#### Abstract

Aiming at the problem of impact angle constraint and input saturation, an integrated guidance and control (IGC) algorithm with impact angle constraint and input saturation is proposed. A three-channel independent design model of missile IGC with impact angle constraint is established, and an extended state observer with fast finite-time convergence is designed to estimate and compensate model errors and coupling relationship between channels. Based on the nonsingular terminal sliding mode control and backstepping control, the IGC three-channel independent design is completed. Nussbaum function and an auxiliary system are introduced to deal with the input saturation. The Lyapunov function is constructed to prove the finite-time convergence of the IGC algorithm. The missile six-degree-of-freedom simulation results show the effectiveness and superiority of the IGC algorithm.

#### 1. Introduction

The traditional design of missile guidance and control system is based on the spectrum separation, the guidance system and control system are designed separately, and the coupling relationship between them is not considered. At the end of the guidance phase, due to the drastic change of the relative motion between the missile and the target, it is difficult for the control system to meet the requirement of fast response of the guidance system. If the velocity and acceleration of the target are large, this design method will lead to system instability and large miss distance [1]. In order to solve this problem, the concept of integrated guidance and control (IGC) design is proposed. The main idea is to design the guidance system and the control system together and consider the coupling relationship between them. The control instruction is generated directly from the relative motion information of the missile and the target [2]. This algorithm makes full use of the comprehensive information such as line-of-sight (LOS) angle, altitude, overload, and so on and can effectively improve the guidance and control performance.

In recent years, many literatures have carried out IGC design, using various control technologies for integrated design, such as small gain theory [3], *θ*-D method [4], sliding mode control [5], adaptive control [6], and so on. In [7], an IGC model is established. The target maneuvering, model nonlinearity, and system disturbance are regarded as the total disturbances. The second-order disturbance observer is used to estimate the disturbances, and the sliding mode control is used for IGC design. The stability of the system is proved by the Lyapunov stability theory.

Because the IGC model has strict feedback form, the backstepping control is widely used in the IGC design. In [8], the nonlinear model of guidance and control system in longitudinal plane is established. The IGC design is carried out based on backstepping control technology, and the adaptive law is used to estimate the uncertainty of the model. However, the backstepping control needs to derive the virtual control quantity, and there exists the problem of exponential expansion. In order to solve this problem, dynamic surface control is used in the IGC design [9, 10], and the problem of multiple derivation of virtual control quantity is solved by introducing first-order filter. In [11], the IGC design is completed by combining dynamic surface control, neural network disturbance observer, and barrier Lyapunov function.

In order to increase the damage efficiency of warhead, many missiles (such as some anti–ship missiles, anti–tank missiles, air defense missiles, etc.) need to hit the target with impact angle constraints [12]. Therefore, the impact angle constraint needs to be considered in IGC design. Based on sliding mode control and dynamic inverse control, a robust dynamic inverse design method is proposed to compensate the system uncertainty in [13]. Then, this method is combined with dynamic surface control to complete the three-dimensional IGC design with impact angle constraint. In order to avoid the high-frequency buffeting, the continuous approximate function is used to replace the symbolic function, but the inherent structure of sliding mode control is changed so that the robustness of the system is reduced. In [14], adaptive control is used to compensate the system interference, and combined with dynamic surface control, the IGC design with impact angle constraint is completed. The traditional IGC design based on dynamic surface control can only guarantee the gradual convergence of the system states to the expected value, and the rapidity of IGC algorithm is reduced due to the introduction of first-order filter.

Due to the physical constraints of the missile actuator, the large maneuvering of the target, the internal disturbance of the system, and the external disturbance, it is easy to increase the amplitude of the control quantity, which may reach the upper bound of the actuator constraint and lead to the control input saturation. The input saturation problem may lead to the degradation of control performance and even some unpredictable results. Therefore, it is necessary to consider the problem of input saturation in IGC design. In order to solve the input saturation of nonlinear systems, Chen et al. [15] regard the part beyond the control quantity as external disturbance and use the finite-time convergent disturbance observer to estimate and compensate the disturbance. Then, a robust tracking controller is designed based on terminal sliding mode control. Liang et al. [16] introduce an auxiliary system into the IGC model in longitudinal channel and employ the Nussbaum function to deal with the input saturation problem. Based on the dynamic surface control, they complete IGC design in longitudinal channel. For the input saturation problem, an improved saturation function and an auxiliary system are used in [17], and the auxiliary system states are applied to the design of integrated control law and stability analysis. At the same time, aiming at the target maneuvering, external disturbance, and model uncertainty, a finite-time convergent disturbance observer is introduced to estimate and compensate.

On the basis of the above analysis and considering the impact angle constraint, input saturation, and model uncertainty, a finite-time convergent IGC algorithm against maneuvering target is presented for STT missile in this paper. The main contributions of this paper are as follows:(1)An IGC design model considering impact angle constraint, model uncertainty, and input saturation is established by combining the missile dynamics model and motion kinematics between the missile and target.(2)An extended state observer (ESO) with finite-time convergence is improved to estimate and compensate the disturbances caused by target maneuvering, system disturbance, and model uncertainty.(3)Based on the improved ESO and backstepping control, a novel IGC algorithm with impact angle constraint is proposed, and the Nussbaum function is used to deal with the input saturation.(4)Based on Lyapunov stability theory and finite-time stability theory, the stability and global finite-time convergence of the proposed IGC algorithm are proved.

#### 2. Problem Formulation and Preliminaries

In this section, the IGC model is presented and the related definition and lemmas are given. The IGC model is established based on the missile dynamics model and motion kinematics between the missile and target. The related definition and lemmas are presented to facilitate the IGC law design and stability analysis.

##### 2.1. Problem Formulation

The relative motion model of the missile and target is established in the three-dimensional inertial coordinate system, as shown in Figure 1. *Oxyz* is the inertial coordinate system; *Ox*_{4}*y*_{4}*z*_{4} is the LOS coordinate system. *M* and *T* represent the missile and target, respectively. and are the LOS elevation angle and azimuth angle, respectively, and is the relative distance between the missile and target. is the projection of on the horizontal plane of the inertial coordinate system; that is, .

The relative motion equation of missile and target in three dimension can be described as [18]where and are the acceleration vectors in LOS coordinate system of missile and target, respectively.

Let be the acceleration vector of missile in ballistic coordinate system. According to the missile dynamic equation, we can obtain and aswhere is the missile mass, is the gravity acceleration, and is the missile trajectory inclination angle. and are lift and lateral forces, which are given aswhere and are attack angle and sideslip angle of missile, respectively, and are lift and lateral force produced by rudder deflection angle, respectively, is the derivative of the lift force coefficient with respect to , is the derivative of the lateral force coefficient with respect to , is the aerodynamics reference area, is the dynamic pressure, and and are unknown and bounded uncertain terms.

Invoking (1) and (2), the relative motion equation is established aswithwhere and are unknown and bounded uncertainties.

*Remark 1. *, , , , , , , and are all bounded. Therefore, it can be known from (5) that and are bounded.

The missile dynamics model can be established as [19]where and are pitch and roll angle, respectively; , , and are roll, yaw, and pitch moments of inertia, respectively; , , and are roll, yaw, and pitch angular rates, respectively; and are the derivatives of roll and pitch moment coefficients with respect to , respectively; and are the derivatives of roll and pitch moment coefficients with respect to , respectively; , , and are the derivatives of roll, yaw, and pitch moment coefficients with respect to , , and , respectively; and , , , , , and are external disturbance and modeling errors.

According to the principle of considering the main factors and treating the secondary factors as uncertainty, the missile dynamics model (6) is simplified towithTaking the longitudinal plane as an example, the missile impact angle is defined as the included angle between the missile and the target velocity vectors at the engagement time, and the impact angle corresponds to the terminal LOS angle as follows [20, 21]:So, the impact angle constraint can be transformed into the terminal LOS angle constraint.

Let , , , and , where is the expected impact LOS elevation angle at the engagement time. The control model of the IGC system in the pitch channel with impact angle constraint can be obtained by combining (4) and (7) as follows:withDue to the large maneuver of the target, the uncertainty of the missile itself, and external interference, the amplitude of the missile control command may become larger, reaching the upper limit of the actuator constraint, and the control quantity may be saturated. The input saturation problem will make the dynamic quality of the guidance system worse, lead to the control performance decline and even destroy the system stability, and lead to system crash. Therefore, it is necessary to consider the saturation of actuators in the IGC design. Considering the input saturation, the control model of the IGC system in the pitch channel (10) can be rewritten aswhere is the actual pitch rudder deflection angle. is defined aswhere is the known upper bound of , that is, the maximum rudder deflection angle.

Considering the actuator saturation, is not differentiable at . In order to enable the backstepping design method to be applied to IGC design, the following hyperbolic tangent function is used to smooth the saturation function [22] as follows:Let ; thenso is bounded. Combined with (12), (14), and (15), the control model of the IGC system in the pitch channel considering input saturation iswhere .

Similar to the control model of the IGC system in the pitch channel, the control model in yaw channel can be established by combining (4) and (7) as follows:withwhere is the expected LOS azimuth angle.

According to (7), the control model in the roll channel can be established aswith

##### 2.2. Preliminaries

For the convenience of the following analysis, the relative definitions and lemmas are introduced in this section.

*Definition 1. *For easy writing, define *,* where is symbolic function, and *.*

Lemma 1 (see [23]). *Assume that there is a Lyapunov function satisfying for any , , ; then the origin of the system is finite-time convergent, and the convergence time satisfies*

Lemma 2 (see [24]). *Assume that a Lyapunov function satisfying for any , , , and then the system states are stable in finite-time, and the convergence time satisfieswhere , . Convergence domain satisfies*

Lemma 3 (see [25]). *For any real number , , there exists a real number such that the following inequality holds:*

Lemma 4 (see [26]). *Suppose that and are smooth functions defined in , and , is a Nussbaum gain function, if the inequality is satisfiedwhere , , , and ; then and are bounded in .*

#### 3. Fast Finite-Time Disturbance Observer

Extended state observer is an effective method to estimate system uncertainties and external disturbances. It regards system uncertainties and external disturbances as total disturbance and extends them to a new variable. Then an observer is designed to estimate the disturbance. ESO has little dependence on the model and can be calculated in real time, so it is widely used. However, the traditional ESO regards the total disturbance as a constant or a slowly varying quantity, so the derivative of the total disturbance is zero. This method hinders the further improvement of ESO. For time-varying disturbance, by increasing the ESO gain, the satisfactory disturbance estimation effect can be obtained. However, when there is measurement noise in the output of the system, high gain will lead to the amplification of measurement noise. Based on the design idea of ESO, a fast finite-time convergent ESO is improved by using the supertwisting algorithm [27], which has the advantages of ESO and sliding mode control. In addition, the supertwisting algorithm can reduce the chattering phenomenon of sliding mode control. The improved ESO can not only estimate the total disturbance, but also has robustness and finite-time convergence.

Consider the following first-order nonlinear system:where is the system input, and is the total disturbance of the system.

*Assumption 1. * is continuously differentiable, and its differential satisfies .

Based on the supertwisting algorithm, a fast finite-time convergent ESO is improved for system (27) as follows:where , , , .

Invoking (27) and (28), we havewhere .

*Remark 2. *Supertwisting algorithm can greatly weaken chattering and has strong robustness and high-precision control performance. However, the algorithm has shortcomings as follows: (1) the control law is continuous function, but not smooth function, affecting control performance; (2) when system states are far from the equilibrium point, the convergence rate is slow. Aiming at these disadvantages, a linear term is introduced to accelerate convergence speed and ensure smoothness of function. From (29), we can see that the linear term in (28) will accelerate convergence speed when system states are far away from equilibrium point. When system approaches equilibrium point, the nonlinear term plays a major role and accelerates convergence speed of system. Therefore, compared with traditional supertwisting ESO, the fast finite-time convergent ESO (28) has faster convergence speed.

Proposition 1. *For system (27) with Assumption 1, the ESO (28) is employed to estimate the disturbance of the system. Its estimation error can finite-time converge to the region as follows:*

*Proof. *Define a new vector asDifferentiating with respect to time giveswithIt is easy to prove that is a Hurwitz matrix, so there exists a positive definite symmetric matrix satisfying , where is a symmetric positive definite matrix.

Construct the Lyapunov function asSince is a positive definite matrix, we haveDifferentiating with respect to time and substituting (32) into it givesNoting that and , invoking (35) and (36) yieldswhere , , .

It can be obtained from Lemma 1 that the proposed ESO (28) is stable, and the observer errors and will converge to a small neighborhood of zero in a finite time, and the convergence time satisfieswhere , . The convergence domain satisfiesThe proof is completed.

Taking the pitch channel as an example, the proposed ESO (28) is used to estimate the uncertainties in the model (12) as follows:where , , , is the estimation of , , and is the estimation of , .

#### 4. IGC Design with Input Saturation

##### 4.1. Design of Pitch Channel Controller and Finite-Time Convergence Analysis

In order to solve the nonlinear saturation problem, the following auxiliary systems are introduced as

Then, the control model of the IGC system in the pitch channel with input saturation and impact angle constraint is obtained as

The model (42) has a strict feedback structure, so the backstepping method combined with terminal sliding mode control can be used to design the controller. The specific design steps of the IGC algorithm are given below: *Step 1*. Select the following nonsingular terminal sliding surface with impact angle constraint [28] as with where , , , and ; and are positive odd numbers. Differentiating with respect to time gives with Substituting (42) into (45), we have The virtual control law is designed as A new virtual control law is introduced to avoid the differential expansion problem caused by the derivation of the control law and to ensure the fast convergence of the system in finite time. is generated by . A low-pass filter is designed to generate and its derivatives as where , .

*Remark 3. *The dynamic surface control used a first-order linear filter to solve the differential expansion problem of backstepping control. The performance of the filter is related to . Reducing can improve the convergence speed and filtering accuracy, but too large will amplify the measurement noise. The filter (49) can improve the convergence speed and filtering accuracy by reducing and , which reduces the selection requirement of , and can meet the requirements of finite-time convergence. *Step 2*. Define the sliding mode surface as Differentiating with respect to time gives The virtual control law is designed as Design the first-order nonlinear filter as where , . *Step 3*. Define the sliding mode surface as Differentiating with respect to time gives The virtual control law is designed as Design the first-order nonlinear filter as where , . *Step 4*. Define the sliding mode surface asDifferentiating with respect to time giveswhere .

The actual control law is designed aswithwhere is the design parameter.

Proposition 2. *For system (17), the designed IGC law (60) can make the states of the pitch channel closed-loop system converge to a smaller neighborhood of the origin in finite time.*

*Proof. *From (43) to (60), we haveThere exists a positive real number such that [29], so we can rewrite (67) asConstruct the Lyapunov function asDifferentiating with respect to time givesBased on Young’s inequality, the inequality (70) can be rewritten aswithLetwhere ; we can obtainAccording to Lemma 3, we havewhere .

Due to , the following inequality holds:Solving the inequality (76) results inAccording to Lemma 4, it can be seen that and are bounded, and further, , , , , , , , and are bounded. The following two forms can be obtained from (75) asFor (78), if , according to Lemma 1, is finite-time convergent, and the convergence region isFor (79), similar to the analysis of (78), can finite-time converge to the regionInvoking (80) and (81), it is obtained that can finite-time converge to the region . Therefore, can finite-time converge to the region , and the convergence regions of and are discussed as follows:(1)If , we can rewrite (43) as Construct the Lyapunov function as Differentiating with respect to time and substituting (82) into it gives From Lemma 2, it can be seen that can converge to region in finite time, where , . By choosing parameters and reasonably, can be achieved. At this time, the convergence domain of is Combining (82), the convergence domain of is(2)If , we can rewrite (43) asIn summary, and can converge to a small neighborhood of the origin in finite time, and the neighborhood isThe proof is complete.

##### 4.2. Design of Yaw and Roll Channel Controller

Referring to the design process of the pitch channel, the integrated guidance control law of the yaw channel is designed as