪1. Hence, (9) can be simplified into(10)q∗=qt-ϑRdom.Equation (10) shows that missile attitude is fed back to the seeker due to the existence of radome error, thereby generating a measurement error of qt. Consequently, a radome error parasitic loop is formed in the guidance loop.The influence of the correction network on DRRTF characteristics fluctuates on a small scale within the common operating frequencies of the seeker (less than 5 Hz), thereby failing to transform its basic amplitude-phase characteristics. Therefore, the analysis model of the disturbance rejection rate can be simplified to demonstrate intuitively its influence on the guidance and control system. Ignoring the impact of time delay process, the correction network, effect of back EMF, high-frequency dynamics, and small quantity and assumingG1(s)=1, G2(s)=1, R≈1, and L≈0, the equivalent gain of the stabilized loop is determined to be K2=G2k2KT/J. In addition, the equivalent gain of the tracking loop is K1=G1k1. Figure 4 presents the educed seeker model with radome effects.Figure 4
Simplified seeker model with radome effects.Figure4 shows that the DRRTF, including the radome slope, can be educed as (11) with Kr=Rdom/(1-Rdom).(11)Rs=Δq˙sϑ˙s=-Kr+sGDs/K1K21/K1K2s2+GDs+K2/K1K2s+1K 1 K 2 ≫ K n is observed when the time delay process and correction network are ignored. To simplify the analysis, on the basis of the seeker control system performance and ignoring the high-frequency dynamics, the DRRTF under the effect of both radome slope and different types of disturbance torque can be simplified as a first-order model, as shown in Table 1, where Ts is the equivalent time constant.Table 1
First-order DRRTF.
Disturbance type DRRTF Equivalent coefficient Spring torque R s s = K s - K r T s s + 1 K s = K n K 1 K 2 Damping torque R v s = s K v - K r T s s + 1 K v = K ω K 1 K 2


### 2.3. Guidance System Model

Figure5 presents a block diagram of the guidance loop with the disturbance rejection and radome parasitic loop. The noise filter is considered a first-order model, the flight control system a third-order model, and the seeker dynamics a first-order model with the equivalent time constant of Ts=Tg/5.Figure 5
Guidance system model with parasitic effects.In Figure5, at is the target acceleration, am denotes the missile acceleration, Vc represents the closing velocity, Vm stands for the missile velocity, TF-t signifies the time to go, Tg refers to the guidance time constant, Tα indicates the missile turning rate time constant, and N symbolizes the effective navigation ratio of the proportional guidance.The dimensionless method is adopted to reduce the parameter. Assumings=s-/Tg, T-F=TF/Tg, t-=t/Tg, and T-α=Tα/Tg, the equivalent model of the dimensionless guidance system can be obtained as shown in Figure 6.Figure 6
Dimensionless guidance system model.


## 2.1. Modeling of the DRRTF

Figure1 shows the basic geometry in pitch plane. In Figure 1, qt represents the LOS angle, qs is the seeker rotation angle, ϑ denotes the missile attitude angle, φr refers to the gimbal angle, and ε stands for the seeker LOS error angle. From geometry, the detector error angle can be written as follows:(1)ε=qt-qsFigure 1
Basic geometry in pitch plane.Figure2 presents a typical control loop model of a gimbaled seeker.Figure 2
Control loop model of a gimbaled seeker.In Figure2, L is the inductance of armature winding, R denotes the total resistance, KT represents the torque constant, KE stands for the back electromotive force (EMF), Tem(t) refers to the electromagnetic torque, TD signifies the total load torque of the motor and load, J indicates the total motor rotational inertia, and q˙ symbolizes the LOS rate required for guidance.Figure2 shows that the model mainly includes the stabilized, tracking, disturbance torque, and back EMF loops. The disturbance torque loop is caused by wire pulling and friction among rotation joints, which happens in the seeker rotation process. In Figure 2, various types of disturbance torque are equivalent to the transfer function GD(s). The disturbance torque is the main factor leading to problems of the disturbance rejection rate of the gimbaled seeker, including the spring and damping torques. A real gimbaled seeker disturbance torque is a nonlinear model. A simplified model is adopted to analyze the influence of disturbance torque on seeker disturbance rejection rate and output accuracy. The expression is as follows [22]:(2)GDs=Kns+Kω,where Kn is the spring torque coefficient and Kω denotes the damping torque coefficient.The occurrence of the disturbance torque couples the missile and seeker motions, thereby causing DRRPL and reducing the accuracy of seeker control. The DRRTFR(s) is defined as follows:(3)Rs=Δq˙sϑ˙s,where Δq˙(s) is the additional rotation angular rate of the platform caused by the missile motion in relation to the inertial space. From the definition, the smaller the amplitude of R(s), the stronger the seeker’s isolation capability to the disturbance of the missile.In accordance with Figure2, assuming H(s)=1 and ignoring the effect of back EMF, the seeker DRRTF can be educed as(4)Rs=Δq˙sϑ˙s=sGDsLs+RsJsLs+R+GDsLs+R+G2sk2KT+G1sG2sk1k2KTWith the damping torqueGD(s)=Kω and spring torque GD(s)=Kn/s models substituted into (4), the DRRTF models under the effect of the damping and spring torques are shown in (5)Rvs=Δq˙vsϑ˙s=sKωLs+RsJsLs+R+KωLs+R+G2sk2KT+G1sG2sk1k2KT(6)Rss=Δq˙ssϑ˙s=KnLs+RsJsLs+R+G2sk2KT+KnLs+R+G1sG2sk1k2KT


## 2.2. Radome Parasitic Effects in a Stabilized Seeker

The existence of radome affects the seeker measurement on the real target position. Figure3 shows the basic missile–target geometry under the existence of the radome error.Figure 3
Basic geometry for radome analysis.In Figure3, Δq denotes the radome refraction angle, ε∗ is the error angle between the seeker centerline and apparent LOS, and q∗ represents the apparent LOS angle caused by the radome error.The refraction angleΔq varies with the seeker gimbal angle φr. For preliminary analysis, the refraction angle is assumed to be linearly proportional to the gimbal angle(7)Δq=Rdomφr,where Rdom is the radome slope. Rdom is generally regarded as a constant despite being variable during flight [23, 24].In accordance with the angular relationships in Figure3, q∗ can be expressed as follows:(8)q∗=qt+qs-ϑRdom.Considering a stable seeker tracking,ε is generally a small parameter. Thus, qs≈qt. Then, (8) can be rewritten as follows:(9)q∗≅qt+qt-ϑRdom=qt1+R-ϑRdom.Generally speaking,Rdom≪1. Hence, (9) can be simplified into(10)q∗=qt-ϑRdom.Equation (10) shows that missile attitude is fed back to the seeker due to the existence of radome error, thereby generating a measurement error of qt. Consequently, a radome error parasitic loop is formed in the guidance loop.The influence of the correction network on DRRTF characteristics fluctuates on a small scale within the common operating frequencies of the seeker (less than 5 Hz), thereby failing to transform its basic amplitude-phase characteristics. Therefore, the analysis model of the disturbance rejection rate can be simplified to demonstrate intuitively its influence on the guidance and control system. Ignoring the impact of time delay process, the correction network, effect of back EMF, high-frequency dynamics, and small quantity and assumingG1(s)=1, G2(s)=1, R≈1, and L≈0, the equivalent gain of the stabilized loop is determined to be K2=G2k2KT/J. In addition, the equivalent gain of the tracking loop is K1=G1k1. Figure 4 presents the educed seeker model with radome effects.Figure 4
Simplified seeker model with radome effects.Figure4 shows that the DRRTF, including the radome slope, can be educed as (11) with Kr=Rdom/(1-Rdom).(11)Rs=Δq˙sϑ˙s=-Kr+sGDs/K1K21/K1K2s2+GDs+K2/K1K2s+1K 1 K 2 ≫ K n is observed when the time delay process and correction network are ignored. To simplify the analysis, on the basis of the seeker control system performance and ignoring the high-frequency dynamics, the DRRTF under the effect of both radome slope and different types of disturbance torque can be simplified as a first-order model, as shown in Table 1, where Ts is the equivalent time constant.Table 1
First-order DRRTF.
Disturbance type DRRTF Equivalent coefficient Spring torque R s s = K s - K r T s s + 1 K s = K n K 1 K 2 Damping torque R v s = s K v - K r T s s + 1 K v = K ω K 1 K 2


## 2.3. Guidance System Model

Figure5 presents a block diagram of the guidance loop with the disturbance rejection and radome parasitic loop. The noise filter is considered a first-order model, the flight control system a third-order model, and the seeker dynamics a first-order model with the equivalent time constant of Ts=Tg/5.Figure 5
Guidance system model with parasitic effects.In Figure5, at is the target acceleration, am denotes the missile acceleration, Vc represents the closing velocity, Vm stands for the missile velocity, TF-t signifies the time to go, Tg refers to the guidance time constant, Tα indicates the missile turning rate time constant, and N symbolizes the effective navigation ratio of the proportional guidance.The dimensionless method is adopted to reduce the parameter. Assumings=s-/Tg, T-F=TF/Tg, t-=t/Tg, and T-α=Tα/Tg, the equivalent model of the dimensionless guidance system can be obtained as shown in Figure 6.Figure 6
Dimensionless guidance system model.


## 3. Lyapunov Stability of the Guidance System

### 3.1. Uniform Asymptotic Stability of the Guidance System

The guidance system shown in Figure6 is a linear time-varying system. In this study, passivity theorems are used to analyze the Lyapunov stability of the equilibrium point of the system. When studying the Lyapunov stability, we consider the case of zero input. Let the reference input at in Figure 6 be zero, and the guidance system is considered as follows:(12)x˙=ft,x.If(13)ft,0=0,∀t≥0,then the origin is the equilibrium point of (12) at t=0. The equilibrium point x=0 of the system (12) is(i)
uniformly stable if eachξ>0 has δ=δξ>0 independent of t0 such that(14)xt0<δ⟹xt<ξ,∀t≥t0≥0and(ii)
uniformly asymptotically stable if it is uniformly stable and has a positive constantc independent of t0 such that for all xt0<c, xt→0 as t→∞, uniformly in t0; that is, for each η>0, T=Tη>0 is observed, such that(15)xt<η,∀t≥t0+Tη,∀xt0<c.Therefore, if the guidance system is uniformly asymptotically stable at some initial momentt0, then the system has Lyapunov stability at all t0 in the defined interval and the convergence of the system will not deteriorate with an increase in t0.


### 3.2. Lyapunov Stability Represented by Passivity Theorems

Passivity theorems involve the concepts of passivity and positive real. The definitions of passivity and positive real are provided in the following. Proof of the relevant lemma and theorem is presented in [19].


#### 3.2.1. Passivity of a Time-Varying Memoryless Function

When the output of the system is independent of the state variables, the system can be expressed asy=ht,u, that is, a memoryless function. If h is a scalar function that satisfies the inequality(16)αu2≤uht,u≤βu2for all t,u, where α and β are real numbers with β≥α, then h belongs to the sector α,β. If ht,u belongs to the sector 0,∞, that is, uht,u≥0, then the system y=ht,u is passive. If ht,u belongs to the sector 0,∞, then y=ht,u is strictly passive.


#### 3.2.2. Strictly Positive Real Transfer Functions

A single-input–single-output linear system transfer functionGs is strictly positive real function if and only if(1)
G∞>0,(2)
all poles ofGs are in Res<0, and(3)
ReGjω>0, ∀ω>0.The second condition requires all poles ofGs to be in the open left-half complex plane, whereas the third condition requires the Nyquist plot of Gjω to lie in the closed right-half complex plane.Lemma1 provides the relationship between positive real and passivity properties.Lemma 1.
The linear time-invariant minimal realization(17)x˙=Ax+Buy=Cx+Duwith G(s)=CsI-A-1B+D is strictly passive if Gs is strictly positive real function.


#### 3.2.3. Passivity Theorems

Consider the feedback connection of Figure7, where Gs is a time-invariant dynamical system and ht,u is a time-varying memoryless function. Theorem 2 presents the method of analyzing the Lyapunov stability of feedback systems; this method uses the passivity properties of the feedback components.Figure 7
Feedback connection.Theorem 2.
Consider the feedback connection of a strictly passive, time-invariant, dynamical systemGs with a passive, time-varying, memoryless function ht,u. Then, the origin of the closed-loop system is uniformly asymptotically stable.Furthermore, ifGs is a strictly positive real transfer function (from Lemma 1), then the dynamical system is strictly passive. According to Theorem 2, the origin of the closed-loop system is uniformly asymptotically stable.


#### 3.2.4. Loop Transformations

WhenGs or ht,u does not satisfy the conditions of Theorem 2, the feedback system can be transformed into an equivalent feedback system, which satisfies the conditions through loop transformations. Suppose Gs is a time-invariant dynamical system and ht,u is a time-varying memoryless function that belongs to the sector α,β. Then, the feedback system can be reconfigured into a feedback system that is connected by G~s and h~t,u through loop transformations, as shown in Figure 8, where h~t,u is a memoryless function that belongs to the sector 0,∞. We can now apply Theorem 2 if G~s satisfies the conditions of the respective theorem. In this case, the systems shown in Figures 7, 8(a), and 8(b) are equivalent in terms of stability.Figure 8
Bock diagrams of equivalent feedback system.
(a)
Equivalent system (b)
Loop transformations


## 3.1. Uniform Asymptotic Stability of the Guidance System

The guidance system shown in Figure6 is a linear time-varying system. In this study, passivity theorems are used to analyze the Lyapunov stability of the equilibrium point of the system. When studying the Lyapunov stability, we consider the case of zero input. Let the reference input at in Figure 6 be zero, and the guidance system is considered as follows:(12)x˙=ft,x.If(13)ft,0=0,∀t≥0,then the origin is the equilibrium point of (12) at t=0. The equilibrium point x=0 of the system (12) is(i)
uniformly stable if eachξ>0 has δ=δξ>0 independent of t0 such that(14)xt0<δ⟹xt<ξ,∀t≥t0≥0and(ii)
uniformly asymptotically stable if it is uniformly stable and has a positive constantc independent of t0 such that for all xt0<c, xt→0 as t→∞, uniformly in t0; that is, for each η>0, T=Tη>0 is observed, such that(15)xt<η,∀t≥t0+Tη,∀xt0<c.Therefore, if the guidance system is uniformly asymptotically stable at some initial momentt0, then the system has Lyapunov stability at all t0 in the defined interval and the convergence of the system will not deteriorate with an increase in t0.


## 3.2. Lyapunov Stability Represented by Passivity Theorems

Passivity theorems involve the concepts of passivity and positive real. The definitions of passivity and positive real are provided in the following. Proof of the relevant lemma and theorem is presented in [19].


### 3.2.1. Passivity of a Time-Varying Memoryless Function

When the output of the system is independent of the state variables, the system can be expressed asy=ht,u, that is, a memoryless function. If h is a scalar function that satisfies the inequality(16)αu2≤uht,u≤βu2for all t,u, where α and β are real numbers with β≥α, then h belongs to the sector α,β. If ht,u belongs to the sector 0,∞, that is, uht,u≥0, then the system y=ht,u is passive. If ht,u belongs to the sector 0,∞, then y=ht,u is strictly passive.


### 3.2.2. Strictly Positive Real Transfer Functions

A single-input–single-output linear system transfer functionGs is strictly positive real function if and only if(1)
G∞>0,(2)
all poles ofGs are in Res<0, and(3)
ReGjω>0, ∀ω>0.The second condition requires all poles ofGs to be in the open left-half complex plane, whereas the third condition requires the Nyquist plot of Gjω to lie in the closed right-half complex plane.Lemma1 provides the relationship between positive real and passivity properties.Lemma 1.
The linear time-invariant minimal realization(17)x˙=Ax+Buy=Cx+Duwith G(s)=CsI-A-1B+D is strictly passive if Gs is strictly positive real function.


### 3.2.3. Passivity Theorems

Consider the feedback connection of Figure7, where Gs is a time-invariant dynamical system and ht,u is a time-varying memoryless function. Theorem 2 presents the method of analyzing the Lyapunov stability of feedback systems; this method uses the passivity properties of the feedback components.Figure 7
Feedback connection.Theorem 2.
Consider the feedback connection of a strictly passive, time-invariant, dynamical systemGs with a passive, time-varying, memoryless function ht,u. Then, the origin of the closed-loop system is uniformly asymptotically stable.Furthermore, ifGs is a strictly positive real transfer function (from Lemma 1), then the dynamical system is strictly passive. According to Theorem 2, the origin of the closed-loop system is uniformly asymptotically stable.


### 3.2.4. Loop Transformations

WhenGs or ht,u does not satisfy the conditions of Theorem 2, the feedback system can be transformed into an equivalent feedback system, which satisfies the conditions through loop transformations. Suppose Gs is a time-invariant dynamical system and ht,u is a time-varying memoryless function that belongs to the sector α,β. Then, the feedback system can be reconfigured into a feedback system that is connected by G~s and h~t,u through loop transformations, as shown in Figure 8, where h~t,u is a memoryless function that belongs to the sector 0,∞. We can now apply Theorem 2 if G~s satisfies the conditions of the respective theorem. In this case, the systems shown in Figures 7, 8(a), and 8(b) are equivalent in terms of stability.Figure 8
Bock diagrams of equivalent feedback system.
(a)
Equivalent system (b)
Loop transformations


## 3.2.1. Passivity of a Time-Varying Memoryless Function

When the output of the system is independent of the state variables, the system can be expressed asy=ht,u, that is, a memoryless function. If h is a scalar function that satisfies the inequality(16)αu2≤uht,u≤βu2for all t,u, where α and β are real numbers with β≥α, then h belongs to the sector α,β. If ht,u belongs to the sector 0,∞, that is, uht,u≥0, then the system y=ht,u is passive. If ht,u belongs to the sector 0,∞, then y=ht,u is strictly passive.


## 3.2.2. Strictly Positive Real Transfer Functions

A single-input–single-output linear system transfer functionGs is strictly positive real function if and only if(1)
G∞>0,(2)
all poles ofGs are in Res<0, and(3)
ReGjω>0, ∀ω>0.The second condition requires all poles ofGs to be in the open left-half complex plane, whereas the third condition requires the Nyquist plot of Gjω to lie in the closed right-half complex plane.Lemma1 provides the relationship between positive real and passivity properties.Lemma 1.
The linear time-invariant minimal realization(17)x˙=Ax+Buy=Cx+Duwith G(s)=CsI-A-1B+D is strictly passive if Gs is strictly positive real function.


## 3.2.3. Passivity Theorems

Consider the feedback connection of Figure7, where Gs is a time-invariant dynamical system and ht,u is a time-varying memoryless function. Theorem 2 presents the method of analyzing the Lyapunov stability of feedback systems; this method uses the passivity properties of the feedback components.Figure 7
Feedback connection.Theorem 2.
Consider the feedback connection of a strictly passive, time-invariant, dynamical systemGs with a passive, time-varying, memoryless function ht,u. Then, the origin of the closed-loop system is uniformly asymptotically stable.Furthermore, ifGs is a strictly positive real transfer function (from Lemma 1), then the dynamical system is strictly passive. According to Theorem 2, the origin of the closed-loop system is uniformly asymptotically stable.


## 3.2.4. Loop Transformations

WhenGs or ht,u does not satisfy the conditions of Theorem 2, the feedback system can be transformed into an equivalent feedback system, which satisfies the conditions through loop transformations. Suppose Gs is a time-invariant dynamical system and ht,u is a time-varying memoryless function that belongs to the sector α,β. Then, the feedback system can be reconfigured into a feedback system that is connected by G~s and h~t,u through loop transformations, as shown in Figure 8, where h~t,u is a memoryless function that belongs to the sector 0,∞. We can now apply Theorem 2 if G~s satisfies the conditions of the respective theorem. In this case, the systems shown in Figures 7, 8(a), and 8(b) are equivalent in terms of stability.Figure 8
Bock diagrams of equivalent feedback system.
(a)
Equivalent system (b)
Loop transformations


## 4. Stability Analysis of the Guidance System

The guidance system shown in Figure6 is transformed into a feedback system, as shown in Figure 7, wherein the linear time-invariant system transfer function is as follows:(18)Gs-=Ns-s-/5+14-NVc/VmRs-T-αs-+1,whereas the time-varying memoryless function is as follows:(19)ht-,u=1T-F-t-.Lett-k be any moment of the terminal guidance. We discuss the stability of guidance systems in the bounded closed domain t-∈0,t-k. Then, ht-,u belongs to the sector 1/T-F,1/T-F-t-k and is not passive. With the use of transformations to reconfigure the system into the form shown in Figure 8(b), where h~t-,u is a time-varying memoryless function that belongs to the sector 0,∞, Gs- is transformed into(20)G~s-=s-s-/5+14-NVc/VmRs-T-αs-+1+N/T-F-t-ks-s-/5+14-NVc/VmRs-T-αs-+1+N/T-F.In accordance with Theorem2, if G~s- is strictly positive real, then the feedback system satisfies the Lyapunov stability condition; that is, the origin of the original PN guidance system is uniformly asymptotically stable over the t-∈0,t-k interval. The following are three conditions for G~s- to be strictly positive real:(i)
Substituting the DRRTF in Table1 into (14) shows that G~∞=1>0 for each DRRTF. Therefore, the first condition is satisfied.(ii)
Expanding the characteristic equation ofG~s- into the following form:(21)ansn+an-1sn-1+⋯+a2s2+a1s1+a0=0Then, for the second condition, the Routh criterion can be used to obtain the constraint inequality that the eigenvalues are all on the open left-half complex plane.(iii)
The third condition holds when the Nyquist plot ofG~jω- lies in the closed right-half complex plane.The previous analysis shows that the parameters that affect the stability of the PN guidance system includeN, Vc/Vm, T-α, T-F, t-k, and Rs-.


## 5. Simulations and Analysis

This section analyzes the influence of different DRRTF models caused by disturbance torque and radome error on the stability of the guidance system through simulations. The typical PN guidance system parameters (N=4, Vc/Vm=1.5, T-α=3, T-F=10, and Tg=0.5), equivalent gains (K1=10, K2=10), and disturbance torque coefficients (Kn=0.001, Kω=0.001) are selected. In accordance with the Routh criterion, the second condition under the abovementioned parameters is satisfied. To examine the third condition, take Kr equal to -0.03 and 0.03, respectively, and the Nyquist plots of G~jω- corresponding to different DRRTF models and t-k are presented in Figure 9.Figure 9
Nyquist plots.
(a)
Spring torque (b)
Damping torqueFigure9 illustrates that the critical stability time t-kc for different DRRTF models is different under the same conditions. Regardless of the disturbance torque type, the critical stability time of the system when Kr=0.03 is greater than that when Kr=-0.03. Therefore, when Kr=-0.03, the constraint on the terminal guidance time when the guidance system satisfies the Lyapunov stability conditions is stricter than that when Kr=0.03.t - u n s = T - F - t - k indicates unstable flight time. Figure 10 shows the uniformly asymptotically stable boundary of the system determined by Ks or Kv and t-uns under different T-F when N=4 and T-α=3. The regions above and below the curve are stable and unstable, respectively.Figure 10
Stable boundary corresponding to differentT-F.
(a)
Spring torque (b)
Damping torqueFigure10 illustrates that different T-F have little effect on the unstable flight time of the guidance system. The time of instability can be delayed by extending the terminal guidance time. Under the action of spring torque, the unstable flight time t-uns increases rapidly with the increase of Ks. When Ks increases to a certain value, the guidance system cannot meet the stability conditions during the entire terminal guidance time. Moreover, when Kr=-0.03, the unstable flight time is higher and the stability regions are smaller than those when Kr=0.03. The change law of the stable boundary of the guidance system under the action of the damping moment is similar to that of the spring torque. Only when Kr=0.03, the unstable flight time decreases slightly and then increases rapidly with the increase of Kv.Figure11 shows the uniformly asymptotically stable boundary of the system determined by Kr and t-uns under different disturbances when T-F=10.Figure 11
Stable boundary corresponding to differentKr.Figure11 illustrates that when Kr<0, the unstable flight time is considerably larger than that when Kr>0 and increases sharply with the increase of Kr. When Kr<0, the tolerance of the guidance system to the radome error is approximately 3%, which is less than that when Kr>0 (higher than 8%). When Kr>0, the unstable flight time decreases slowly as Kr increases, which is beneficial to the guidance system. However, when Kr exceeds a certain limit, the guidance system immediately becomes unstable, indicating that the tolerance of the guidance system to the radome error is also limited.Figure12 presents the uniformly asymptotically stable boundary of the system determined by T-α and Ks or Kv under different Kr when T-F=10 and t-uns=7.Figure 12
Stable boundary corresponding to differentT-α.
(a)
Spring torque (b)
Damping torqueFigure12 shows that the stability regions of the guidance system decrease with the increase of T-α. T-α is related to the altitude of the missile’s flight, and T-α is relatively large at high altitude. If Tg is small, then the guidance system will likely lose stability when the missile is flying at a high altitude. Therefore, the decelerating response of the guidance system in high airspace should be given considerable attention. Moreover, under the action of spring torque, the stability regions when Kr<0 are smaller and the stability regions when Kr>0 are larger than that when Kr=0. Under the action of damping torque, positive and negative Kr lead to a stability regions reduction. When Kr decreases to a certain value and T-α is large, the guidance system will not be stable. Therefore, the disturbance rejection rate index should be strictly constrained in engineering practice.From the above analysis, the stability regions for a negative slope are smaller than that for a positive slope, and lager values ofT-α lead to a smaller range of stability. These trends are consistent with those obtained in [8] using the Routh criterion. However, there will be differences in the stability margin.


## 6. Conclusions

The influence of the disturbance rejection rate and radome error slope on the Lyapunov stability of the PN guidance system is analyzed by passivity theorems. Results of the simulation show that different DRRTF models have different effects on the stability of the guidance system. The stable boundary and tolerance of the PN guidance system to the spring torque, damping torque, and radome slope differ significantly. The unstable flight time caused by negative radome slope is higher than that due to positive radome slope. The constraint of disturbance rejection rate index when the radome slope is negative and is stricter than that when radome slope is positive. These results provide a theoretical basis for the overall design and engineering application of guidance systems and radar seekers.



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*Source: 1890426-2018-06-20.xml*