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A nonsingular fast terminal sliding mode guidance law with an impact angle constraint is proposed to solve the problem of missile guidance accuracy and impact angle constraint for maneuvering targets. Aiming at the singularity problem of the terminal sliding mode, a fast terminal sliding mode surface with finite-time convergence and impact angle constraint is designed based on fixed-time convergence and piecewise sliding mode theory. To weaken chattering and suppress interference, a second-order sliding mode supertwisting algorithm is improved. By designing the parameter adaptive law, an adaptive smooth supertwisting algorithm is designed. This algorithm can effectively weaken chattering without knowing the upper bound information of interference, and it converges faster. Based on the proposed adaptive supertwisting algorithm and the sliding mode surface, a guidance law with the impact angle constraint is designed. The global finite-time convergence of the guidance law is proved by constructing the Lyapunov function. The simulation results verify the effectiveness of the proposed guidance law, and compared with the existing terminal sliding mode guidance laws, the proposed guidance law has higher guidance precision and angle constraint accuracy.

In modern warfare, many missiles (such as some antiship missiles, antitank missiles, and air defense missiles) need to hit the target with certain impact angles to increase the damage effectiveness of the warheads. Therefore, the impact angle constraint is a problem that needs to be considered in the design of the guidance law [

Sliding mode control is widely used in the design of the guidance law because of its invariability to interference in the sliding mode. In [

For the disturbance problems such as target maneuvering and system disturbance, there are currently three methods for processing most documents: (1) designing the disturbance observer to estimate the disturbance in real time and online [

In order to solve the above problems, this paper improves an adaptive smooth supertwisting algorithm, which solves the problems of slow convergence speed and the unsmooth control law of the traditional supertwisting algorithm and greatly weakens the chattering problem of the sliding mode control. At the same time, the parameter adaptive law is designed against the disturbance without knowing the upper bound information of the disturbance. Based on the idea of fixed-time convergence and piecewise sliding surface, a nonsingular fast terminal sliding surface with the impact angle constraint is designed. A nonsingular fast terminal sliding mode guidance law with the impact angle constraint is proposed based on the adaptive supertwisting algorithm. The global finite-time convergence is proved by constructing the Lyapunov function. Finally, the effectiveness and superiority of the guidance law are verified by simulation experiments.

In the inertial coordinate system, the relative motion relationship between the missile and the target is established as shown in Figure

Relationship of missile-to-target motion.

According to the relative motion relationship of the missile and the target, the relative motion equations of the missile and the target can be obtained as follows:

Differentiating

The impact angle is the angle between the missile and the target velocity vector at the time of guidance terminal, and the impact angle constraint problem can be transformed into the terminal LOS angle constraint problem [

For the convenience of analysis and proof, the following lemmas are introduced.

Assume that there is a smooth function _{1} > 0 and 0 < _{1} < 1,

Assume that Lyapunov function _{1} > 0, _{2} > 0, and 0 < _{1} < 1; then, the system can converge to the origin in finite time, and the convergence time satisfies

For the nonlinear system _{1} > 0, _{2} > 0, 0 < _{1} < 1, and _{2} > 1, the system is stable in finite time, and the convergence time satisfies

In addition, if the system has a small disturbance, that is,

For the following first-order system,

The supertwisting algorithm can greatly reduce chattering and has strong robustness and high precision control performance [

The parameter adaptive law is designed as follows:

Substituting (

It can be seen from (

Due to the measurement noise of the system, the state of the system cannot reach the equilibrium point completely. In order to avoid the parameter increasing to infinity, the term

It is obvious that _{1} and _{2} will gradually increase, making the system state to converge. When the system state converges to _{1} and _{2} will decrease gradually. If _{1} and _{2} decrease to the point where the interference cannot be eliminated, the system state will deviate from _{1} and _{2} will gradually increase under the effect of the adaptive law, making the system state converge to _{1} and _{2} will gradually decrease. Therefore, _{1} and _{2} are globally bounded.

For the total disturbance

Under Assumption

Define a new state vector as

From (

Differentiating (

Define the following Lyapunov function:

It is easy to prove that _{1} is unbounded radially, i.e.,

Differentiating

If we define

According to (

According to Lemma

Under the control law (

According to (

Differentiating

According to (

When

It is known that

According to Lemma

When _{1} and _{2} decrease to the point where the interference cannot be eliminated, the system state will deviate from _{1} and _{2} will increase again under the effect of the adaptive law, making the system state converge to

The terminal sliding mode control adopts the nonlinear function as the sliding mode surface, which can make the system states converge in finite time, but the method has singular problems. In order to avoid singular problems, based on the piecewise sliding surface [

Differentiating

Condition (1) can ensure that

According to the above conditions, this paper selects function

Substituting (

The equivalent guidance law is designed as

Substituting (

In order to counteract the disturbance, suppress chattering, and accelerate the convergence speed of the sliding surface, based on the adaptive smooth, fast supertwisting algorithm proposed in the second section, an auxiliary guidance law is designed as

The parameter adaptive law is designed as

Combining with (

For the convenience of description, the design guidance law (

In order to test the performance of the designed guidance law, ASNTSMG, this section conducts simulation analysis based on ballistic simulation in different scenarios. The initial position of the missile is (0 m, 0 m), and the initial position of the target is (1000 m, 5000 m). The missile’s velocity is ^{2}, and the maximum available overload of the missile is 20 g. The parameters of ASNTSMG are set as follows:

In order to verify the superiority of the designed guidance law, this section also carries out the nonsingular fast terminal sliding mode guidance law (NFTSMG) proposed in [

The parameters are set as follows:

The expression of SONTSMG is

The parameters are set as follows:

The average overload

Attack moving target with different impact angle constraints: set _{m0} = 45°. The target makes sinusoidal maneuver, and its acceleration is _{t} = 30sin (^{2}, and _{t0} = 150°. The simulation results are shown in Figure

It can be seen from Figures

Figure _{m0}, the longer the saturation time, which is mainly due to the larger overload needed in the earlier stage, which makes the missile meet the requirements of angle constraint and guidance accuracy. When

Simulation results of Case

Comparative simulation of ASNTSMG, NFTSMG, and SONTSMG: the relevant initial parameters are set to _{m0} = 45°, _{t0} = 180°. The movement of the target is set as follows:

Cosine motion: _{t} = 30cos (^{2}

Square wave motion: _{t} = 30sgn (sin (^{2}

The simulation results are shown in Figures

Figure

Table

According to the analysis of the simulation results of two cases, ASNTSMG can hit the target precisely with the expected impact angle under different expected LOS angles and target maneuvering conditions. Compared with the existing guidance laws NFTSMG and SONTSMG, ASNTSMG can effectively attack the target with less impact angle error, miss distance, and energy consumption, which verifies the effectiveness and superiority of ASNTSMG.

Missile and target motion trajectory. (a) Target cosine motion. (b) Target square wave motion.

LOS angle. (a) Target cosine motion. (b) Target square wave motion.

LOS angular rate. (a) Target cosine motion. (b) Target square wave motion.

Missile overload. (a) Target cosine. (b) Target square wave motion.

Simulation results of different guidance laws.

Target movement | Guidance law | Attack time (s) | Miss distance (m) | Angle error (deg) | _{me} (g) |
---|---|---|---|---|---|

Cosine maneuver | NFTSMG | 16.24 | 0.95 | 0.59 | 3.88 |

SONTSMG | 16.12 | 0.76 | 0.21 | 3.58 | |

ASNTSMG | 16.05 | 0.39 | 0.02 | 3.43 | |

Square wave maneuver | NFTSMG | 15.54 | 1.03 | 0.88 | 3.99 |

SONTSMG | 15.47 | 0.86 | 0.58 | 3.48 | |

ASNTSMG | 15.41 | 0.43 | 0.03 | 3.36 |

In this paper, a nonsingular fast terminal sliding mode guidance law is proposed to solve the problem of guidance accuracy and impact angle constraint. Through theoretical analysis and simulation verification, the following conclusions can be obtained:

The proposed adaptive smooth supertwisting algorithm can effectively counteract the disturbance of the system and accelerate the convergence speed of the system without knowing the upper bound of the disturbance.

The designed nonsingular terminal sliding mode surface can realize the fast finite-time convergence of the system states and ensure the impact angle constraint and guidance accuracy requirements.

This guidance law can attack the target precisely under the conditions of different expected LOS angles and target maneuvers. Compared with the existing nonsingular fast terminal sliding mode guidance law and second-order nonsingular terminal sliding mode guidance law, this law has higher guidance accuracy and angle constraint accuracy and consumes less energy.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.