Integrated Guidance and Control of Multiple Interceptor Missiles Based on Improved Distributed Cooperative Control Strategy

ABSTRACT

In this study, an improved cooperative integrated guidance and control (IGC) design method is proposed based on distributed networks to address the guidance and control problem of multiple interceptor missiles. An IGC model for a leading interceptor is constructed based on the relative kinematic relations between missiles and a target and the kinematic equations of the missiles in a pitch channel. The unknown disturbances of the model are estimated using a finite-time disturbance observer (FTDO). Then, the control algorithm for the leading interceptor is designed according to the disturbance estimation and nonsingular fast dynamic surface sliding mode control (SMC). To enhance the rate of convergence of the cooperative control commands for the interceptors, an improved cooperative control strategy is proposed based on the leader-follower distributed network. Consequently, the two velocity components of the interceptor in the pitch channel can be obtained, which are subsequently converted to the total velocity and flight path angle commands of the interceptor using kinematic relations. The control algorithm for the following interceptor is similarly designed using an FTDO and dynamic surface SMC. The effectiveness of the improved distributed cooperative control strategy for multiple interceptors is validated through simulations.

KEYWORDS:
Integrated guidance and control; Finite-time disturbance observer (FTDO); Nonsingular fast dynamic surface sliding mode; Distributed network; Improved cooperative control strategy

INTRODUCTION

With the advancement of anti-missile technology, cooperative multi-missile attack and defense is attracting increasing attention owing to its unique strengths. As a result, the development of cooperative multi-missile guidance and control technology, which is a key element for ensuring the attack and defense performance of a weapon system, has gained momentum. Through coordination between missiles, cooperative engagement integrates multiple interceptor missiles as a united combat group that is information-sharing, function-complementary, and tactics-cooperative. Using the group advantage, a multi-missile system can execute a multi-layer all-around attack on an enemy’s defense system or a target with overall-promoted penetration capabilities and carries out tasks that are difficult for a single interceptor missile to perform. Therefore, it is of practical significance to study the cooperative guidance and control of multiple interceptors (Zhao and Yang 2017; Daughtery and Qu 2014).

With respect to the cooperative guidance and control of multi-interceptors, Jeon et al. (2010) and Lee et al. (2007) propose a guidance law with controllable attack time and angle-of-attack constraint and apply it to the salvo attack of anti-ship missiles. Based on this idea, researchers subsequently introduced other guidance and control methods, including sliding mode control (SMC) (Hail and Balakrishnan 2012; Cho et al. 2016), optimal control (Nikusokhan and Nobahari 2016), differential game (Kang and Kim 2011), and dynamic surface control (Wang et al. 2015). These methods rely on specifying attack time before launching to achieve coordination. As no information exchange occurs between missiles during flight, these methods apparently have temporal limitations. With progress in the consensus of multi-agent systems, researchers have begun to use the consensus theory to study the cooperative guidance and control of multi-interceptors. Using the coordination strategy under the cooperative guidance framework, Kumar and Ghose (2014) adjust missile trajectories so that the coordination variable of each missile can approach the expected coordination variable for realizing cooperative guidance. Zhao et al. (2014) apply leader-follower formation control to the cooperative guidance of multi-interceptors by employing an analogous leader-follower cooperative guidance framework. Sun et al. (2014) and Zhao et al. (2016) explore the guidance and control law of the leader-follower topology considering time delay and topology switch. By constructing an integrated cost function for multiple missiles, Shaferman and Shima (2015) design a cooperative guidance law for multiple missiles for intercepting a maneuvering target. However, the application of this function is faced with multiple constraints, as each missile requires the global information of all participating partners. Balhance et al. (2017) study the cooperative guidance law using the optimal control theory and improves communication between multiple missiles. These consensus-based cooperative guidance and control approaches typically employ the regular form of the cooperation strategy and fail to consider the convergence rate of interceptors to cooperative control commands.

Conventionally, in the design of the guidance and control system of interceptors, a control loop is set as a fast loop whereas a guidance loop is set as a slow loop. The basis of these methods is to design two subsystems independently, without considering the coupling between the two loops. On the contrary, integrated guidance and control (IGC) design depends on the control force generated from the engagement information between an interceptor and a target and the kinematic information of the interceptor per se to drive the interceptor to hit the target. This can ensure the stability of interceptor flight attitude and improve guidance accuracy (Menon and Ohlmeyer 2001; Shtessel et al. 2009). In recent years, researchers have integrated IGC design with various control theories, such as SMC (Shtessel et al. 2009; Lee et al. 2016), back-stepping control (Seyedipour et al. 2017; Ran et al. 2014), optimal control (Vaddi et al. 2009), and predictive control (Shamaghdari et al. 2015), for creating a suite of aircraft IGC design methods with diverse features. Even though it is well known that the advantages of IGC methods can be utilized to enhance the stability of multi-interceptor systems, research in this area is still inadequate at the moment.

Therefore, by adopting the above-discussed IGC method, an improved distributed cooperative IGC algorithm for multiple interceptor missiles featuring the leader-follower topology is proposed in this study based on the distributed network cooperative control strategy and SMC theory, with the aim of improving the rate of convergence of inceptor missiles to cooperative control commands.

IGC MODEL FOR THE LEADING INTERCEPTOR

The engagement geometry of the leading interceptor and target in the pitch channel are shown in Fig. 1.

Figure 1
Engagement geometry of the leading interceptor and target.

In Fig. 1, M and T denote the interceptor missile and target, respectively; a_m4ε and a_tε are the accelerations of the missile and target, respectively; V_m and V_t represent missile velocity and target velocity, respectively; θ_m and θ_t are the flight path angles of the missile and target, respectively; q_ε is the missile/target LOS angle; R is the distance between the missile and target. The relative kinematic model of the interceptor in the longitudinal plane can be obtained as (Eq. 1):

(1) $\left\{\begin{array}{c}\dot{\mathit{r}}={\mathit{V}}_{\mathit{t}}\cos \left({\mathit{q}}_{\epsilon }-{\mathit{\theta }}_{\mathit{t}}\right)-{\mathit{V}}_{\mathit{m}}\cos \left({\mathit{q}}_{\epsilon }-{\mathit{\theta }}_{\mathit{m}}\right)\hfill \\ \mathit{r}{\dot{\mathit{q}}}_{\epsilon }={\mathit{V}}_{\mathit{m}}\sin \left({\mathit{q}}_{\epsilon }-{\mathit{\theta }}_{\mathit{m}}\right)-{\mathit{V}}_{\mathit{t}}\sin \left({\mathit{q}}_{\epsilon }-{\mathit{\theta }}_{\mathit{t}}\right)\hfill \end{array}\right.$

The kinematic model of the leading interceptor in the pitch channel is expressed as (Eq. 2):

(2) $\left\{\begin{array}{c}\dot{\mathit{\alpha }}={\mathit{\omega }}_{\mathit{z}}-\frac{{\mathit{qSC}}_{\mathit{y}}^{\mathit{\alpha }}\mathit{\alpha }\cos \mathit{\alpha }}{{\mathit{mV}}_{\mathit{m}}}+{\mathit{d}}_{\mathit{\alpha }}\hfill \\ {\mathit{\omega }}_{\mathit{z}}=\frac{\mathit{qS}{\overline{\mathit{Lm}}}_{\mathit{z}}^{\mathit{\alpha }}\mathit{\alpha }}{{\mathit{J}}_{\mathit{z}}}+\frac{\mathit{qS}{\overline{\mathit{Lm}}}_{\mathit{z}}^{{\mathit{\omega }}_{\mathit{z}}}{\mathit{\omega }}_{\mathit{Z}}}{{\mathit{J}}_{\mathit{z}}}+\frac{{\mathit{M}}_{\mathit{z}}}{{\mathit{J}}_{\mathit{z}}}+{\mathit{d}}_{{\mathit{\omega }}_{\mathit{z}}}\hfill \\ {\mathit{a}}_{\mathit{m}3\epsilon }=\frac{{\mathit{qSC}}_{\mathit{y}}^{\mathit{\alpha }}\mathit{\alpha }}{\mathit{m}}\hfill \end{array}\right.$

where S is the reference area of the missile; m the missile mass; α is the angle of attack (AOA); ω_z is the pitch angular velocity; q is the dynamic pressure; d_α and d_ωz are the perturbation and uncertain disturbance in each part of the system, respectively; J_z is the rotational inertia of the missile; C^a_y, m^a_z, and m^ωz_z denote the aerodynamic force and moment coefficients; and M_z is the control moment of the missile in the pitch channel.

Based on Eqs. 1 and 2, if we define x₁ = q̇_ε, x₂ = q̇_ε, x₃ = α, and x₄ = ω_z, the nonlinear IGC model of the leading interceptor in the pitch channel can be written as (Eq. 3):

(3) $\left\{\begin{array}{c}{\dot{\mathit{x}}}_1={\mathit{x}}_2\hfill \\ {\dot{\mathit{x}}}_2={\mathit{a}}_{22}{\mathit{x}}_2+{\mathit{a}}_{23}{\mathit{x}}_3+\frac{{\mathit{a}}_{\mathit{t}\epsilon }}{\mathit{r}}\hfill \\ {\dot{\mathit{x}}}_3={\mathit{a}}_{33}{\mathit{x}}_3+{\mathit{x}}_4+{\mathit{d}}_{\mathit{\alpha }}\hfill \\ {\dot{\mathit{x}}}_4={\mathit{a}}_{43}{\mathit{x}}_3+{\mathit{a}}_{44}{\mathit{x}}_4+{\mathit{b}}_4\mathit{u}+{\mathit{d}}_{{\mathit{\omega }}_{\mathit{z}}}\hfill \\ {\mathit{y}}_2={\mathit{x}}_2\hfill \end{array}\right.$

where ${\mathit{a}}_{22}=-\frac{2\dot{\mathit{r}}}{\mathit{r}},{\mathit{a}}_{23}=-\frac{{\mathit{qSC}}_{\mathit{y}}^{\mathit{\alpha }}}{\mathit{mr}},{\mathit{a}}_{33}=-\frac{{\mathit{qSC}}_{\mathit{y}}^{\mathit{\alpha }}}{{\mathit{mV}}_{\mathit{M}}},{\mathit{a}}_{43}=\frac{\mathit{qS}{\overline{\mathit{L}}}_{\mathit{z}}^{\mathit{\alpha }}}{{\mathit{J}}_{\mathit{z}}},{\mathit{a}}_{44}=\frac{\mathit{qS}{\overline{\mathit{LM}}}_{\mathit{z}}^{{\mathit{\omega }}_{\mathit{z}}}}{{\mathit{J}}_{\mathit{z}}},{\mathit{b}}_4=\frac{1}{{\mathit{J}}_{\mathit{z}}},\mathrm{and}\mathit{u}={\mathit{M}}_{\mathit{z}}$.

Assumption 1: The LOS angle of the interceptor varies insignificantly during the terminal phase of guidance. In addition, the angle between the SOL direction and velocity direction is small; hence, a_m3ε ≈ a_m4ε.

Assumption 2: The unknown disturbances in the IGC system, d_α and , are continuously differentiable and have a bounded first derivative, i.e., d_i < L, where L is a positive constant.

FTDO DESIGN

An FTDO estimates the nonlinear uncertain terms in the system model and feeds their estimated values into the control system for compensation. To eliminate the effects of the unknown disturbances involved in model (Eq. 3), viz., a_tε, d_α, and d_ωz, on the control system of the leading interceptor, an FTDO is designed to estimate these terms.

Definingv_r = ṙ and ve = rq̇_ε, from Eq. 1, we have (Eq. 4)

(4) ${\dot{\mathit{v}}}_{\epsilon }=\dot{\mathit{r}}{\dot{\mathit{q}}}_{\epsilon }+\dot{\mathit{r}}{\ddot{\mathit{q}}}_{\epsilon }=\dot{\mathit{r}}{\dot{\mathit{q}}}_{\epsilon }-2\dot{\mathit{r}}{\dot{\mathit{q}}}_{\epsilon }+{\mathit{a}}_{\mathit{t}\epsilon }-{\mathit{a}}_{\mathit{m}4\epsilon }=-\frac{{\mathit{v}}_{\mathit{r}}{\mathit{v}}_{\epsilon }}{\mathit{r}}+{\mathit{a}}_{\mathit{t}\epsilon }-{\mathit{a}}_{\mathit{m}4\epsilon }$

The longitudinal acceleration of the target, a_tε, can be estimated using the following FTDO (Eq. 5):

(5) $\left\{\begin{array}{c}{\dot{\mathit{z}}}_{20}={\mathit{v}}_{20}+{\mathit{g}}_2,{\dot{\mathit{z}}}_{21}={\mathit{v}}_{21},{\dot{\mathit{z}}}_{22}={\mathit{v}}_{22}\hfill \\ {\mathit{v}}_{20}=-{\mathit{\lambda }}_{20}{\mathit{L}}_2^{1/3}{\left|{\mathit{z}}_{20}-{\mathit{v}}_{\epsilon }\right|}^{2/3}\mathrm{sgn}\left({\mathit{z}}_{20}-{\mathit{v}}_{\epsilon }\right)-{\mathit{\sigma }}_{20}\left({\mathit{z}}_{20}-{\mathit{v}}_{\epsilon }\right)+{\mathit{z}}_{21}\hfill \\ {\mathit{v}}_{21}=-{\mathit{\lambda }}_{21}{\mathit{L}}_2^{1/2}{\left|{\mathit{z}}_{21}-{\mathit{v}}_{20}\right|}^{1/2}\mathrm{sgn}\left({\mathit{z}}_{21}-{\mathit{v}}_{20}\right)-{\mathit{\sigma }}_{21}\left({\mathit{z}}_{21}-{\mathit{v}}_{20}\right)+{\mathit{z}}_{22}\hfill \\ {\mathit{v}}_{22}=-{\mathit{\lambda }}_{22}{\mathit{L}}_2{\left|{\mathit{z}}_{22}-{\mathit{v}}_{21}\right|}^{{\mathit{q}}_2/{\mathit{p}}_2}\mathrm{sgn}\left({\mathit{z}}_{22}-{\mathit{v}}_{21}\right)-{\mathit{\sigma }}_{22}\left({\mathit{z}}_{22}-{\mathit{v}}_{21}\right)\hfill \\ {\mathit{z}}_{20}={\tilde{\mathit{v}}}_{\epsilon },{\mathit{z}}_{21}={\tilde{\mathit{a}}}_{\mathit{t}\epsilon },{\mathit{z}}_{22}={\widetilde{\dot{\mathit{a}}}}_{\mathit{t}\epsilon }\hfill \end{array}\right.$

where g₂ = - (v_r v_ε/r) - (qSC^a_y/m)a; ṽ_ε and ã_tε are the disturbance estimates of v_ε and a_tε, respectively; λ₂₀, λ₂₁, λ₂₂, σ₂₀, σ₂₁, and σ₂₂ are the design parameters of the FTDO; q₂ and p₂ are terminal parameters with 0 < p₂ < q₂.

According Shtessel et al. (2007), the FTDO error system is stable during finite time; thus, we define the estimation error of the target acceleration as e₂₁ = z₂₁ - a_tε.

Similarly, the disturbances of the leading interceptor in the AOA loop and pitch angular velocity loop, d_α and d_ωz, can be estimated by (Eqs. 6 and 7):

(6) $\left\{\begin{array}{c}{\dot{\mathit{z}}}_{30}={\mathit{v}}_{30}+{\mathit{g}}_3,{\dot{\mathit{z}}}_{31}={\mathit{v}}_{31},{\dot{\mathit{z}}}_{32}={\mathit{v}}_{32}\hfill \\ {\mathit{v}}_{30}=-{\mathit{\lambda }}_{30}{\mathit{L}}_3^{1/3}{\left|{\mathit{z}}_{30}-{\mathit{x}}_3\right|}^{2/3}\mathrm{sgn}\left({\mathit{z}}_{30}-{\mathit{x}}_3\right)-{\mathit{\sigma }}_{30}\left({\mathit{z}}_{30}-{\mathit{x}}_3\right)+{\mathit{z}}_{31}\hfill \\ {\mathit{v}}_{31}=-{\mathit{\lambda }}_{31}{\mathit{L}}_3^{1/2}{\left|{\mathit{z}}_{31}-{\mathit{v}}_{30}\right|}^{1/2}\mathrm{sgn}\left({\mathit{z}}_{31}-{\mathit{v}}_{30}\right)-{\mathit{\sigma }}_{31}\left({\mathit{z}}_{31}-{\mathit{v}}_{30}\right)+{\mathit{z}}_{32}\hfill \\ {\mathit{v}}_{32}=-{\mathit{\lambda }}_{32}{\mathit{L}}_3{\left|{\mathit{z}}_{32}-{\mathit{v}}_{31}\right|}^{{\mathit{q}}_3/{\mathit{p}}_3}\mathrm{sgn}\left({\mathit{z}}_{32}-{\mathit{v}}_{31}\right)-{\mathit{\sigma }}_{32}\left({\mathit{z}}_{32}-{\mathit{v}}_{31}\right)\hfill \\ {\mathit{z}}_{30}={\tilde{\mathit{x}}}_3,{\mathit{z}}_{31}={\tilde{\mathit{d}}}_{\mathit{\alpha }},{\mathit{z}}_{32}={\widetilde{\dot{\mathit{d}}}}_{\mathit{\alpha }}\hfill \end{array}\right.$

(7) $\left\{\begin{array}{c}{\dot{\mathit{z}}}_{40}={\mathit{v}}_{40}+{\mathit{g}}_4,{\dot{\mathit{z}}}_{41}={\mathit{v}}_{41},{\dot{\mathit{z}}}_{42}={\mathit{v}}_{42}\hfill \\ {\mathit{v}}_{40}=-{\mathit{\lambda }}_{40}{\mathit{L}}_4^{1/3}{\left|{\mathit{z}}_{40}-{\mathit{x}}_4\right|}^{2/3}\mathrm{sgn}\left({\mathit{z}}_{40}-{\mathit{x}}_4\right)-{\mathit{\sigma }}_{40}\left({\mathit{z}}_{40}-{\mathit{x}}_4\right)+{\mathit{z}}_{41}\hfill \\ {\mathit{v}}_{41}=-{\mathit{\lambda }}_{41}{\mathit{L}}_4^{1/2}{\left|{\mathit{z}}_{41}-{\mathit{v}}_{40}\right|}^{1/2}\mathrm{sgn}\left({\mathit{z}}_{41}-{\mathit{v}}_{40}\right)-{\mathit{\sigma }}_{41}\left({\mathit{z}}_{41}-{\mathit{v}}_{40}\right)+{\mathit{z}}_{42}\hfill \\ {\mathit{v}}_{42}=-{\mathit{\lambda }}_{42}{\mathit{L}}_4{\left|{\mathit{z}}_{42}-{\mathit{v}}_{41}\right|}^{{\mathit{q}}_4/{\mathit{p}}_4}\mathrm{sgn}\left({\mathit{z}}_{42}-{\mathit{v}}_{41}\right)-{\mathit{\sigma }}_{42}\left({\mathit{z}}_{42}-{\mathit{v}}_{41}\right)\hfill \\ {\mathit{z}}_{40}={\tilde{\mathit{x}}}_4,{\mathit{z}}_{41}={\tilde{\mathit{d}}}_{{\mathit{\omega }}_{\mathit{z}}},{\mathit{z}}_{32}={\widetilde{\dot{\mathit{d}}}}_{{\mathit{\omega }}_{\mathit{z}}}\hfill \end{array}\right.$

where g₃ = a₃₃ x₃ + x₄; g₄ = a₄₃ x₃ + a₄₄ x₄ + b₄ u; d̃_a and d̃_ωz are the estimated values of d_α and d_ωz with estimation errors of e₃₁ = z₃₁ - _dα and e₄₁ = z₄₁ - d_ωz, respectively.

DESIGN OF THE NONSINGULAR FAST DYNAMIC SMC CONTROLLER

The interceptor IGC model is a mismatched uncertain system; we designed a nonsingular fast dynamic surface SMC model as the control algorithm for the leading interceptor, based on the IGC model (Eq. 3) and FTDO estimations (Eqs. 5 to 7).

• According to dynamic surface SMC design, the second subsystem in Eq. 3 is considered. x_2d is defined as the system tracking command signal.

The first dynamic error surface is defined as (Eq. 8):

(8) ${\mathit{s}}_2=-{\mathit{a}}_{23}^{-1}\left({\mathit{x}}_2-{\mathit{x}}_{2\mathit{d}}\right)$

The derivative of s₂ is determined to obtain the error dynamic equation (Eq. 9):

(9) ${\dot{\mathit{s}}}_2={\mathit{a}}_{23}^{-2}{\dot{\mathit{a}}}_{23}\left({\mathit{x}}_2-{\mathit{x}}_{2\mathit{d}}\right)-{\mathit{a}}_{23}^{-1}\left({\mathit{a}}_{22}{\mathit{x}}_2+\frac{{\mathit{a}}_{\mathit{t}\epsilon }}{\mathit{r}}-{\dot{\mathit{x}}}_{2\mathit{d}}\right)-{\mathit{x}}_3$

The FTDO-estimated ã_tε from Eq. 5 is substituted into Eq. 9 to acquire the virtual control input of the first dynamic surface (Eq. 10):

(10) ${\mathit{x}}_3^{*}={\mathit{a}}_{23}^{-2}{\dot{\mathit{a}}}_{23}\left({\mathit{x}}_2-{\mathit{x}}_{2\mathit{d}}\right)+{\mathit{k}}_2{\mathit{s}}_2-{\mathit{a}}_{23}^{-1}\left({\mathit{a}}_{22}{\mathit{x}}_2+\frac{{\tilde{\mathit{a}}}_{\mathit{t}\epsilon }}{\mathit{r}}-{\dot{\mathit{x}}}_{2\mathit{d}}\right)$

where k₂ > 0. To prevent the increase in computational complexity owing to the ‘explosion of terms’ when finding the derivative of the virtual control input, x^*₃ is fed through a first-order low-pass filter to obtain the filtered virtual control input (Eq. 11):

(11) ${\mathit{\tau }}_3{\dot{\overline{\mathit{x}}}}_3^{*}+{\overline{\mathit{x}}}_3^{*}={\mathit{x}}_3^{*},{\overline{\mathit{x}}}_3^{*}\left(0\right)={\mathit{x}}_3^{*}\left(0\right)$

where τ₃ is the time constant of the filter. Hence, the derivative of the virtual control input after error surface filtering is (Eq. 12):

(12) ${\dot{\overline{\mathit{x}}}}_3^{*}=-{\mathit{\tau }}_3^{-1}\left({\overline{\mathit{x}}}_3^{*}-{\mathit{x}}_3^{*}\right)$

• The second dynamic error surface is defined as (Eq. 13):

(13) ${\mathit{s}}_3={\mathit{x}}_3-{\overline{\mathit{x}}}_3^{*}$

The derivative of s₃ is determined to obtain the error dynamic equation (Eq. 14):

(14) ${\dot{\mathit{s}}}_3={\dot{\mathit{x}}}_3-{\dot{\overline{\mathit{x}}}}_3^{*}={\mathit{a}}_{33}{\mathit{x}}_3+{\mathit{x}}_4+{\mathit{d}}_{\mathit{\alpha }}-{\dot{\overline{\mathit{x}}}}_3^{*}$

Based on the design approach of the first dynamic surface, the FTDO-estimated d̃_a from Eq. 6 is substituted into Eq. 14 to acquire the virtual control input for the second dynamic surface (Eq. 15):

(15) ${\mathit{x}}_4^{*}=-{\mathit{a}}_{33}{\mathit{x}}_3-{\tilde{\mathit{d}}}_{\mathit{\alpha }}+{\dot{\overline{\mathit{x}}}}_3^{*}-{\mathit{k}}_3{\mathit{s}}_3$

where k₃ > 0. x^*₄ is fed through a low-pass filter to obtain (Eq. 16):

(16) ${\mathit{\tau }}_4{\dot{\overline{\mathit{x}}}}_4^{*}+{\overline{\mathit{x}}}_4^{*}={\mathit{x}}_4^{*},{\overline{\mathit{x}}}_4^{*}\left(0\right)={\mathit{x}}_4^{*}\left(0\right)$

where τ₄ is the time constant of the filter. Hence, the derivative of the virtual control input after error surface filtering is (Eq. 17):

(17) ${\dot{\overline{\mathit{x}}}}_4^{*}=-{\mathit{\tau }}_4^{-1}\left({\overline{\mathit{x}}}_4^{*}-{\mathit{x}}_4^{*}\right)$

• The third dynamic error surface is defined as (Eq. 18):

(18) ${\mathit{s}}_4={\mathit{x}}_4-{\overline{\mathit{x}}}_4^{*}$

The derivative of s₄ is determined to obtain the error dynamic equation (Eq. 19):

(19) ${\dot{\mathit{s}}}_4={\dot{\mathit{x}}}_4-{\dot{\overline{\mathit{x}}}}_4^{*}={\mathit{a}}_{43}{\mathit{x}}_3+{\mathit{a}}_{44}{\mathit{x}}_4+{\mathit{b}}_4\mathit{u}+{\mathit{d}}_{{\mathit{\omega }}_{\mathit{z}}}-{\dot{\overline{\mathit{x}}}}_4^{*}$

To prevent singularity in the leading interceptor system and to converge to the equilibrium position within limited time, we design a nonsingular fast sliding mode reaching law (Eq. 20):

(20) $\dot{\mathit{s}}=-{\left|{\mathit{x}}_{\mathit{a}}\right|}^{\mathit{p}/\mathit{q}-1}\left[{\mathit{k}}_{\mathit{a}}\mathit{s}+{\mathit{k}}_{\mathit{b}}{\left|\mathit{s}\right|}^{\partial }\mathrm{sgn}\left(\mathit{s}\right)\right]$

where p, q ∈N⁺ and odd, 1 < p/q < 2, k_a > 0, k_b > 0, and 0 < ∂ < 1.

Based on Eqs. 19 and 20 and the FTDO-estimated d̃_ωz from Eq. 7, the improved nonsingular fast dynamic surface SMC law of the leading interceptor is (Eq. 21):

(21) $\mathit{u}={\mathit{b}}_4^{-1}\left[-{\mathit{a}}_{43}{\mathit{x}}_3-{\mathit{a}}_{44}{\mathit{x}}_4-{\tilde{\mathit{d}}}_{{\mathit{\omega }}_{\mathit{z}}}+{\dot{\overline{\mathit{x}}}}_4^{*}-{\mathit{k}}_4{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}{\mathit{s}}_4-{\mathit{k}}_5{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}{\left|{\mathit{s}}_4\right|}^{\partial }\mathrm{sgn}\left({\mathit{s}}_4\right)\right]$

where k₄ > 0, k₅ > 0, 1 < p/q < 2, and 0 < ∂ < 1.

STABILITY ANALYSIS

It is assumed that the estimation errors of the FTDO system satisfy (Eq. 22):

(22) $\left|{\mathit{e}}_{21}\right|<{\mathit{N}}_2,\left|{\mathit{e}}_{31}\right|<{\mathit{N}}_3,\left|{\mathit{e}}_{41}\right|<{\mathit{N}}_4$

where N₂, N₃, and N₄ are positive constants.

Filtering errors are defined as follows (Eq. 23):

(23) ${\mathit{y}}_3={\overline{\mathit{x}}}_3^{*}-{\mathit{x}}_3^{*},{\mathit{y}}_4={\overline{\mathit{x}}}_4^{*}-{\mathit{x}}_4^{*}$

The derivatives of y₃ and y₄ are determined to obtain the dynamic errors of filtering (Eq. 24):

(24) ${\dot{\mathit{y}}}_3=-{\mathit{\tau }}_3^{-1}{\mathit{y}}_3-{\dot{\mathit{x}}}_3^{*},{\dot{\mathit{y}}}_4=-{\mathit{\tau }}_4^{-1}{\mathit{y}}_4-{\dot{\mathit{x}}}_4^{*}$

Based on Eqs. 8 to 18 and 23, we have:

(25) $\left\{\begin{array}{c}{\mathit{x}}_2=-{\mathit{a}}_{23}{\mathit{s}}_2+{\mathit{x}}_2\mathit{d}\hfill \\ {\mathit{x}}_3={\mathit{s}}_3+{\overline{\mathit{x}}}_3^{*}={\mathit{s}}_3+{\mathit{y}}_3+{\mathit{x}}_3^{*}\hfill \\ {\mathit{x}}_4={\mathit{s}}_4+{\overline{\mathit{x}}}_4^{*}={\mathit{s}}_4+{\mathit{y}}_4+{\mathit{x}}_4^{*}\hfill \end{array}\right.$

Based on Eqs. 3, 8 to 18, and 23 to 25, we have (Eq. 26 to 28):

(26) ${\dot{\mathit{s}}}_2={\mathit{a}}_{23}{\dot{\mathit{a}}}_{23}\left({\mathit{x}}_2-{\mathit{x}}_{2\mathit{d}}\right)-{\mathit{a}}_{23}^{-1}\left({\mathit{a}}_{22}{\mathit{x}}_2+\frac{{\mathit{a}}_{\mathit{t}\epsilon }}{\mathit{r}}-{\dot{\mathit{x}}}_{2\mathit{d}}\right)-{\mathit{s}}_3-{\mathit{y}}_3-{\mathit{x}}_3^{*}=-{\mathit{s}}_3-{\mathit{y}}_3-{\mathit{k}}_2{\mathit{s}}_2+{\tilde{\mathit{e}}}_{21}$

where ${\tilde{\mathit{e}}}_{21}=\frac{\mathit{m}}{{\mathit{qSC}}_{\mathit{y}}^{\mathit{\alpha }}}\left({\tilde{\mathit{a}}}_{\mathit{t}\epsilon }-{\mathit{a}}_{\mathit{t}\epsilon }\right)$

It is assumed that |ẽ₂₁| < Ñ₂, where Ñ₂ is a positive constant.

(27) ${\dot{\mathit{s}}}_3={\mathit{a}}_{33}{\mathit{x}}_3+{\mathit{s}}_4+{\mathit{y}}_4+{\mathit{x}}_4^{*}+{\mathit{d}}_{\mathit{\alpha }}-{\dot{\mathit{x}}}_3^{*}={\mathit{s}}_4+{\mathit{y}}_4-{\mathit{k}}_3{\mathit{s}}_3-{\mathit{e}}_{31}$

(28) ${\dot{\mathit{s}}}_4={\mathit{a}}_{43}{\mathit{x}}_3+{\mathit{a}}_{44}{\mathit{x}}_4+{\mathit{b}}_4\mathit{u}+{\mathit{d}}_{{\mathit{\omega }}_{\mathit{z}}}-{\mathit{x}}_4^{*}=-{\mathit{k}}_4{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}{\mathit{s}}_4-{\mathit{k}}_5{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}{\left|{\mathit{s}}_4\right|}^{\partial }\mathrm{sgn}\left({\mathit{s}}_4\right)-{\mathit{e}}_{41}$

As all variables and their derivatives in the system model are bounded, there exist continuous functions z̃₃ < 0 and z̃₄ < 0 such that variables ẋ* ₃ and ẋ* ₄ satisfy (Eq. 29):

(29) $\begin{array}{c}\left|{\mathit{x}}_3^{*}\right|\le {\tilde{\mathit{z}}}_3\left({\mathit{s}}_2,{\mathit{s}}_3,{\mathit{y}}_3,{\mathit{y}}_4,{\mathit{e}}_{21},{\mathit{e}}_{31},{\mathit{k}}_2,{\mathit{k}}_3,{\mathit{x}}_{2\mathit{d}},{\dot{\mathit{x}}}_{2\mathit{d}},{\ddot{\mathit{x}}}_{2\mathit{d}}\right)\hfill \\ \left|{\dot{\mathit{x}}}_4^{*}\right|\le {\tilde{\mathit{z}}}_4\left({\mathit{s}}_2,{\mathit{s}}_3,{\mathit{s}}_4,{\mathit{y}}_3,{\mathit{y}}_4,{\mathit{e}}_{21},{\mathit{e}}_{31},{\mathit{k}}_2,{\mathit{k}}_3,{\mathit{x}}_{2\mathit{d}},{\dot{\mathit{x}}}_{2\mathit{d}},{\ddot{\mathit{x}}}_{2\mathit{d}}\right)\hfill \end{array}$

Given constants χ and R^*, where χ > 0 and R* > 0, the following compact sets are defined: B₂ = [x_2d, ẋ₂d, ẋ̇_2d]T, x_2d + ẋ_2d + ẋ̇_2d ≤ χ and B₂ = [s₂, s₃, s₄, y₃, y₄, e₂₁, e₃₁]T, s₂ 2, _{s2 3}, _{s2 4}, _{y2 3}, _{y2 4} ≤ R *. Moreover, we know that B₁ × B₂ is a compact set. Let constants M₃ and M₄ be the maxima of ˜z₃ and ˜z₄ on B₁ × B₂, respectively, where M₃ > 0 and M₄ > 0. We have (Eq. 30):

(30) $\left|{\dot{\mathit{x}}}_3^{*}\right|\le {\mathit{M}}_3,\left|{\dot{\mathit{x}}}_4^{*}\right|\le {\mathit{M}}_4$

Based on Eqs. 25 to 28, the following can be obtained (Eqs. 31 to 35):

(31) ${\mathit{s}}_2{\dot{\mathit{s}}}_2={\mathit{s}}_2\left(-{\mathit{s}}_3-{\mathit{y}}_3-{\mathit{k}}_2{\mathit{s}}_2+{\tilde{\mathit{e}}}_{21}\right)\le \left(\frac{3}{2}-{\mathit{k}}_2\right){\mathit{s}}_2^2+\frac{1}{2}{\mathit{s}}_3^2+\frac{1}{2}{\mathit{y}}_3^2+\frac{1}{2}{\tilde{\mathit{N}}}_2^2$

(32) ${\mathit{s}}_3{\dot{\mathit{s}}}_3={\mathit{s}}_3\left({\mathit{s}}_4+{\mathit{y}}_4-{\mathit{k}}_3{\mathit{s}}_3-{\mathit{e}}_{31}\right)\le \left(\frac{3}{2}-{\mathit{k}}_3\right){\mathit{s}}_3^2+\frac{1}{2}{\mathit{s}}_4^2+\frac{1}{2}{\mathit{y}}_4^2+\frac{1}{2}{\mathit{N}}_3^2$

(33) $\begin{array}{c}{\mathit{s}}_4{\dot{\mathit{s}}}_4={\mathit{s}}_4\left(-{\mathit{k}}_4{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}{\mathit{s}}_4-{\mathit{k}}_5{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}{\left|{\mathit{s}}_4\right|}^{\partial }\mathrm{sgn}\left({\mathit{s}}_4\right)-{\mathit{e}}_{41}\right)\le -{\mathit{k}}_4{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}{\mathit{s}}_4^2-{\mathit{k}}_5\left|{\mathit{x}}_4\right|{}^{\mathit{p}/\mathit{q}-1}\left|{\mathit{s}}_4\right|^{\partial +1}+\frac{1}{2}{\mathit{s}}_4^2+\frac{1}{2}{\mathit{e}}_{41}^2\le \hfill \\ \le \left(\frac{1}{2}-{\mathit{k}}_4{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}+\frac{3}{2}{\mathit{k}}_5{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}\right){\mathit{s}}_4^2+\frac{1}{2}{\mathit{N}}_4^2+\frac{1}{2}{\mathit{k}}_5{\left|{\mathit{x}}_4\right|}^{\mathit{p}/\mathit{q}-1}\hfill \end{array}$

(34) ${\mathit{y}}_3{\dot{\mathit{y}}}_3={\mathit{y}}_3\left(-{\mathit{\tau }}_3^{-1}{\mathit{y}}_3-{\dot{\mathit{x}}}_3^{*}\right)\le -{\mathit{\tau }}_3^{-1}{\mathit{y}}_3^2+\frac{1}{2}\left({\mathit{y}}_3^2+{\mathit{M}}_3^2\right)$

(35) ${\mathit{y}}_4{\dot{\mathit{y}}}_4={\mathit{y}}_4\left(-{\mathit{\tau }}_4^{-1}{\mathit{y}}_4-{\dot{\mathit{x}}}_4^{*}\right)\le -{\mathit{\tau }}_4^{-1}{\mathit{y}}_4^2+\frac{1}{2}\left({\mathit{y}}_4^2+{\mathit{M}}_4^2\right)$

According to the nonlinear IGC model in Eq. 3, the Lyapunov function is considered as:

(36) $\mathit{V}=\frac{1}{2}\left({\mathit{s}}_2^2+{\mathit{s}}_3^2+{\mathit{s}}_4^2+{\mathit{y}}_3^2+{\mathit{y}}_4^2\right)$

The derivatives of both sides of Eq. (36) are determined to obtain (Eq. 37):

(37) $\begin{array}{c}\dot{\mathit{V}}={\mathit{s}}_2{\dot{\mathit{s}}}_2+{\mathit{s}}_3{\dot{\mathit{s}}}_3+{\mathit{s}}_4{\dot{\mathit{s}}}_4+{\mathit{y}}_3{\dot{\mathit{y}}}_3+{\mathit{y}}_4{\dot{\mathit{y}}}_4\le \left(\frac{3}{2}-{\mathit{k}}_2\right){\mathit{s}}_2^2+\frac{1}{2}{\mathit{s}}_3^2+\frac{1}{2}{\mathit{y}}_3^2+\frac{1}{2}{\tilde{\mathit{N}}}_2^2+\left(\frac{3}{2}-{\mathit{k}}_3\right){\mathit{s}}_3^2+\frac{1}{2}{\mathit{s}}_4^2+\frac{1}{2}{\mathit{y}}_4^2+\frac{1}{2}{\mathit{N}}_3^2\hfill \\ +\left(\frac{1}{2}-{\mathit{k}}_4+\frac{3}{2}{\mathit{k}}_5\right){\mathit{s}}_4^2+\frac{1}{2}{\mathit{N}}_4^2+\frac{1}{2}{\mathit{k}}_5-{\mathit{\tau }}_3^{-1}{\mathit{y}}_3^2+\frac{1}{2}\left({\mathit{y}}_3^2+{\mathit{M}}_3^2\right)+-{\mathit{\tau }}_4^{-1}{\mathit{y}}_4^2+\frac{1}{2}\left({\mathit{y}}_4^2+{\mathit{M}}_4^2\right)\hfill \end{array}$

The design parameters should satisfy (Eq. 38):

(38) $\left\{\begin{array}{c}{\mathit{k}}_2\ge \frac{3}{2}+\frac{1}{2}\mathit{\kappa },{\mathit{k}}_3\ge 2+\frac{1}{2}\mathit{\kappa },{\mathit{k}}_4\ge \frac{3}{2}{\mathit{k}}_5+1+\frac{1}{2}\mathit{\kappa }\hfill \\ {\mathit{\tau }}_3^{-1}\ge 1+\frac{1}{2}\mathit{\kappa },{\mathit{\tau }}_4^{-1}\ge 1+\frac{1}{2}\mathit{\kappa }\hfill \end{array}\right.$

where κ is a constant, and κ > 0. Thus, we have:

(39) $\dot{\mathit{V}}\le \mathit{\kappa }\mathit{V}+\stackrel{⌢}{\mathit{A}}$

where $\stackrel{⌢}{\mathit{A}}=\frac{1}{2}{\tilde{\mathit{N}}}_2^2+\frac{1}{2}{\mathit{N}}_3^2+\frac{1}{2}{\mathit{N}}_4^2+\frac{1}{2}{\mathit{M}}_3^2+\frac{1}{2}{\mathit{M}}_4^2+\frac{1}{2}{\mathit{k}}_5$

From Eq. 39, we finally have (Eq. 40):

(40) $\mathit{V}\left(\mathit{t}\right)\le \left\{\left[\mathit{\kappa }\mathit{V}\left(0\right)-\stackrel{⌢}{\mathit{A}}\right]{\mathit{e}}^{-\mathit{\kappa }\mathit{t}}+\stackrel{⌢}{\mathit{A}}\right\}/\mathit{\kappa }$

Therefore, system convergence can be ensured by appropriately adjusting design parameters k₂, k₃, k₄, k₅, τ₃, and τ₄. If k₂, k₃, k₄, and k₅ are increased while τ₃ and τ₄ are reduced, a sufficiently large κ can be ensured so that filtering error and error surface are sufficiently small. This ensures control accuracy.

DESIGN OF DISTRIBUTED NETWORK BASED COOPERATIVE CONTROL STRATEGY

The problem of the cooperative control of multiple intelligent agents can be described as a graph, which can be analyzed using graph theory. For the guidance and control of multiple interceptor missiles, one missile acquires the state information of other missiles through information exchange to achieve coordination among them. Therefore, a leader-follower topological structure that consists of one leading interceptor and n following interceptors can be built. This structure is described using an undirected graph. Regarding each missile as a communication node, the information exchange between missiles is expressed as G = {V, E, A}, where V = {v₁, v₂, v₃, ....., v_n} represents the set formed by all missile nodes, E denotes the links between nodes, and A = [a_ij] ∈Rⁿ ×ⁿ is the adjacency matrix of the undirected graph. If information exchange exists between nodes i and j, then a_ij > 0; otherwise, a_ij = 0. L is the Laplace matrix of G, the elements of which satisfy (Eq. 41):

(41) $\left\{\begin{array}{c}{\overline{\mathit{l}}}_{\mathit{ii}}=\sum _{\mathit{j}=1,\mathit{j}\ne \mathit{i}}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\hfill \\ {\overline{\mathit{l}}}_{\mathit{ij}}=-{\overline{\mathit{a}}}_{\mathit{ij}},\mathit{j}\ne \mathit{i}\hfill \end{array}\right.$

In a multi-missile topology with a leading interceptor, the leading interceptor features an independent state that does not change with followers. The purpose of including the leader state into the synchronization algorithm as part of the cooperative control strategy is that follower states should approach the leader state.

Let B = diag {b₁, b₂, ....b_n} represent whether each follower can acquire the information of the leader, where b_i > 0, i ∈ {1, 2, 3, ...., n} implies yes and b_i = 0, i ∈ {1, 2, 3, ...., n} implies no.

Based on the fixed leader-follower topology, an improved cooperative control strategy for multiple interceptors can be designed using the distributed network method, as follows (Eq. 42):

(42) ${\mathit{v}}_{\mathit{i}}{\overline{\mathit{k}}}_{\mathit{i}1}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_0-{\mathit{x}}_{\mathit{i}}\right)\right)+{\overline{\mathit{k}}}_{\mathit{i}2}{\left|{\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_0-{\mathit{x}}_{\mathit{i}}\right)\right|}^{{\partial }_{\mathit{i}}}\mathrm{sgn}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_0-{\mathit{x}}_{\mathit{i}}\right)\right)+{\dot{\mathit{x}}}_0$

where x₀ denotes the position of the leader; x_i, i ∈ {1, 2, 3, ...., n} denotes the position of a follower; v_i = ẋ_i denotes the velocity of a follower; ∂_i > 0; k_i1 > 0 and k_i2 > 0 are constants.

Proof:

Lemma 1(Zou et al. 2010): Laplace matrix L - has the following properties:

• If G - is connected, the eigenvalue of L - is , which is referred to as the algebraic connectivity of the graph; a larger value indicates a more connected network.

• 0 is one eigenvalue of L - , and the corresponding eigenvector is 1.

Defining an error variable, e_i = x_i - x ₀, we have (Eq. 43):

(43) ${\dot{\mathit{e}}}_{\mathit{i}}={\dot{\mathit{x}}}_{\mathit{i}}-{\dot{\mathit{x}}}_0={\overline{\mathit{k}}}_{\mathit{i}1}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right)+{\overline{\mathit{k}}}_{\mathit{i}2}{\left|{\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right|}^{{\partial }_{\mathit{i}}}\mathrm{sgn}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_0-{\mathit{x}}_{\mathit{i}}\right)\right)$

The Lyapunov function is defined as (Eq. 44):

(44) $\mathit{V}=\frac{1}{2}{\mathit{e}}^{\mathit{T}}\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)\mathit{e}$

where e = [e₁, e₂, ...,e_n]^T.

We define k₁ = min {k -i ₁}, k₂ = min {k_i2}, and ∂ = min{∂_i}. The derivatives of Eq. 44 are determined to obtain (Eq. 45):

(45) $\begin{array}{c}\dot{\mathit{V}}={\dot{\mathit{e}}}^{\mathit{T}}\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)\mathit{e}=-{\sum }_{\mathit{i}=1}^{\mathit{n}}\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]{\dot{\mathit{e}}}_{\mathit{i}}=-{\sum }_{\mathit{i}=1}^{\mathit{n}}\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]\left[{\overline{\mathit{k}}}_{\mathit{i}1}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right)\right.+\hfill \\ +{\overline{\mathit{k}}}_{\mathit{i}2}{\left|{\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right|}^{{\partial }_{\mathit{i}}}\mathrm{sgn}\left.\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_0-{\mathit{x}}_{\mathit{i}}\right)\right.\right]=-{\sum }_{\mathit{i}=1}^{\mathit{n}}{\overline{\mathit{k}}}_{\mathit{i}1}{\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]}^2-{\sum }_{\mathit{i}=1}^{\mathit{n}}{\overline{\mathit{k}}}_{\mathit{i}2}{\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]}^{1+{\partial }_{\mathit{i}}}\le \hfill \\ \le -{\overline{\mathit{k}}}_1{{\sum }_{\mathit{i}=1}^{\mathit{n}}\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]}^2-{\overline{\mathit{k}}}_2{{\sum }_{\mathit{i}=1}^{\mathit{n}}\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]}^{1+\partial }\le -{\overline{\mathit{k}}}_1{\sum }_{\mathit{i}=1}^{\mathit{n}}{\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]}^2-{\overline{\mathit{k}}}_2{\left({\sum }_{\mathit{i}=1}^{\mathit{n}}{\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]}^2\right)}^{\frac{1+\partial }{2}}\hfill \end{array}$

Considering V(e) ≠ 0, Eq. 45 can be rewritten as:

(46) $\frac{{\sum }_{\mathit{i}=1}^{\mathit{n}}{\left[{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{a}}_{\mathit{ij}}\left({\mathit{e}}_{\mathit{j}}-{\mathit{e}}_{\mathit{i}}\right)-{\overline{\mathit{b}}}_{\mathit{i}}{\mathit{e}}_{\mathit{i}}\right]}^2}{\mathit{V}\left(\mathit{e}\right)}=\frac{{\mathit{e}}^{\mathit{T}}{\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)}^{\mathit{T}}\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)\mathit{e}}{\frac{1}{2}\left({\mathit{e}}^{\mathit{T}}{\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)}^{\mathit{T}}\mathit{e}\right)}\ge 2{\mathit{\lambda }}_{\min }\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)$

From Eqs. 46 to 47:

(47) $\dot{\mathit{V}}\left(\mathit{t}\right)\le -{\overline{\mathit{k}}}_1\left[2{\mathit{\lambda }}_{\min }\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)\right]-{\overline{\mathit{k}}}_2{\left[2{\mathit{\lambda }}_{\min }\left(\overline{\mathit{L}}+\overline{\mathit{B}}\right)\right]}^{\frac{1+\partial }{2}}$

It is known from Lemma 1 that V(t) can converge within finite time. That is, followers can converge to the state of the leader, achieving successful cooperative guidance and control for multiple interceptors.

REALIZATION OF THE DISTRIBUTED NETWORK BASED COOPERATIVE CONTROL STRATEGY

The essence of multi-interceptor cooperative attack is to coordinate the positions of following interceptors and the leading interceptor. Therefore, to realize the distributed network synchronization strategy, each interceptor in the network must follow the velocity commands provided by the synchronization strategy given by Eq. 42.

The kinematic relations of the interceptors in the network that participate in coordinated attack are (Eq. 48):

(48) $\left\{\begin{array}{c}{\dot{\mathit{x}}}_{\mathit{i}}={\mathit{V}}_{\mathit{i}}\cos {\mathit{\theta }}_{\mathit{i}}\hfill \\ {\dot{\mathit{y}}}_{\mathit{i}}={\mathit{V}}_{\mathit{i}}\sin {\mathit{\theta }}_{\mathit{i}}\hfill \end{array}\right.$

whereẋ_i and ẏ_i represent the two velocity components within the inertial frame of interceptor i in the pitch channel.

Based on the distributed network synchronization strategy given in Eq. 42, the missile velocity reference commands are:

(49) $\left\{\begin{array}{c}\hfill {\overline{\mathit{V}}}_{\mathit{mxi}}={\overline{\mathit{k}}}_{\mathit{i}1}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_{\mathit{m}}-{\mathit{x}}_{\mathit{i}}\right)\right)\\ \hfill +{\overline{\mathit{k}}}_{\mathit{i}2}{\left|{\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_{\mathit{m}}-{\mathit{x}}_{\mathit{i}}\right)\right|}^{{\partial }_{\mathit{i}}}\\ \hfill \mathrm{sgn}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_{\mathit{m}}-{\mathit{x}}_{\mathit{i}}\right)\right)+{\dot{\mathit{x}}}_{\mathit{m}}\\ \hfill {\overline{\mathit{V}}}_{\mathit{myi}}={\overline{\mathit{k}}}_{\mathit{i}1}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{y}}_{\mathit{j}}-{\mathit{y}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{y}}_{\mathit{m}}-{\mathit{y}}_{\mathit{i}}\right)\right)\\ \hfill +{\overline{\mathit{k}}}_{\mathit{i}2}{\left|{\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{y}}_{\mathit{j}}-{\mathit{y}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{y}}_{\mathit{m}}-{\mathit{y}}_{\mathit{i}}\right)\right|}^{{\partial }_{\mathit{i}}}\\ \hfill \mathrm{sgn}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{y}}_{\mathit{j}}-{\mathit{y}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{y}}_{\mathit{m}}-{\mathit{y}}_{\mathit{i}}\right)\right)+{\dot{\mathit{y}}}_{\mathit{m}}\end{array}\right.$

Using Eqs. 48 and 49, the total velocity and flight path angle of the interceptor are obtained as (Eq. 50):

(50) $\left\{\begin{array}{c}{\overline{\mathit{V}}}_{\mathit{mi}}^{*}=\sqrt{{\left({\overline{\mathit{V}}}_{\mathit{mxi}}\right)}^2+{\left({\overline{\mathit{V}}}_{\mathit{myi}}\right)}^2}\hfill \\ {\overline{\mathit{\theta }}}_{\mathit{mi}}^{*}=\mathrm{arctam}\left(\frac{{\overline{\mathit{V}}}_{\mathit{myi}}}{{\overline{\mathit{V}}}_{\mathit{mxi}}}\right)\hfill \end{array}\right.$

The signal must be filtered to obtain the derivative of the total velocity and flight path angle, x - . . Let x - and x -* ₁be the actual command and ideal command of the required signal, respectively. We obtain (Eq. 51):

(51) $\ddot{\overline{\mathit{x}}}=-2{\mathit{\zeta }}_{\mathit{n}}{\mathit{\omega }}_{\mathit{n}}\dot{\overline{\mathit{x}}}-{\mathit{\omega }}_{\mathit{n}}^2\overline{\mathit{x}}+{\mathit{\omega }}_{\mathit{n}}^2{\overline{\mathit{x}}}_1^{*}$

where ζ_n and ω_n are the damping and bandwidth of the filter, respectively. The use of the filter can effectively address the differentiation problem of the command signals without affecting the amplitudes of the commands and their derivatives.

DESIGN OF FOLLOWING INTERCEPTOR CONTROLLER

It can be seen that the commands provided by the cooperative control strategy can be converted to velocity and flight path angle commands. The controller for following interceptors adopts the dynamic surface SMC algorithm to achieve command signal tracking for the missiles in the cooperative network. Assuming that missile velocities are controllable and air resistance and gravity can be neglected, the flight velocity of a follower can be expressed as (Eq. 52):

(52) ${\dot{\mathit{V}}}_{\mathit{m}}=\frac{\cos {\mathit{\alpha }}_{\mathit{i}}}{\mathit{m}}{\mathit{P}}_{\mathit{i}}$

where P_i is engine thrust.

From Eq. 53, an error surface is defined as:

(53) ${\mathit{s}}_{\mathit{v}}={\mathit{V}}_{\mathit{m}}-{\overline{\mathit{V}}}_{\mathit{mi}}$

where V_mi is the filtered total velocity of the interceptor. The derivative of s_v is obtained as (Eq. 54):

(54) $\dot{\mathit{s}}{}_{\mathit{v}}=\frac{\cos {\mathit{\alpha }}_{\mathit{i}}}{\mathit{m}}{\mathit{P}}_{\mathit{i}}-{\dot{\overline{\mathit{V}}}}_{\mathit{mi}}$

where V̇_mi is the derivative of V_mi .

The sliding mode reaching law provided below is used to ensure that interceptor velocity can rapidly follow the system command.

(55) $\dot{\mathit{s}}=-{\mathit{k}}_{\mathit{a}}\mathit{s}-{\mathit{k}}_{\mathit{b}}{\left|\mathit{s}\right|}^{\mathit{\lambda }}\mathrm{sgn}\left(\mathit{s}\right)$

Based on Eqs. 53 to 56, the engine thrust of a follower is (Eq. 56):

(56) ${\mathit{P}}_{\mathit{i}}=\frac{\mathit{m}}{\cos {\mathit{\alpha }}_{\mathit{i}}}\left[{\dot{\overline{\mathit{V}}}}_{\mathit{mi}}-{\mathit{k}}_{\mathit{v}1}{\mathit{s}}_{\mathit{v}}-{\mathit{k}}_{\mathit{v}2}{\left|{\mathit{s}}_{\mathit{v}}\right|}^{{\mathit{\lambda }}_{\mathit{v}}}\mathrm{sgn}\left({\mathit{s}}_{\mathit{v}}\right)\right]$

Let x_i1 = θ_mi, x_i2 = α_i, and x_i3 = ω_i. The kinematic equation of interceptor i in the pitch channel is (Eq. 57):

(57) $\left\{\begin{array}{c}{\dot{\mathit{x}}}_{\mathit{i}1}={\mathit{a}}_{\mathit{i}23}{\mathit{x}}_{\mathit{i}2}+{\mathit{d}}_{\mathit{i}1}\hfill \\ {\dot{\mathit{x}}}_{\mathit{i}2}={\mathit{a}}_{\mathit{i}33}{\mathit{x}}_{\mathit{i}2}+{\mathit{d}}_{\mathit{i}2}\hfill \\ {\dot{\mathit{x}}}_{\mathit{i}3}={\mathit{a}}_{\mathit{i}43}{\mathit{x}}_{\mathit{i}2}+{\mathit{a}}_{\mathit{i}44}{\mathit{x}}_{\mathit{i}3}+{\mathit{b}}_{\mathit{i}3}{\mathit{u}}_{\mathit{i}}+{\mathit{d}}_{\mathit{i}3}\hfill \end{array}\right.$

where ${\mathit{a}}_{\mathit{i}23}=\frac{{\mathit{q}}_{\mathit{i}}{\mathit{SC}}_{\mathit{yi}}^{{\mathit{\alpha }}_{\mathit{i}}}}{{\mathit{mV}}_{\mathit{m}}},{\mathit{a}}_{\mathit{i}33}=-\frac{{\mathit{q}}_{\mathit{i}}{\mathit{SC}}_{\mathit{yi}}^{{\mathit{\alpha }}_{\mathit{i}}}}{{\mathit{mV}}_{\mathit{m}}},{\mathit{a}}_{\mathit{i}43}=\frac{{\mathit{q}}_{\mathit{i}}\mathit{S}{\overline{\mathit{Lm}}}_{\mathit{zi}}^{{\mathit{\alpha }}_{\mathit{i}}}}{{\mathit{J}}_{\mathit{z}}},{\mathit{a}}_{\mathit{i}44}=\frac{{\mathit{q}}_{\mathit{i}}\mathit{S}{\overline{\mathit{Lm}}}_{\mathit{zi}}^{{\mathit{\omega }}_{\mathit{zi}}}}{{\mathit{J}}_{\mathit{z}}},{\mathit{b}}_{\mathit{i}4}=\frac{1}{{\mathit{J}}_{\mathit{z}}}$

Following the FTDO used above for the leader, an FTDO is designed to estimate disturbance d_i1 (Eq. 58):

(58) $\left\{\begin{array}{c}{\dot{\mathit{z}}}_{\mathit{i}10}={\mathit{v}}_{\mathit{i}10}+{\mathit{a}}_{\mathit{i}23}{\mathit{x}}_{\mathit{i}2},{\dot{\mathit{z}}}_{\mathit{i}11}={\mathit{v}}_{\mathit{i}11},{\dot{\mathit{z}}}_{\mathit{i}12}={\mathit{v}}_{\mathit{i}22}\hfill \\ {\mathit{v}}_{\mathit{i}10}=-{\mathit{\lambda }}_{\mathit{i}10}{\mathit{L}}_{\mathit{i}1}^{1/3}{\left|{\mathit{z}}_{\mathit{i}10}-{\mathit{x}}_{\mathit{i}1}\right|}^{2/3}\mathrm{sgn}\left({\mathit{z}}_{\mathit{i}10}-{\mathit{x}}_{\mathit{i}1}\right)+{\mathit{z}}_{\mathit{i}11}\hfill \\ {\mathit{v}}_{\mathit{i}11}=-{\mathit{\lambda }}_{\mathit{i}11}{\mathit{L}}_{\mathit{i}1}^{1/2}{\left|{\mathit{z}}_{\mathit{i}11}-{\mathit{v}}_{\mathit{i}10}\right|}^{1/2}\mathrm{sgn}\left({\mathit{z}}_{\mathit{i}11}-{\mathit{v}}_{\mathit{i}10}\right)+{\mathit{z}}_{\mathit{i}12}\hfill \\ {\mathit{v}}_{\mathit{i}12}=-{\mathit{\lambda }}_{\mathit{i}12}{\mathit{L}}_{\mathit{i}1}{\left|{\mathit{z}}_{\mathit{i}12}-{\mathit{v}}_{\mathit{i}11}\right|}^{{\mathit{q}}_{\mathit{i}1}/{\mathit{p}}_{\mathit{i}1}}\mathrm{sgn}\left({\mathit{z}}_{\mathit{i}12}-{\mathit{v}}_{\mathit{i}11}\right)\hfill \\ {\mathit{z}}_{\mathit{i}10}={\tilde{\mathit{x}}}_{\mathit{i}1},{\mathit{z}}_{\mathit{i}11}={\tilde{\mathit{d}}}_{\mathit{i}1},{\mathit{z}}_{\mathit{i}12}={\dot{\tilde{\mathit{d}}}}_{\mathit{i}1}\hfill \end{array}\right.$

where the estimated value of d_i1 is d̃_i1 and the estimation errors are e_i11 = z_i11 - d_i1.

Similarly, unknown disturbances d_i2 and d_i3 are estimated with estimation errors of e_i12 = z_{i 12} - d_{i 2} and e_{i 13} = z_i13 - d_i3, respectively.

Based on the FTDO estimates and kinematic model in the pitch channel, the controller for following interceptors is designed using the dynamic surface SMC law.

• The first dynamic error surface is defined as (Eq. 59):

(59) ${\mathit{s}}_{\mathit{i}1}={\mathit{a}}_{\mathit{i}23}^{-1}\left({\mathit{x}}_{\mathit{i}1}-{\overline{\mathit{\theta }}}_{\mathit{m}}\right)$

The derivative of s_i1 is obtained (Eq. 60):

(60) ${\mathit{s}}_{\mathit{i}1}={\mathit{x}}_{\mathit{i}2}+{\mathit{a}}_{\mathit{i}23}^{-1}\left({\mathit{d}}_{\mathit{i}1}-{\dot{\overline{\mathit{\theta }}}}_{\mathit{m}}\right)$

Based on the dynamic surface design method and FTDO, the virtual control input of the first dynamic surface is (Eq. 61):

(61) ${\mathit{x}}_{\mathit{i}2}^{*}={\mathit{a}}_{\mathit{i}2}^{-1}\left(-{\tilde{\mathit{d}}}_{\mathit{i}1}+{\dot{\overline{\mathit{\theta }}}}_{\mathit{m}}\right)-{\mathit{k}}_{\mathit{i}1}{\mathit{s}}_{\mathit{i}1}$

where k_i1 > 0. To prevent the increase in computational complexity owing to the ‘explosion of terms’ while finding the derivative of the virtual control input, x * ₃ is fed through a first-order low-pass filter to obtain the filtered virtual control input (Eq. 62):

(62) ${\mathit{\tau }}_{\mathit{i}2}{\dot{\overline{\mathit{x}}}}_{\mathit{i}2}^{*}+{\overline{\mathit{x}}}_{\mathit{i}2}^{*}={\mathit{x}}_{\mathit{i}2}^{*},{\overline{\mathit{x}}}_{\mathit{i}2}^{*}\left(0\right)={\mathit{x}}_{\mathit{i}2}^{*}\left(0\right){\dot{\overline{\mathit{x}}}}_{\mathit{i}2}^{*}=-{\mathit{\tau }}_{\mathit{i}2}^{-1}\left({\overline{\mathit{x}}}_{\mathit{i}2}^{*}-{\mathit{x}}_{\mathit{i}2}^{*}\right)$

where τ_i2 > 0 is the time constant of the filter.

The second dynamic error surface is defined as (Eq. 63):

(63) ${\mathit{s}}_{\mathit{i}2}={\mathit{x}}_{\mathit{i}2}-{\overline{\mathit{x}}}_{\mathit{i}2}^{*}$

The derivate of s_i2 is determined (Eq. 64):

(64) ${\dot{\mathit{s}}}_{\mathit{i}2}={\mathit{a}}_{\mathit{i}33}{\mathit{x}}_{\mathit{i}2}+{\mathit{x}}_{\mathit{i}3}+{\mathit{d}}_{\mathit{i}2}-{\dot{\overline{\mathit{x}}}}_{\mathit{i}2}^{*}$

Similar to the first dynamic surface, the virtual control input of the second dynamic surface is (Eq. 65):

(65) ${\mathit{x}}_{\mathit{i}3}^{*}=-{\mathit{a}}_{\mathit{i}33}{\mathit{x}}_{\mathit{i}2}-{\tilde{\mathit{d}}}_{\mathit{i}2}+{\dot{\overline{\mathit{x}}}}_{\mathit{i}2}^{*}-{\mathit{k}}_{\mathit{i}2}{\mathit{s}}_{\mathit{i}2}$

where k_i2 > 0. x^*_i3 is fed through the following low-pass filter to obtain (Eq. 66):

(66) $\begin{array}{c}{\mathit{\tau }}_{\mathit{i}3}{\dot{\overline{\mathit{x}}}}_{\mathit{i}3}^{*}+{\overline{\mathit{x}}}_{\mathit{i}3}^{*}={\mathit{x}}_{\mathit{i}3}^{*},{\overline{\mathit{x}}}_{\mathit{i}3}^{*}\left(0\right)={\mathit{x}}_{\mathit{i}3}^{*}\left(0\right)\hfill \\ {\dot{\overline{\mathit{x}}}}_{\mathit{i}3}^{*}=-\mathit{\tau }{}_{\mathit{i}3}{}^{-1}\left({\overline{\mathit{x}}}_{\mathit{i}3}^{*}-{\mathit{x}}_{\mathit{i}3}^{*}\right)\hfill \end{array}$

where τ_i3 > 0 is the time constant of the filter.

• The third dynamic error surface is defined as (Eq. 67):

(67) ${\mathit{s}}_{\mathit{i}3}={\mathit{x}}_{\mathit{i}3}-{\overline{\mathit{x}}}_{\mathit{i}3}^{*}$

The derivative of s_i3 is determined (Eq. 68):

(68) ${\dot{\mathit{s}}}_{\mathit{i}3}={\mathit{a}}_{\mathit{i}43}{\mathit{x}}_{\mathit{i}2}+{\mathit{a}}_{\mathit{i}44}{\mathit{x}}_{\mathit{i}3}+{\mathit{b}}_{\mathit{i}3}{\mathit{u}}_{\mathit{i}}+{\mathit{d}}_{\mathit{i}3}-{\dot{\overline{\mathit{x}}}}_{\mathit{i}3}^{}$

The following sliding mode reaching law is used to ensure a high rate of convergence for the system (Eq. 69):

(69) $\dot{\mathit{s}}=-{\mathit{k}}_{\mathit{a}}\mathit{s}-{\mathit{k}}_{\mathit{b}}{\left|\mathit{s}\right|}^{\mathit{\lambda }}\mathrm{sgn}\left(\mathit{s}\right)$

The dynamic surface SMC law for following interceptors is (Eq. 70):

(70) ${\mathit{u}}_{\mathit{i}}={\mathit{b}}_{\mathit{i}3}^{-1}\left[-{\mathit{a}}_{\mathit{i}43}{\mathit{x}}_{\mathit{i}2}-{\mathit{a}}_{\mathit{i}44}{\mathit{x}}_{\mathit{i}3}-{\tilde{\mathit{d}}}_{\mathit{i}3}+{\dot{\overline{\mathit{x}}}}_{\mathit{i}3}^{*}-{\mathit{k}}_{\mathit{i}31}{\mathit{s}}_{\mathit{i}3}-{\mathit{k}}_{\mathit{i}32}{\left|{\mathit{s}}_{\mathit{i}3}\right|}^{{\mathit{\lambda }}_{\mathit{i}3}}\mathrm{sgn}\left({\mathit{s}}_{\mathit{i}3}\right)\right]$

where k_i31 > 0, k_i32 > 0, and 0 < λ_i3 < 1.

The stability of the control algorithm for followers can be proved using Eqs. 20 to 40.

SIMULATION VALIDATION

To validate the effectiveness of the improved distributed cooperative IGC algorithm, we assume a communication topology in which the leader can communicate with three other followers and neighboring communication exists between the followers, as shown in Fig. 2. The initial conditions of the leader, followers, and target are listed in Table 1.

Figure 2
Communication topology of the leader and followers.

Table 1
Initial conditions of the leading interceptor, following interceptors, and target.

No.	ID	Parameter	Value	Parameter	Value	Parameter	Value
1	Leader	xm 0	0 m	ym 0	3300 m	Vm 0	400 m/s
2	Missile 1	x 1 m 0	100 m	y 1 m 0	2400 m	V 1 m 0	400 m/s
3	Missile 2	x 2 m 0	300 m	y 2 m 0	2700 m	V 2 m 0	400 m/s
4	Missile 3	x 3 m 0	200 m	y 3 m 0	2600 m	V 3 m 0	400 m/s
5	Target	xt 0	3000 m	yt 0	380 m	Vt 0	240 m/s

The parameters of the FTDO are provided below.

1. FTDO parameters of the leader: λ₂₀ = λ₃₀ = λ₄₀ = 2, λ₂₁ = λ₃₁ = λ₄₁ = 1.5, λ₂₂ = λ₃₂ = λ₄₂ = 2, q₂ = q₃ = q₄ = 0.5, p₂ = p₃ = p₄ = 8, σ₂₀ = σ₂₁ = σ₂₂ = σ₃₀ = σ₃₁ = σ₃₂ = σ₄₀ = σ₄₁ = σ₄₂ = 0.1, L₂ = 100, L₃ = 10, and L₄ = 50.

2. FTDO parameters of the followers: λ_i10 = λ_i11 = λ_i12 = 2, λ_i11 = λ_i21 = λ_i31 = 1.5, λ_i12 = λ_i22 = λ_i32 = 1.5, q_i1 = q_i2 = q_i3 0.5, p_i1 = p_i2 = p_i3 = 8, and L_i1 = L_i2 = L_i3 = 10.

Parameters of the dynamic surface SMC algorithm:

1. Parameters for the leader control law: k₂ = 5, k₃ = 10, k₄ = 12, k₅ = 16, ∂ = 0.6, and x_2d = 0.

2. Parameters for the follower control law: k_v1 = k_{v 2} = 2.5, k_{i 1} = 5, k_{i 2} = 2.5, k_{i 3} = 10, k_{i 31} = 40, k_{i 32} = 5, and λ_{i 1} = λ_{i 3} = λ_v = 0.6.

Parameters for the filter: τ₃ = τ₄ = τ_i2 = τ_{i 3} = 0.002, ζ_n = 0.8, and ω_n = 40.

Parameters for the distributed cooperative control strategy:; ki₁ = 0.2, k_{i 2} = 0.5, and ∂_i = 0.6.

It is assumed that the disturbance in the system is d₃ = d₄ = d_i1 = d_{i 2} = d_i3 = 0.1sin(t) and the target acceleration is a = 5 m/s².

The regular and improved distributed cooperative control strategies are examined in the simulations. The formula for the former is (Eq. 71):

(71) ${\mathit{v}}_{\mathit{i}}={\overline{\mathit{k}}}_{\mathit{j}1}\left({\sum }_{\mathit{j}=1}^{\mathit{n}}{\overline{\mathit{a}}}_{\mathit{ij}}\left({\mathit{x}}_{\mathit{j}}-{\mathit{x}}_{\mathit{i}}\right)+{\overline{\mathit{b}}}_{\mathit{i}}\left({\mathit{x}}_0-{\mathit{x}}_{\mathit{i}}\right)\right)+{\dot{\mathit{x}}}_0$

where; k_j1 > 0.8.

Simulation results are obtained using the parameters given above, and they are compared in Figs. 3 to 12.

Figure 3
Trajectories of the interceptors and target obtained using the improved cooperative control strategy.

Figure 4
Velocity curves of the following interceptors obtained using the improved cooperative control strategy.

Figure 5
Flight path angles of the interceptors obtained using the improved cooperative control strategy.

Figure 6
AOA curves of the interceptors obtained using the improved cooperative control strategy.

Figure 7
Pitch angular velocities of the interceptors obtained using the improved cooperative control strategy.

Figure 8
Trajectories of the interceptors and target obtained using the regular cooperative control strategy.

Figure 9
Velocity curves of the following interceptors obtained using the regular cooperative control strategy.

Figure 10
Flight path angles of the interceptors obtained using the regular cooperative control strategy.

Figure 11
AOA curves of the interceptors obtained using the regular cooperative control strategy.

Figure 12
Pitch angular velocities of the interceptors obtained using the regular cooperative control strategy.

The trajectories of the interceptors and target obtained using the two strategies are shown in Figs. 3 and 8. A comparison between the figures shows that when the improved strategy is used, follower trajectories gradually approach the target trajectory with a higher rate of convergence and smoother trajectory curves, and followers are able to hit the target by following the leader.

The velocity curves of the interceptors obtained using the two strategies are shown in Figs. 4 and 9. Convergence is reached at 6-8 s using the improved strategy, whereas it is reached at 8-10 s using the regular strategy. Comparisons between Figs. 5 and 10, 6 and 11, and 7 and 12 suggest that the improved strategy proposed in this study provides a higher rate of convergence and smoother transition with strong robustness.

CONCLUSION

An improved distributed network cooperative IGC algorithm is developed based on the leader-follower topology to address the multi-interceptor IGC problem. The controller for the leading interceptor is designed based on an FTDO and the nonsingular fast dynamic surface SMC law, whose stability is proved using the Lyapunov principle. An improved multi-interceptor cooperative control strategy is proposed based on distributed network cooperative control. The following interceptor controller is similarly designed using an FTDO and dynamic surface SMC. The algorithm is validated using simulations. It is demonstrated that the developed algorithm can meet the cooperative guidance and control requirements of multiple interceptors while increasing the rate of convergence for the interceptors that react to cooperative control commands.

• FUNDING
  Aeronautical Science Foundation of China [http://dx.doi.org/10.13039/501100004750] Grant No 2016ZC12005

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Edited by

Section Editor: Luiz Martins-Filho

Publication Dates

• Publication in this collection
  02 May 2019
• Date of issue
  2019

History

• Received
  14 Feb 2018
• Accepted
  17 Apr 2018