Research Article Research on Three-Dimensional Integrated Guidance and Control Design with Multiple Constraints

In this paper, a ﬁ xed-time convergence-integrated guidance and control (IGC) method is proposed, which considers terminal line-of-sight (LOS) angle constraint, full-state constraints, and input saturation. Firstly, an IGC design model considering the full coupling of three channels is constructed, and a ﬁ xed-time convergence disturbance observer is used to estimate and compensate the unknown disturbances in the model. Secondly, based on the ﬁ xed-time stability theory, sliding mode control, and backstepping control, a novel IGC scheme is carried out, and the second-order instruction ﬁ lter is used to restrict the system states and control instructions. Furthermore, the ﬁ xed-time stability of the close-loop system is proved and the expression of convergence time upper bound is given. Finally, comparison simulation and Monte Carlo simulation verify the e ﬀ ectiveness and superiority of the proposed IGC algorithm.


Introduction
The guidance and control system (GCS) is the core system of missile. The classical GCS is designed individually, which includes guidance loop and control loop. In contrast, the IGC design scheme takes guidance loop and control loop as a whole. Since the IGC design method can take advantage of the coupling relationship between guidance system and control system, as well as the comprehensive information, such as the information on the relative motion of the missile and the target, the attitude angle, and overload of the missile, it can significantly enhance the missile performance compared with the classical scheme [1,2].
There were many theories and methods for IGC scheme, including small gain theory [3], predictive control [4], variable structure control [5], adaptive control [6], and back-Many missiles, such as antitank missiles and antiaircraft missiles, can increase their warhead damage by controlling the terminal impact angles when attacking the target [11]. Therefore, terminal angle constraint such as LOS angle constraint can be considered in IGC design to improve the combat effectiveness of missile. In [12], a robust IGC design method with terminal LOS angle constraint was obtained based on dynamic inverse control, and the system uncertainties were restrained by introducing sliding mode control. The sign function was replaced by continuous high gain saturation function to solve the chattering problem inherent in sliding mode control. But it also changed the original structure of sliding mode control and reduced the robustness. In [13], an IGC scheme with terminal LOS angle constraint was designed based on dynamic surface control, and adaptive control was adopted to compensate system disturbances. However, dynamic surface control-based IGC design could only ensure that the system states converge asymptotically, but the missile attack time is limited.
Because of the aerodynamic structure limitation, the missile system state variables, such as attack angle and Euler angular rates, should change within certain ranges during the flight. In the meanwhile, because of the actuator physical limitation, the control ability generated by the actuator is limited. Taking pneumatic rudder as an example, both positive and negative rudder deflection angles have upper bounds. If rudder deflection angles and system state variables exceed the allowed ranges, the missile guidance and control performance will be negatively affected. Even the missile will be uncontrollable and unstable, leading to offtarget [14]. Therefore, state constraints and control constraint should be considered in the IGC design. In [15], dynamic surface control was used to carry out an IGC system, and the model uncertainty and disturbance were estimated by a proposed finite-time convergence disturbance observer. In addition, the Nussbaum function was used to deal with the control constraint, which was also known as input saturation. In [16], aiming at state constraints, an IGC system was designed based on dynamic surface control and barrier Lyapunov function. In order to solve the state constraints, a saturation function was introduced into the design to ensure that the state variables did not change beyond the constraint ranges.
The above analysis shows that most of the literatures on IGC design research aim to solve unitary constraint problem, such as terminal LOS angle constraint, input saturation, state constraints, and finite-time convergence. However, the missile GCS was faced with a variety of constraints. As far as we know, there are few literatures dealing with the problems of input saturation, full-state constraints, terminal LOS angle constraint, and global finite-time convergence in 3D IGC scheme. For this issue, a fixed-time IGC algorithm for STT missile is proposed in this paper, which considers terminal LOS angle constraint, input saturation, and full-state constraints. The main contributions are as follows: (a) A 3D IGC design model with multiple constraints is established rather than the single plane model established in [5] (b) A sliding mode surface which can strictly converge in fixed-time is designed to solve terminal angle constraint rather than common asymptotic stability (c) The second-order instruction filter is constructed to estimate the derivative of the virtual commands, which effectively relieve the computation burden in backstepping design, and guarantee virtual commands in prescribed ranges. Furthermore, an auxiliary system is designed to eliminate the tracking error of the filter (d) Different from the general asymptotic or finite-time convergence IGC algorithms proposed in [15][16][17], a 3D IGC algorithm is proposed to achieve global fixed-time convergence in the presence of extern disturbances, terminal angle constraint, full-state constraints, and input saturation

Problem Description and Preliminaries
2.1. Problem Description. Figure 1 establishes the relative motion of missile and target in 3D inertial coordinate system Oxyz, where Ox 4 y 4 z 4 is the LOS coordinate system. M represent the missile and T represent target. ε is LOS elevation angle and η is the azimuth angle. R is relative distance of missile and target. Define a m 4 = ½a mx 4 a my 4 a mz 4 T and a t 4 = ½a tx 4 a ty 4 a tz 4 T which are acceleration vectors of missile and target in Ox 4 y 4 z 4 , respectively. The missile and target relative motion equations [18] are Define a m 2 = ½a mx 2 a my 2 a mz 2 T , which is missile acceleration vector in ballistic coordinate system. It can be obtained from the missile dynamic equations as International Journal of Aerospace Engineering where m is missile mass, θ m is fight path angle, α is attack angle, β is sideslip angle, γ V is bank angle, Y is aerodynamic lift force, Z is lateral force, P is thrust force, and g is gravitational acceleration. From the coordinate transformation, we can get where ψ Vm is the heading angle of missile. The aerodynamic forces Y and Z are modeled as where Y δ z and Z δ y are lift force and lateral force produced by the rudder angles, C α y and C β z are aerodynamic coefficient with respect to α and β, Q = ρV 2 m /2 is dynamic pressure, S stands for the reference area, ρ is atmospheric density, and d Y and d Z are the unknown bounded uncertainties including modeling error and model uncertainties, here treated as disturbance variables.
Since the rudder deflection angles contribute little to the aerodynamic forces, Y δ z and Z δ y can be regarded as small quantities [19].
In this paper, the IGC design applies only to the terminal guidance stage. Therefore, assuming that the missile flies without power and the flight speed is approximately invariant, the velocity change rate is regarded as interference. For STT missile, its required bank angle control is zero. So it can be assumed to be a small quantity, and sin ðγ V Þ ≈ 0 and cos ðγ V Þ ≈ 1. Define new state variables x 1 = ½_ ε _ η T and x # 2 = ½α β T . Combined with (1) to (4), the relative motion equation in 3D space can be described as with As is well known that R is bounded, and target acceleration and aerodynamic parameters (Y δ z , Z δ y , d Y , and d Z ) are bounded, so d 1 ðtÞ is bounded.
According to missile dynamic equations [20], the missile control model can be described as where γ is roll angle, J x , J y , and J z are the inertia components, ω x , ω y , and ω z are Euler angular rates, m α x , m δ y y , and m δ z z are moment aerodynamic coefficients with respect to α, β, δ x , δ y , and δ z , which δ x , δ y , and δ z are rudder angles, Δ α , Δ β , Δ γ , Δ ω x , Δ ω y , and Δ ω z are external disturbances and modeling errors, and ϑ is missile pitch angle.
The missile terminal LOS angle constraint [21] can be described as Define new state variables (5) and (7), the IGC design model with terminal LOS angle constraint is established as with , Due to the existence of factors such as large target maneuvers, unsatisfactory initial states in the shift phase of middle and final guidance, external interference, and system uncertainty, the state variables such as α, β, γ, ω x , ω y , and ω z may exceed the allowable ranges, which will worsen the dynamic quality of the system, affect the control performance, and even lead to control divergence.
In this paper, the research goal is to design an IGC algorithm that makes the missile system state variables and rudder angles not exceed the allowed ranges and the LOS angles and angular rates converge in fixed-time.
Lemma 3 (see [22,23]). For the system _ y = −a 1 sgn b 1 ðyÞ − a 2 sgn b 2 ðyÞ, and a 1 > 0, a 2 > 0, 0 < b 1 < 1, and b 2 > 1, then the system is stable in fixed-time and the convergence time satisfies In addition, if the system has a small disturbance, that is _ y = −a 1 sgn b 1 ðyÞ − a 2 sgn b 2 ðyÞ + ς, ς is a small positive number, then the system can converge to the neighborhood of the origin Ω = fjyj ≤ 2ϑja 1 ϑ b 1 + a 2 b 1 ϑ b 2 = ςg in fixed-time, and the convergence time satisfies Lemma 4 (see [24]). Assuming that the Lyapunov function Vð xÞ satisfies _ VðxÞ ≤ −a 1 V b 1 ðxÞ − a 2 V b 2 ðxÞ, and a 1 > 0, a 2 > 0, 0 < b 1 < 1, and b 2 > 1, then the system is stable in fixed-time and the convergence time satisfies Lemma 5 (see [25]). For any real number x i , i = 1, 2, ⋯, n, there are real numbers 0 < a < 1 and b > 1, such that the following equations are true International Journal of Aerospace Engineering According to Lemma 5, the following equations hold

IGC Design and Stability Analysis
The IGC model (9) established in this paper is a strictfeedback nonlinear system with uncertainty, so the backstepping control design method is used to accomplish the IGC scheme. The backstepping control is improved based on fixed-time stability theory to let the system states fixed-time convergence. A second-order instruction filter is introduced into backstepping control to overcome "differential expansion." On the one hand, the derivatives of virtual control quantities are realized through integration, and the noise influence on the system is effectively reduced. On the other hand, the system states and actual control instructions are constrained to ensure that they do not exceed the allowable ranges.

Novel IGC Law Design.
For the missile attacking the target accurately with desired terminal LOS angle, that is, the system states x 0 and x 1 fixed-time converge to zero, based on Lemma 5 and piecewise sliding mode surface idea, a fixed-time convergence sliding mode is designed as 0i , x 0i j j≥ δ, where a 1 > 1, 0 < a 2 = p 1 /p 2 < 1, k i = diag ðk i1 , k i2 Þ is the positive definite matrix to be designed, i = 1, 2, δ > 0, p 1 , and p 2 are odd numbers, The time derivative of (16) is given by Based on fixed-time stability theory, backstepping control, and the above sliding mode surface, the design process of the novel IGC scheme with multiple constraints are as follows: Step 1. Define the tracking error as The time derivative of s 1 is given by Design the virtual control law of x # 2 as where b 1 > 1, 0 < b 2 = p 3 /p 4 < 1, p 3 and p 4 are odd numbers, k 3 and k 4 are positive definite diagonal matrixes to be designed, and v 1 is filter tracking error compensation, which is used to compensate the influence of instruction filter on virtual control law, and its definition will be given later.d 1 is the estimation of d 1 . Here, the observer proposed in [26] is used to estimate the disturbances, and the specific expression is where λ 1i > 0 ði = 1, 2, 3Þ,x 1 = x 1 −x 1 , andx 1 is the estimation of x 1 . In order to avoid "differential expansion" and considering the limitation of virtual control variables, a new virtual control law was introduced by referring to the dynamic surface design method. x # 2d and its derivative _ x # 2d are obtained by passing through a second-order instruction filter. The state space expression of the filter is where S M ð⋅Þ is the saturation function, which is defined as It can be seen from (24) and (25) that the calculation of _ x d does not need a differentiator, thus avoiding the problem of "differential expansion." If x c is bounded, then both x d 5 International Journal of Aerospace Engineering and _ x d are continuously bounded. By selecting a larger natural frequency ω n , jx d − x c j can be made small. However, if it is too large, it will bring high frequency noise to the system, so compromise should be considered when selecting parameters.
The expression of v 1 is where l 1 is the positive definite diagonal matrix to be designed and μ 1 is a smaller positive number.
Step 2. Define the tracking error as s 2 = x 2 − x 2d and x 2d = ½ x # 2d ; 0. The time derivative of s 2 is given by The virtual control quantity of x 3 is designed as where k 5 and k 6 are positive definite diagonal matrixes to be designed, χ = ½g T 1 s 1 0 T .
Let x 3c pass through the second-order filter to get x 3d and _ x 3d . The expression of v 2 is Step 3. The tracking error is defined as s 3 = x 3 − x 3d . The time derivative of s 3 is given by The virtual control law of u is designed as Let u c pass through the second-order filter to get the actual IGC law u. The expression of v 3 is The IGC algorithm structure is illustrated in Figure 2.

Stability Analysis
Theorem 6. For the system (9), the proposed novel IGC law from (20) to (33) can guarantee that the system states x 0 and x 1 converge to the origin neighborhood in fixed-time, and the closed-loop IGC system is stable in spite of disturbances.
Proof. From (21) to (32), we can obtain whered i = d i −d i ði = 1, 2, 3Þ and s # 2 = ½s 21 s 22 T . Construct the Lyapunov function The time derivative of V is obtained as International Journal of Aerospace Engineering According to (15), it can be obtained as Based on the Young inequality, we can get

International Journal of Aerospace Engineering
And the convergence time satisfies Combining with (46) and (35), s 1 can converge in fixedtime t 1 to the region s 1 ≤ ffiffiffiffiffiffi ffi 2ϑ 1 p . Define Δ = ffiffiffiffiffiffi ffi 2ϑ 1 p , then js 1i j ≤ Δ, i = 1, 2. According to the region where the value of x 0i is located, the convergence region of x 0i and x 1i can be obtained as follows.
(a) When jx 0i j > δ, it can be obtained from (16) that According to Lemma 4, x 0i can converge to the region Ω 2i = fjx 0i j ≤ 2ϑ 2i jk 1i ϑ and convergence time satisfies (b) When jx 0i j < δ, it can be obtained from (16) that Based on the above analyses, x 0i and x 1i can converge to the neighborhood 11 International Journal of Aerospace Engineering with fixed-time. In addition, we can get the convergence time which satisfies It can be seen from (53) that the upper bound of convergence time t m increases with the increase of k, k 1i , k 2i , a 1 , and b 1 and increases with the decrease of a 2 and b 2 . According to (43), k is related to the control parameter matrices k i ði = 3, 4, ⋯, 8Þ and l j ðj = 1, 2, 3Þ. Therefore, t m only depends on the control parameters, independent of the system states.
Although the convergence speed of the IGC system states will increase as the parameters a 1 , b 1 , k i ði = 1, 2, ⋯, 8 Þ, and l j ðj = 1, 2, 3Þ increase and decrease as the parameters a 2 and b 2 increase, excessive pursuit of convergence speed will produce large control instructions, requiring the missile to provide a large overload, and the missile available overload is limited, so compromise should be considered in parameter selection.
This completes the proof.

Numerical Simulation Results and Analysis
The missile 6-DOF simulation is implemented to illustrate the performance of the proposed IGC (PIGC) design method. Table 1 shows the initial conditions of missile and target. Table 2 shows the missile dynamic parameters. The states and actuator constraints are described in Table 3. The lateral acceleration of the target is set as a tz2 = 10 sin ðπt/2Þ m/s 2 . Set the desired terminal LOS angles ε d = −70°and η d = 0°.

Case 1: Comparison Simulation with Existing Methods.
The robust IGC design method (RIGC) proposed in [12] and the new dynamic surface control based IGC design method (NDIGC) proposed in [15] are introduced for comparative simulation analysis.
Missile and target trajectories, LOS angular rates, and relative distances are shown in Figures 3 and 4. The missile trajectories are relatively smooth under the three IGC laws.
PIGC can make the LOS angular rates converge stably to near zero within fixed-time t m and has a significant advantage in zeroing angular rates, thus leading to a small miss distance (see Table 4).
LOS angles are presented in Figure 5. It can be seen that the LOS angles can gradually converge to the expected value under the three IGC laws. The LOS angles of PIGC converge fastest. RIGC adopts sliding mode control to suppress disturbances such as model uncertainty and target maneuver. When there are large model uncertainties and disturbances, the convergence performance of RIGC is greatly affected. NDIGC selects linear sliding mode surface, which can only ensure the states convergent asymptotically. Therefore, PIGC is superior in terms of angle constraints and convergence.
The attack angles, sideslip angles, and roll angles curves are given in Figure 6. The three Euler angular rates are 13 International Journal of Aerospace Engineering presented in Figure 7. The angles and angular rates can converge to near zero under three IGC algorithms. Compared with PIGC scheme, larger attack angle, sideslip angle, and roll angle result in larger pitch, yaw, and roll angular rates under RIGC and NDIGC. Moreover, the PCIGC can ensure that the system states do not exceed the constraint ranges.
The rudder angle curves are shown in Figure 8. Compared with RIGC, NDIGC and PIGC can restrict the rudder angles within the constraints. PIGC, in particular, can ensure that the rudder angles converge steadily to near zero in finite-time.
The simulation results in Figures 3, 5, and 6 seem divergent at the intercept time. This is because the relative motions of the missile and target change dramatically at the end, so the LOS angular rate changes dramatically and diverges at the last moment of the simulation. At the same time, as the system state constraints such as attack angle are not considered and corresponding virtual control instructions are not restricted, system states of RIGC and NDIGC cannot converge stably and diverge at the end of simulation. Table 4 shows the simulation results of three IGC laws in terms of intercept times, miss distances, and LOS angle errors. It can be seen that PIGC has the smallest intercept time, miss distance, and angle errors. PIGC can impact the target accurately with shorter time, smaller miss distance, and higher angle constraint accuracy.
From the simulation results, it can be concluded that PIGC can effectively solve terminal angle constraints, state constraints, and control constraints and has fixed-time convergence, which is superior to NDIGC and RIGC.

Case 2: Monte Carlo
Simulation. In this case, a missile intercepts a ground target. In case 1, it is assumed that  14 International Journal of Aerospace Engineering the missile velocity is constant. But in a real scenario, the missile's speed would certainly change because of aerodynamic drag and gravity. Therefore, in order to be closer to engineering practice, the simulation under timevarying missile velocity will be considered in this case. At the same time, In order to further verify the robustness of PIGC algorithm, Monte Carlo simulation was carried out with 100 simulation times. Simulation conditions are as follows: (1) The aerodynamic coefficients vary from -10% to +10% of the nominal value and follow normal distribution (2) The mass and moment of inertia vary from -10% to +10% of the nominal value and follow normal distribution (3) The initial position coordinates y m and z m of the missile vary randomly between 4500 m~5500 m and 0 m~500 m, respectively, subjected to normal distribution. The target position is (5000 m, 0 m, 1000 m) (4) The missile initial fight path angle θ m and heading angle ψ Vm vary randomly between 0 deg~10 deg and -5 deg~0 deg, respectively, subjected to normal distribution (5) The mathematical expression of aerodynamic resistance is         As shown in Figures 9 and 10, perfect attacking target with small miss distance is achieved in each scenario. The mean and standard deviation of miss distance are found as 1.2639 m and 0.5019 m, respectively. It can be seen in Figure 11 that the missile's speed is reduced by air resistance.
In Figures 12 and 13, the LOS angles finite-time converge to the desired angles, and the convergence time is all less than t m that embodies the fixed-time convergence characteristic of PIGC. In Figures 14-22, the system states and rudder angles are all finite-time converge to near zero and never violate the constraints. In addition, there are problems of nonsmoothness at about 10 to 11 seconds in Figures 14,19,and 22 and at about 7 to 8 seconds in Figures 15,16,17,18,20,and 21. These were because the nonsingular terminal surface (20) proposed in this paper is a piecewise function with switching term, so when the sliding mode surface is in the switching position, the control instructions have small fluctuations; thus, the curves of the system states will appear not smooth.
It can be seen from these figures that PIGC achieves satisfactory performance under different initial fight conditions and uncertainties of the missile-related parameters. Monte Carlo simulation shows that the proposed IGC algorithm is robust.

Conclusion
In this paper, a novel fixed-time IGC algorithm has been proposed for terminal angle constraint, full-state constraints, and control constraint. The IGC design model considering multiple constraints is established, and the disturbance observer is used to estimate and compensate the model dis-turbance. The second-order instruction filter is introduced to overcome the "differential expansion," system state constraints, and control constraint. The fixed-time convergence and the stability of the IGC scheme are strictly proved. Simulation results verify the effectiveness and superiority of the proposed IGC algorithm.
The IGC algorithm proposed in this paper contains many design parameters, which can be further studied on parameter optimization.

Data Availability
All data are included within this paper.

Conflicts of Interest
The authors declare that they have no conflicts of interest.