Integrated Guidance and Control with Input Saturation and Impact Angle Constraint

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.&e Lyapunov function is constructed to prove the finite-time convergence of the IGC algorithm.&e missile six-degree-of-freedom simulation results show the effectiveness and superiority of the IGC algorithm.


Introduction
e 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. e main idea is to design the guidance system and the control system together and consider the coupling relationship between them. e control instruction is generated directly from the relative motion information of the missile and the target [2]. is 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. e target maneuvering, model nonlinearity, and system disturbance are regarded as the total disturbances. e second-order disturbance observer is used to estimate the disturbances, and the sliding mode control is used for IGC design. e 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. e 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]. erefore, 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]. en, this method is combined with dynamic surface control to complete the threedimensional 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. e 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. e input saturation problem may lead to the degradation of control performance and even some unpredictable results. erefore, 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.
en, 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. e 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.

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

Problem Formulation.
e 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. ε and η are the LOS elevation angle and azimuth angle, respectively, and R is the relative distance between the missile and target. r is the projection of R on the horizontal plane of the inertial coordinate system; that is, r � R cos ε. e relative motion equation of missile and target in three dimension can be described as [18] 2 _ R_ ε + R€ ε + R _ η 2 sin ε cos ε � a ty 4 − a my 4 , where m is the missile mass, g is the gravity acceleration, and θ m is the missile trajectory inclination angle. Y and Z are lift and lateral forces, which are given as 2 Discrete Dynamics in Nature and Society where α and β are attack angle and sideslip angle of missile, respectively, Y δ z and Z δ y are lift and lateral force produced by rudder deflection angle, respectively, C α y is the derivative of the lift force coefficient with respect to α, C β z is the derivative of the lateral force coefficient with respect to β, S is the aerodynamics reference area, Q � ρV 2 m /2 is the dynamic pressure, and d Y and d Z are unknown and bounded uncertain terms.
Invoking (1) and (2), the relative motion equation is established as where d ε and d η are unknown and bounded uncertainties.
where ϑ and c are pitch and roll angle, respectively; J x 1 , J y 1 , and J z 1 are roll, yaw, and pitch moments of inertia, respectively; ω x 1 , ω y 1 , and ω z 1 are roll, yaw, and pitch angular rates, respectively; m α x 1 and m α z 1 are the derivatives of roll and pitch moment coefficients with respect to α, respectively; m Discrete Dynamics in Nature and Society coefficients with respect to δ x , δ y , and δ z , respectively; and Δ α , Δ β , Δ c , Δ ω x 1 , Δ ω y 1 , and Δ ω z 1 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 to Taking the longitudinal plane as an example, the missile impact angle θ d 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 where ε d is the expected impact LOS elevation angle at the engagement time. e control model of the IGC system in the pitch channel with impact angle constraint can be obtained by combining (4) and (7) as follows: Due 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. e 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. erefore, 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 as where sat(δ z ) is the actual pitch rudder deflection angle. sat(δ z ) is defined as where δ z max is the known upper bound of δ z , that is, the maximum rudder deflection angle.
Considering the actuator saturation, sat(δ z ) is not differentiable at |δ z | � δ z max . 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 so d z is bounded. Combined with (12), (14), and (15), the control model of the IGC system in the pitch channel considering input saturation is 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: with Discrete Dynamics in Nature and Society where η d is the expected LOS azimuth angle. According to (7), the control model in the roll channel can be established as

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

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. en 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. is 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. e 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 u is the system input, and d is the total disturbance of the system. Assumption 1. d is continuously differentiable, and its differential satisfies | _ d| ≤ ς. Based on the supertwisting algorithm, a fast finite-time convergent ESO is improved for system (27) as follows: Invoking (27) and (28), we have 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 e 1 in (28) will accelerate convergence speed when system states are far away from equilibrium point. When system approaches equilibrium point, the nonlinear term [e 1 ] a plays a major role and accelerates convergence speed of system. erefore, compared with traditional supertwisting ESO, the fast finitetime 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 as Differentiating e with respect to time gives with It is easy to prove that A is a Hurwitz matrix, so there exists a positive definite symmetric matrix P satisfying A T P + PA � Q, where Q is a symmetric positive definite matrix.
Construct the Lyapunov function as Since P is a positive definite matrix, we have Differentiating V 1 with respect to time and substituting (32) into it gives Noting that |e 1 | ≤ ‖e‖ 1/a and ‖ς‖ � | _ d| ≤ ς, invoking (35) and (36) yields where min (P)‖P‖. It can be obtained from Lemma 1 that the proposed ESO (28) is stable, and the observer errors e 1 and e 2 will converge to a small neighborhood of zero in a finite time, and the convergence time satisfies where θ 1 ∈ (0, λ 1 ), θ 2 ∈ (0, λ 2 ). e convergence domain satisfies e 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: Discrete Dynamics in Nature and Society _ x 12 � a 22 x 12 + a 23 x 13 + k 11 x 12 a + x 12 ,

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 en, the control model of the IGC system in the pitch channel with input saturation and impact angle constraint is obtained as e model (42) has a strict feedback structure, so the backstepping method combined with terminal sliding mode control can be used to design the controller. e 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 λ 1 � (3 − b)σ b− 1 /2, λ 2 � (b − 1)σ b− 3 /2, σ > 0, and 0 < b � p 1 /p 2 < 1; p 1 and p 2 are positive odd numbers. Differentiating s 11 with respect to time gives e virtual control law x 13c is designed as A new virtual control law x 13d 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. x 13d is generated by x 13c . A low-pass filter is designed to generate x 13d and its derivatives _ x 13d as where y 11 � x 13d − x 13c , x 13d (0) � x 13c (0).

Remark 3.
e dynamic surface control used a first-order linear filter τ 1 _ x 13d � y 1 to solve the differential expansion problem of backstepping control. e performance of the filter is related to τ 1 . Reducing τ 1 can improve the convergence speed and filtering accuracy, but too large will amplify the measurement noise. e filter (49) can improve the convergence speed and filtering accuracy by reducing τ 1 and c, which reduces the selection requirement of τ 1 , and can meet the requirements of finite-time convergence.
Step 2. Define the sliding mode surface as Differentiating s 12 with respect to time gives Design the first-order nonlinear filter as where Step 3. Define the sliding mode surface as Differentiating s 13 with respect to time gives e virtual control law x 15c is designed as Design the first-order nonlinear filter as where y 13 Step 4. Define the sliding mode surface as Differentiating s 14 with respect to time gives where g ′ (δ z ) � ((zg(δ z ))/zδ z ) � (4/(e (δ z /δ z max ) + e (δ z /δ zmax ) ) 2 ). e actual control law δ zc is designed as where c χ > 0 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 have _ s 11 � a 22 x 12 + a 23 s 12 + y 11 + x 13c + d ε + ρ 1 + ρ 2 φ ′ x 11 x 12 � − l 11 s 11 − l 12 s 11 c + d ε + a 23 s 12 + y 11 , _ s 12 � s 13 + y 12 + x 14c + a 33 x 13 Discrete Dynamics in Nature and Society ere exists a positive real number ξ 1i such that | _ x 1ic | ≤ ξ 1i [29], so we can rewrite (67) as Construct the Lyapunov function as Differentiating V 2 with respect to time gives 1+c + a 23 s 11 y 11 + s 12 y 21 + bs 13 y 31 Based on Young's inequality, the inequality (70) can be rewritten as with where k > 0; we can obtain According to Lemma 3, we have where ζ � ζ 1 + ζ 2 . Due to V 2 ≥ 0, the following inequality holds: Solving the inequality (76) results in According to Lemma 4, it can be seen that V 2 and χ 1 are bounded, and further, s 1 , s 2 , s 3 , s 4 , y 1 , y 2 , y 3 , and ζ are bounded. e following two forms can be obtained from (75) as For (78), if k − ζ/V 2 > 0, according to Lemma 1, V 2 is finite-time convergent, and the convergence region is For (79), similar to the analysis of (78), V 2 can finite-time converge to the region Invoking (80) and (81), it is obtained that V 2 can finitetime converge to the region Δ � min(Δ 1 , Δ 2 ). erefore, s 11 can finite-time converge to the region |s 11 | ≤ � � Δ √ � ϕ, and the convergence regions of x 11 and x 12 are discussed as follows: (1) If |x 1 | ≥ σ, we can rewrite (43) as Construct the Lyapunov function as Differentiating V 2 with respect to time and substituting (82) into it gives From Lemma 2, it can be seen that x 1 can converge to region Ω 1 � |x 11 | � ϑ 1 | θ 1 ϑ b 1 + θ 2 ϑ 1− b 1 < 2ϕ in finite time, where θ 1 ∈ (0, 2ρ 2 ), θ 2 ∈ (0, 2ρ 1 ). By choosing parameters k 1 and k 2 reasonably, ϑ 1 ≤ δ can be achieved. At this time, the convergence domain of x 1 is Combining (82), the convergence domain of x 12 is (2) If |x 1 | ≤ σ, we can rewrite (43) as In summary, x 1 and x 12 can converge to a small neighborhood of the origin in finite time, and the neighborhood is e proof is complete.

Simulation Analysis
In this section, the performance of the proposed integrated guidance and control design method (PIGC) is verified based on the missile six-degree-of-freedom simulation. In the inertial coordinate system, the initial missile and target positions are set to (0 m, 5000 m, 0 m) and (8000 m, 1000 m, 500 m), and the speed of the missile is v m � 800 m/s. e missile's initial ballistic inclination angle, ballistic deflection angle, angle of attack, sideslip angle, roll angle, pitch angle rate, yaw angle rate, and roll angle rate are θ m0 � − 10°, ψ vm0 � − 5°, α 0 � 10°, β 0 � 5°, ω x10 � 20°/s, ω y10 � − 20°/s, and ω z10 � 30°/s. e termination condition of the simulation is that y of the missile in the inertial coordinate system is less than or equal to 0. e simulation step size is 0.01 s, and the fourth-order Runge-Kutta method is used to solve the model. e controller parameters are selected as ρ 1 � 0.6, ρ 2 � 1, b � 3/5, c � 0.5, l 11 � l 12 � 3, l 21   It can be seen from Figure 2 that, for different expected LOS angles, the missile can effectively hit the target, and as the initial LOS angle deviation increases, the ballistic curvature also increases accordingly. It can be seen from As can be seen from Figures 4 and 5, the missile's attack angle, sideslip angle, roll angle, and attitude angular velocity can converge to near zero in finite time. However, there is a certain oscillation phenomenon in the early stage, which is mainly due to the finite-time convergence of the LOS angle and LOS angular rate. So, the missile needs a larger overload, that is, a larger rudder deflection angle. However, the initial rudder deflection angle is zero, so the rudder deflection angle in the early stage will change greatly, which will cause a large change for system states such as the attack angle. With the increase of the initial LOS angle error, the convergence time of the states also increases, which is mainly because the convergence time is related to the initial value of the LOS angle error. e larger the initial error, the longer the convergence time.
As can be seen from Figure 6, the rudder deflection angle does not exceed the maximum limit and gradually converged to near zero in the later stages. From the simulation results, it can be seen that under different impact angle constraints, PIGC can accurately hit the target with the expected LOS angle, which shows the effectiveness of PIGC in dealing with the problem of impact angle constraints and input saturation.  Table 1.
As can be seen from Figure 7, under the three IGC algorithms, the missile can smoothly fly towards the target. It can be seen from Figure 8 that the three IGC algorithms can gradually make the LOS angles ε and η gradually converge near the expected values ε d and η d . Among them, PIGC has the fastest convergence speed, reflecting its fast finite-time convergence characteristics. e LOS angles of the DIGC and RIGC have certain divergence trends at the end of the guidance phase, especially the DIGC divergence is more obvious. is is mainly because PIGC uses ESO to compensate the system disturbance and uncertainty, which enhances the robustness of the system. However, the robustness of DIGC and RIGC at the end of guidance phase cannot effectively counteract interference, and therefore divergences occur. e PIGC has the smallest miss distances of the three IGC algorithms, followed by RIGC, indicating that PIGC has higher guidance precision.
It can be seen from Figures 9 and 10 that under PIGC and RIGC, the attack angle, sideslip angle, roll angle, and attitude angular velocity of the missile can converge to near zero in finite time. PIGC converges faster and its convergence accuracy is higher. As can be seen from Figure 11, the rudder deflection angles of the three channels of PIGC do not exceed the maximum limit, and they gradually converge to near zero in the later stages. is is because PIGC has considered and dealt with the problem of input constraint. DIGC and RIGC's pitch and yaw channel rudder deflection angles have reached the upper limit of amplitude during the initial stage of guidance phase. Table 1 shows the simulation results of attack times, LOS angle errors, and miss distances under three IGC algorithms. It can be seen that compared to DIGC and RIGC, PIGC's attack time, LOS angle errors, and miss distance are smallest, and it has high guidance precision and angular constraint precision.
Based on the analysis of the simulation results of the two simulation cases, it can be concluded that, under different impact angle constraints, PIGC can accurately hit the target with expected impact angle, and the miss distance is within      Discrete Dynamics in Nature and Society 1 m. In addition, the state variables such as the LOS angle and the attack angle can be quickly converged in finite-time, and the rudder deflection angle is within the set maximum amplitude range. Compared with DIGC and RIGC, PIGC can hit the target with shorter time, smaller miss distance, and higher angle constraint precision, and its state variables converge faster. Simulation results show the effectiveness and superiority of PIGC.

Conclusion
In this paper, a three-channel IGC algorithm with finite-time convergence is proposed. rough theoretical analysis and simulation verification, the following conclusions are reached: (1) e proposed IGC algorithm can achieve finite-time convergence of the system states. Under different impact angle constraints, the missile can be guaranteed to hit the target accurately with expected impact angle. Compared with the existing IGC algorithms, it has higher guidance precision and better convergence performance. (2) e Nussbaum gain function and an auxiliary system are used to effectively handle the problem of input saturation, which can ensure that the rudder deflection angle does not exceed the set range.