Asymmetric integral barrier Lyapunov function-based three-dimensional integrated guidance and control design with field-of-view constraint and input saturation

Abstract

This paper proposes an adaptive integrated guidance control (IGC) scheme for the flight vehicle intercepting the maneuvering target with asymmetric field-of-view angle constraint, input magnitude and rate saturation, and system uncertainties. A novel nonlinear three-dimensional IGC model is established in the body line-of-sight (LOS) coordinate system by employing the dual-integral control law. Based on this model, the input magnitude and rate saturation for the conventional IGC system can be converted to the limitations of the new augmented states. Asymmetric integral barrier Lyapunov function and dynamic surface control approach are applied to deal with the asymmetric state constraints of the proposed IGC model, and adaptive control laws are designed to compensate for model uncertainties. Furthermore, based on the Lyapunov stability theory, it is proved that all signals in the closed-loop system are bounded while the constrained states are not violated. Finally, the effectiveness and robustness of the proposed IGC scheme are illustrated with numerical simulations.

Keywords

Field-of-view constraint three-dimensional IGC design model integral barrier Lyapunov function body-LOS coordinate system strap-down seeker

Introduction

Over the past decades, the Integrated Guidance and Control (IGC) approach has been the center of attention due to its advantages of considering the coupling between two subsystems and allowing the direct generation of control commands.^1,2 However, when a vehicle equipped with a strap-down seeker intercepts a maneuvering target, due to its narrow field of view, the performance of the IGC system will degrade or even lead to system instability once the target exceeds the seeker’s maximum detection. Therefore, the increased requirements for the safe operation of IGC systems have prompted research into field-of-view (FOV) constrained controllers.

Many fruitful design methods on FOV-constrained controllers have been obtained, such as proportional navigation (PN)-based methods,^3,4 optimal control methods,^5,6 sliding mode control (SMC) methods,^7,8 and backstepping approaches.^9,10 However, the FOV angle considered in the above studies is simplified as the look angle between the velocity of the vehicle and line-of-sight (LOS), which is not suitable for the vehicle equipped with the strap-down seeker.¹¹ In contrast, the body-LOS angle can not only be directly measured but also make full use of the information of the missile’s attitude dynamics. Based on the decoupling model, Zhao et al.^12,13 design the IGC laws considering the FOV constraints in the longitudinal plane using the BLF and DSC method, as well as the integral barrier Lyapunov function (IBLF) and SMC theory, respectively. As an extension, Guo et al.¹⁴ investigate an IGC law with FOV limit in 3D engagement geometry. By using of the reinforcement learning idea of adaptive dynamic programming, Guo et al.¹⁵ propose a novel IGC strategy for high precision strike of the vehicle with FOV constraint in the longitudinal plane. In addition, a low-order IGC design model with FOV constraint is established by Zhang et al.,¹⁶ then presented constrained IBLF-based 3D IGC scheme can guarantee the FOV constraint of the seeker based on the designed low-order IGC model and the roll channel model. It should be pointed out that the FOV constraints in the aforementioned studies are symmetric and this is just a special situation of the FOV constraints studied. However, since seeker is usually mounted on one side of the vehicle head to prevent severe aerodynamic heating, it is necessary to investigate the asymmetric FOV constraints. Chao et al.¹⁷ extend the method proposed by Zhang et al.¹⁶ with using a modified DSC and asymmetric BLF to a broader application such as when the asymmetric side-window constraint is considered.

There is no doubt that the aforementioned studies have made significant contributions to IGC research. However, the flight control system is suffering from multiple constraints due to complex flight conditions and inherent physical limitations of the actuator, where actuator input saturation is another important issue for the flight control system. A saturation function^18–20 is always introduced to describe the input saturation nonlinearity, where the negative effects of saturation are compensated by building an auxiliary system. Besides, Zhang et al.²¹ construct an anti-windup compensator to deal with the input saturation. Note that all of these studies were conducted without considering the rate limitation of the actuator in control strategy design. If the derivative of the signal exceeds the maximum capability of the actual actuator, it may result in severe failure of the actuator or loss of system stability while attempting to generate the required control law. Therefore, there is a growing body of literature that recognizes the importance of input magnitude and its rate constraint. It can be seen that the input magnitude and rate constraint of practical systems are handled by modeling the nonlinearity of input magnitude and rate constraint with saturation functions.^22,23 Without modeling the nonlinearity of input, there are other methods to tackle the saturation of system input, such as the anti-windup model for rigid spacecraft attitude control, ^24,25 a disturbance observer-based control scheme for flexible spacecraft²⁶ and the command filter.²⁷ Besides, Liu et al.²⁸ investigate a new neural adaptive control algorithm for strict-feedback systems with input magnitude and rate constraint, where the input constraint problem is creatively transformed into the issue of state constraint.

Motivated by the aforementioned studies, a novel 3D adaptive IGC law based on the backstepping method and asymmetric integral barrier Lyapunov function (AIBLF) technology is proposed, which is subject to the asymmetric FOV constraint and actuator magnitude and rate limit simultaneously. The main contributions are as follows:

(1) A new 3D IGC model is derived by introducing the strap-down decoupling model and designing dual-integral control law, such that both the FOV constraint and input magnitude and rate limit are transformed to be partial state constraints of the new IGC system. Then the issue of the IGC system with the FOV constraint and input magnitude and rate saturation can be guaranteed as long as the constrained states of the new 3D IGC system are not violated;

(2) A new controller is proposed for the 3D IGC model by introducing the AIBLF, which guarantees that all signals in the closed-loop system are bounded while the constrained states are not violated. That is, from the point of state-constrained tracking control, both the FOV constraint and actuator saturation of the IGC system are handled simultaneously. Compared with the existing symmetric FOV constraint problem of the IGC system, the proposed IGC method can not only realize the asymmetric FOV constraint but also ensure input saturation through model transformation. Besides numerical simulation results verify the effectiveness and superiority of the proposed IGC scheme.

The remainder of this article is organized as follows. Section “Problem formulation” presents the new 3D IGC model. The detailed design process of proposed IGC law and stability analysis are presented in Section “AIBLF-based 3D IGC scheme design and stability analysis.” The simulations and comparison results are shown in Section “Numerical simulation result.” Section “Conclusion” concludes the paper.

Problem formulation

A new 3D IGC model is given, followed by some necessary lemmas, assumptions, and design objectives are presented in this section.

The 3D engagement interception geometry of a flight vehicle $M$ against a moving target $T$ is described in the inertial coordinate frame $Oxyz$ in Figure 1, where $R$ denotes the relative range between vehicle and target; $O x_{1} y_{1} z_{1}$ , $O x_{4} y_{4} z_{4}$ , and $O x_{5} y_{5} z_{5}$ represent the body coordinate system, the LOS coordinate system, and the body-LOS coordinate system,²⁹ respectively; $r$ is the projection of $R$ in the plane $Oxz$ ; $γ_{b}$ is the roll angle of LOS; $ε$ and $η$ denote elevation and azimuth of the LOS angles, $ε_{b}$ and $η_{b}$ represent the body-LOS angles in elevation and azimuth directions, respectively. For simplicity, the body-LOS angles $ε_{b}$ and $η_{b}$ are utilized to approximate the FOV angles in this paper.

Figure 1.

The 3D interception geometry of the vehicle and the target.

The 3D IGC model¹⁴ in the body-LOS coordinate system is derived as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} + d_{1}, \\ {\overset{\cdot}{x}}_{2} = f_{2} + g_{2} x_{3}^{*} + d_{2}, \\ {\overset{\cdot}{x}}_{3} = f_{3} + g_{3} x_{4} + d_{3}, \\ {\overset{\cdot}{x}}_{4} = f_{4} + g_{4} u + d_{4}, \end{matrix}

(1)

with $x_{1} = [ε_{b}, η_{b}]^{T}, x_{2} = [\overset{\cdot}{ε}, \overset{\cdot}{η}]^{T}, x_{3} = [α, β, γ]^{T}, x_{3}^{*} = [α, β]^{T}, x_{4} = [ω_{x}, ω_{y}, ω_{z}]^{T}$ , $u = [δ_{x}, δ_{y}, δ_{z}]^{T}$ ,

\begin{matrix} f_{2} = - [\begin{matrix} \frac{- 2 \overset{\cdot}{R} \overset{\cdot}{ε} - R {\overset{\cdot}{η}}^{2} \sin ε \cos ε}{R} - \frac{P \sin α}{mR} \\ \frac{- 2 \overset{\cdot}{R} \overset{\cdot}{η} \cos ε + 2 R \overset{\cdot}{η} \overset{\cdot}{ε} \sin ε}{R \cos ε} - \frac{P \cos α \sin β}{mR \cos ε} \end{matrix}], \\ g_{2} = [\begin{matrix} - \frac{{qSC}_{y}^{α}}{mR} & 0 \\ 0 & \frac{{qSC}_{z}^{β}}{mr} \end{matrix}], f_{3} = [\begin{matrix} - \frac{1}{m V_{M} \cos β} ({qSC}_{y}^{α} α + P \sin α) \\ \frac{1}{m V_{M}} ({qSC}_{z}^{β} β - P \cos α \sin β) \\ 0 \end{matrix}], \\ g_{3} = [\begin{matrix} - \tan β \cos α & \tan β \sin α & 1 \\ \sin α & \cos α & 0 \\ 1 & - \tan ϑ \cos γ & \tan ϑ \sin γ \end{matrix}], \\ f_{4} = [\begin{matrix} \frac{J_{y} - J_{z}}{J_{x}} ω_{y} ω_{z} \\ \frac{J_{z} - J_{x}}{J_{y}} ω_{x} ω_{z} + \frac{1}{J_{y}} {qSLm}_{y}^{β} β \\ \frac{J_{x} - J_{y}}{J_{z}} ω_{x} ω_{y} + \frac{1}{J_{z}} {qSLm}_{z}^{α} α \end{matrix}], \\ g_{4} = [\begin{matrix} \frac{1}{J_{x}} {qSLm}_{x}^{δ_{x}} & 0 & 0 \\ 0 & \frac{1}{J_{y}} {qSLm}_{y}^{δ_{y}} & 0 \\ 0 & 0 & \frac{1}{J_{z}} {qSLm}_{z}^{δ_{z}} \end{matrix}], \\ d_{1} = - [\begin{matrix} \overset{\cdot}{ϑ} \\ \overset{\cdot}{ψ} \end{matrix}], d_{2} = [\begin{matrix} \frac{a_{ty 4}}{R} - \frac{Δ_{y}}{mR} \\ - \frac{a_{tz 4}}{R \cos ε} + \frac{Δ_{z}}{mR \cos ε} \end{matrix}], d_{3} = [\begin{matrix} Δ_{α} \\ Δ_{β} \\ Δ_{γ} \end{matrix}], \\ d_{4} = [\begin{matrix} Δ_{ω_{x}} \\ Δ_{ω_{y}} \\ Δ_{ω_{z}} \end{matrix}], \end{matrix}

where $α$ , $β$ , and $γ$ are attack angle, sideslip angle, and roll angle; $ϑ$ and $ψ$ are the pitch angle and the yaw angle; $m$ is the mass of the vehicle and $P$ is the push force; $C_{y}^{α}$ and $C_{z}^{β}$ are lift force derivative; $q$ , $L$ , and $S$ are the dynamic pressure, reference length, and aerodynamic reference area; $a_{ty 4}, a_{tz 4}$ are the components of the projection of target acceleration vector in the LOS coordinate system; $ω_{i}, δ_{i}, J_{i} (i = x, y, z)$ are angular rates, the deflection angles, and moments of inertia, $m_{x}^{δ_{x}}$ , $m_{y}^{β}$ , $m_{y}^{δ_{y}}$ , $m_{z}^{α}$ , and $m_{z}^{δ_{z}}$ are the aerodynamic moment coefficients; $Δ_{i} (i = y, z, α, β, γ, ω_{x}, ω_{y}, ω_{z})$ are aerodynamic force coefficients, modeling errors, and external disturbances.

In this paper, the following dual-integral²⁸ is considered to deal with the input saturation

{\begin{matrix} u (t) = u (0) + \int_{0}^{t} ξ (τ) d τ, \\ ξ (t) = υ (0) + \int_{0}^{t} υ (τ) d τ, \end{matrix}

(2)

in which $ξ = {[ξ_{x}, ξ_{y}, ξ_{z}]}^{T} \in R^{3}$ and $υ = {[υ_{x}, υ_{y}, υ_{z}]}^{T} \in R^{3}$ are two virtual control law to be designed later, the initial values $ξ (0)$ and $υ (0)$ are selected in the required range.

For convenience, new augmented state variables and control variable are denoted as $x_{5} = {[δ_{x}, δ_{y}, δ_{z}]}^{T}$ , $x_{6} = {[ξ_{x}, ξ_{y}, ξ_{z}]}^{T}$ , and $υ = {[υ_{x}, υ_{y}, υ_{z}]}^{T}$ . Combined with (1) the novel 3D IGC model for the vehicle equipped with strap-down seeker in the presence of FOV constraint and actuator input saturation can be expressed as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} + d_{1}, \\ {\overset{\cdot}{x}}_{2} = f_{2} + g_{2} x_{3}^{*} + d_{2}, \\ {\overset{\cdot}{x}}_{3} = f_{3} + g_{3} x_{4} + d_{3}, \\ {\overset{\cdot}{x}}_{4} = f_{4} + g_{4} x_{5} + d_{4}, \\ {\overset{\cdot}{x}}_{5} = x_{6}, \\ {\overset{\cdot}{x}}_{6} = υ, \end{matrix}

(3)

where $x_{i} (i = 1, . . ., 6)$ are the system states and $υ$ is the present control input signal of novel 3D IGC system (3). Besides, the augmented states $x_{5}$ and $x_{6}$ in the novel 3D IGC model (3) represent the input $u$ and input rates $\overset{\cdot}{u}$ of the original IGC system (1), respectively. From (3), it is clear that the FOV constraint and input saturation can be guaranteed by limiting the states $x_{1}$ , $x_{5}$ , and $x_{6}$ of novel 3D IGC system, that is, the states of novel system (3) should satisfy the following constraints

{\begin{matrix} x_{1} \in L_{1} : {x_{1} \in R^{2} | - k_{a 1} < x_{1 i} < k_{b 1}}, i = 1, 2, \\ x_{m} \in L_{m} : {x_{m} \in R^{3} | - k_{am} < x_{mj} < k_{bm}}, j = 1, 2, 3, m = 5, 6, \end{matrix}

(4)

where $k_{ai}$ and $k_{bi} (i = 1, 5, 6)$ are positive constants, representing the lower bound and the upper bound for the FOV angle of the strap-down seeker and that for the input $u$ and its rate $\overset{\cdot}{u}$ of the original 3D IGC design system (1).

Then the control objective of this paper is described to design an adaptive control law $υ$ for the IGC system (3) such that constrained states $x_{i} (i = 1, 5, 6)$ are never violated and all signals in (3) are stabilized despite the uncertainties existing in the whole engagement.

For the established novel 3D IGC model (3), the following assumptions, lemmas, and property are needed.

Assumption 1. ³⁰ The initial value of system state $x_{1}$ satisfies

- k_{a 1} < x_{1 i} (0) < k_{b 1}, i = 1, 2 .

(5)

Assumption 2. ¹⁸There exist unknown positive constants $d_{i, max} (i = 1, 2, 3, 4)$ , such that the disturbances $d_{i} (i = 1, 2, 3, 4)$ in the IGC model (3) satisfy

∥ d_{i} ∥ \leq d_{i, max}, i = 1, 2, 3, 4 .

(6)

Assumption 3. ¹⁸ The full states of IGC system (3) can be measured.

Assumption 4. ¹⁹ The control gain matrixes $g_{i} (i = 2, 3, 4)$ are invertible.

To handle the asymmetric state limitation, asymmetric smooth saturation function $T_{a} (x)$ ¹⁰ is introduced as

T_{a} (x_{,} ι_{a}, ι_{b}) = \frac{e^{x} - e^{- x}}{\frac{e^{x}}{ι_{b}} + \frac{e^{- x}}{ι_{a}}},

(7)

in which $ι_{a}$ and $ι_{b}$ are positive constants, and the function $T_{a} (x, ι_{a}, ι_{b})$ has the following property

Property 1. ¹⁰ The inequality $- ι_{a} < T_{a} (x) < ι_{b}$ holds.

Lemma 1. ²⁸ The following AIBLF candidate is considered as

L (z, x_{d}) = \int_{0}^{z} \frac{σ {(k_{a} + k_{b})}^{2}}{(k_{a} + σ + x_{d}) (k_{b} - σ - x_{d})} d σ,

(8)

where $z = x - x_{d}$ , $x_{d}$ is the virtual control law and $x$ satisfies $- k_{a} < x < k_{b}$ , $k_{a}$ and $k_{b}$ are positive constants. The derivative of L with respect to time is given

\overset{\cdot}{L} (z, x_{d}) = \frac{\partial L (z, x_{d})}{\partial z} \overset{\cdot}{z} + \frac{\partial L (z, x_{d})}{\partial x_{d}} {\overset{\cdot}{x}}_{d},

(9)

where $\frac{\partial L (z, x_{d})}{\partial z} = ϕ_{L} (z, x_{d}) z$ , $\frac{\partial L (z, x_{d})}{\partial x_{d}} = (ϕ_{L} (z, x_{d}) + ψ_{L} (z, x_{d})) z,$ in which $ϕ_{L} (z, x_{d}) = \frac{{(k_{a} + k_{b})}^{2}}{(k_{a} + z + x_{d}) (k_{b} - z - x_{d})}$ and $ψ_{L} (z, x_{d}) = \frac{k_{a} + k_{b}}{z} \ln \frac{(k_{a} + x_{d}) (k_{b} - z - x_{d})}{(k_{b} - x_{d}) (k_{a} + z + x_{d})}$ are well-defined in the neighborhood of $z = 0$ .

Lemma 2. ²⁸For any positive constants $k_{a}$ and $k_{b}$ , the following inequality

\frac{1}{2} z^{2} \leq L (z, x_{d}) \leq z^{2} ϕ (z, x_{d})

(10)

holds with $x$ satisfying $- k_{a} < x < k_{b}$ .

Remark 1. Assumption 1 is reasonable since the target should be locked by the strap-down seeker at the beginning of the homing phase.

AIBLF-based 3D IGC scheme design and stability analysis

In this section, the AIBLF-based 3D IGC law with constrained FOV and input saturation are presented based on the novel IGC system (3) as well as the stability analysis is given by Lyapunov theory.

AIBLF-based 3D IGC scheme design

Define the tracking errors as follows

{\begin{matrix} z_{1} = x_{1}, \\ z_{i} = x_{i} - x_{id}, i = 2, . . ., 6, \end{matrix}

(11)

where $x_{id}$ is the virtual control law.

In order to ensure constrained FOV, input saturation, and uncertainties, the AIBLF-based 3D IGC law is given as

{\begin{matrix} x_{2 c} = - (ϕ_{1} + K_{1}) z_{1} - ϕ_{1} {\hat{D}}_{1} z_{1}, \\ x_{3 c}^{*} = g_{2}^{- 1} (- K_{2} z_{2} - f_{2} - {\hat{D}}_{2} z_{2} - ϕ_{1} z_{1} + {\overset{\cdot}{x}}_{2 d}), \\ x_{4 c} = g_{3}^{- 1} (- K_{3} z_{3} - f_{3} - {\hat{D}}_{3} z_{3} - g_{2}^{T} z_{2} + [\begin{matrix} {\overset{\cdot}{x}}_{3 d}^{*} \\ 0 \end{matrix}]), \\ x_{5 c} = g_{4}^{- 1} (- K_{4} z_{4} - f_{4} - {\hat{D}}_{4} z_{4} + {\overset{\cdot}{x}}_{4 d} - g_{3}^{T} z_{3}), \\ x_{6 c} = - (ϕ_{5} + K_{5}) z_{5} - ϕ_{5}^{- 1} ψ_{5} {\overset{\cdot}{x}}_{5 d} - ϕ_{5}^{- 1} g_{4}^{T} z_{4}, \\ v = - K_{6} z_{6} - ϕ_{6}^{- 1} ψ_{6} {\overset{\cdot}{x}}_{6 d} - ϕ_{6}^{- 1} ϕ_{5} z_{5}, \end{matrix}

(12)

where $x_{ic} (i = 2, . . ., 6)$ are the nominal virtual control laws; $K_{i} = diag {k_{i 1}, k_{i 2}} (i = 1, 2)$ , $K_{j} = djag {k_{j 1}, k_{j 2}, k_{j 3}} (j = 3, 4, 5, 6)$ denote gain matrices and $k_{i 1}, k_{i 2}, k_{j 1}, k_{j 2}, k_{j 3}$ are positive parameters to be designed; $ϕ_{1} = diag {ϕ_{L} (z_{11}, x_{1 d 1}), ϕ_{L} (z_{12}, x_{1 d 2})}$ , $z_{1 i}$ and $x_{1 di} (i = 1, 2)$ are the ith elements of the vectors $z_{1}$ and $x_{1 d}$ ; $ϕ_{m} = diag {ϕ_{L} (z_{m 1}, x_{md 1}), ϕ_{L} (z_{m 2}, x_{md 2}), ϕ_{L} (z_{m 3}, x_{md 3})}$ , $ψ_{m} = diag {ψ_{L} (z_{m 1}, x_{md 1}), ψ_{L} (z_{m 2}, x_{md 2}), ψ_{L} (z_{m 3}, x_{md 3})}$ , $z_{mi}$ and $x_{mdi} (i = 1, 2, 3, m = 5, 6)$ are the ith elements of $z_{m}$ and $x_{md}$ ; ${\hat{D}}_{i} (i = 1, 2, 3, 4)$ are adaptive parameters with the updating laws designed as

{\begin{matrix} {\overset{\cdot}{\hat{D}}}_{1} = - μ_{1} {\hat{D}}_{1} + γ_{1} z_{1}^{T} ϕ_{1}^{2} z_{1}, {\hat{D}}_{1} (0) \geq 0, \\ {\overset{\cdot}{\hat{D}}}_{i} = - μ_{i} {\hat{D}}_{i} + γ_{i} z_{i}^{T} z_{i}, {\hat{D}}_{i} (0) \geq 0, i = 2, . . ., 4, \end{matrix}

(13)

where $μ_{i}$ and $γ_{i} (i = 1, 2, 3, 4)$ are positive design parameters. Define the estimation errors of $D_{i} (i = 1, 2, 3, 4)$ as ${\tilde{D}}_{i}$ , satisfying ${\tilde{D}}_{i} = D_{i} - {\hat{D}}_{i}$ , in which $D_{i} = | d_{i \max} |^{2}$ .

To avoid the problem explosion of complexity in the traditional backstepping approach, the following first-order filters are employed in each step

{\begin{matrix} Γ_{i} {\overset{\cdot}{x}}_{id} + x_{id} = x_{ic}, x_{id} (0) = x_{ic} (0), i = 2, 3, 4, \\ Γ_{i} {\overset{\cdot}{x}}_{id} + x_{id} = Tanh (x_{ic}), x_{id} (0) = Tanh (x_{ic} (0)), i = 5, 6, \end{matrix}

(14)

where $Γ_{i} = diag {τ_{i 1}, τ_{i 2}} (i = 2, 3)$ and $Γ_{j} = diag {τ_{j 1}, τ_{j 2}, τ_{j 3}} (j = 4, 5, 6)$ are the time constant matric of the filters, as well as $τ_{i 1}, τ_{i 2}, τ_{j 1}, τ_{j 2}, τ_{j 3}$ are positive design parameters; $Tanh (x_{mc}) = {[T_{a} (x_{mc 1}), T_{a} (x_{mc 2}), T_{a} (x_{mc 3})]}^{T} (m = 5, 6)$ denotes the nominal virtual control law that has been saturated. Then the virtual control law $x_{md} (m = 5, 6)$ generated from (14) always satisfies the inequality $- ι_{am} \leq x_{mdi} \leq ι_{bm} (i = 1, 2, 3)$ because of the property 1 and the Theorem 1.¹⁰ The tracking errors of the filters are defined as

y_{i} = x_{id} - x_{ic}, i = 2, . . ., 6 .

(15)

Stability analysis

The stability proof is presented based on Lyapunov theory and the control objectives are analyzed in this section. Choosing the Lyapunov function candidate

\begin{matrix} V = \sum_{i = 2}^{4} \frac{1}{2} z_{i}^{T} z_{i} + \sum_{j = 1}^{4} \frac{1}{2 γ_{j}} {\tilde{D}}_{j}^{2} + \sum_{k = 2}^{6} \frac{1}{2} y_{k}^{T} y_{k} + L_{1} (z_{1}, x_{1 d}) \\ + L_{5} (z_{5}, x_{5 d}) + L_{6} (z_{6}, x_{6 d}), \end{matrix}

(16)

where the barrier Lyapunov function $L_{1} (z_{1}, x_{1 d})$ , $L_{5} (z_{5}, x_{5 d})$ , $L_{6} (z_{6}, x_{6 d})$ are considered as

L_{1} (z_{1}, x_{1 d}) = \sum_{i = 1}^{2} \int_{0}^{z_{1 i}} \frac{σ {(k_{a 1} + k_{b 1})}^{2}}{(k_{a 1} + σ) (k_{b 1} - σ)} d σ,

(17)

and

\begin{matrix} L_{m} (z_{m}, x_{md}) = \sum_{j = 1}^{3} \int_{0}^{z_{mj}} \frac{σ {(k_{am} + k_{bm})}^{2}}{(k_{am} + σ + x_{mdi}) (k_{bm} - σ - x_{mdi})} d σ, \\ m = 5, 6, \end{matrix}

(18)

where $z_{1 i} (i = 1, 2)$ denote the ith elements of the vectors $z_{1}$ , $z_{mj}$ and $x_{mdj} (j = 1, 2, 3, m = 5, 6)$ denote the jth elements of the vectors $z_{m}$ and $x_{md}$ .

Then the main theorem of this paper is established.

Theorem 1. For the novel 3D IGC system (3) with FOV constraint, input saturation and the uncertainties, under the assumptions 1–4, considering the nominal control laws (12), adaptive laws (13), and filters (14), if $V (0) < ϱ, ϱ$ is any positive constant, the following conclusions can be obtained by choosing the appropriate design parameters $K_{i} (i = 1, . . ., 6), μ_{j}, γ_{j} (j = 1, . . ., 4), Γ_{k} (k = 2, . . ., 6)$ .

(1) The constrained states $x_{i} (i = 1, 5, 6)$ are never violated.

(2) All tracking errors $z_{i} (i = 1, . . ., 6)$ are eventually converge to the bounded sets.

(3) All signals of the closed-loop system are bounded.

Proof. Differentiating (15) with respect to time, the derivatives of filter errors are obtained

{\begin{matrix} {\overset{\cdot}{y}}_{k} = - Γ_{k}^{- 1} y_{k} - {\overset{\cdot}{x}}_{kc}, k = 2, 3, 4, \\ {\overset{\cdot}{y}}_{k} = - Γ_{k}^{- 1} y_{k} - {\overset{\cdot}{x}}_{kc} + δ_{k}, k = 5, 6, \end{matrix}

(19)

where the $δ_{k} = Γ_{k}^{- 1} (Tanh (x_{kc}) - x_{kc}), k = 5, 6$ .

Taking the time derivative of $V$ with (9), (11), (12), (13), (17), (18), and (19), we can obtain

\begin{matrix} \overset{\cdot}{V} = \sum_{i = 2}^{4} z_{i}^{T} {\overset{\cdot}{z}}_{i} + \sum_{j = 1}^{4} \frac{1}{γ_{j}} {\tilde{D}}_{j} {\overset{\cdot}{\tilde{D}}}_{j} + \sum_{k = 2}^{6} y_{k}^{T} {\overset{\cdot}{y}}_{k} + {\overset{\cdot}{L}}_{1} + {\overset{\cdot}{L}}_{5} + {\overset{\cdot}{L}}_{6} \\ \leq - z_{1}^{T} ϕ_{1} K_{1} z_{1} - \sum_{i = 2}^{4} z_{i}^{T} K_{i} z_{i} - z_{5}^{T} ϕ_{5} K_{5} z_{5} - z_{6}^{T} ϕ_{6} K_{6} z_{6} \\ + \sum_{j = 1}^{4} - \frac{μ_{j}}{2 γ_{j}} {\tilde{D}}_{j}^{2} + \frac{μ_{j}}{2 γ_{j}} D_{j}^{2} - z_{1}^{T} ϕ_{1}^{2} z_{1} - z_{5}^{T} ϕ_{5}^{2} z_{5} + z_{1}^{T} ϕ_{1} y_{2} \\ + z_{2}^{T} g_{2} y_{3} + z_{3}^{T} g_{3} y_{4} + z_{4}^{T} g_{4} y_{5} + z_{5}^{T} ϕ_{5} y_{6} + 1 \\ + \sum_{k = 2}^{4} y_{k}^{T} (- Γ_{k}^{- 1} y_{k} - {\overset{\cdot}{x}}_{kc}) + \sum_{k = 5}^{6} y_{k}^{T} (- Γ_{k}^{- 1} y_{k} - {\overset{\cdot}{x}}_{kc} + δ_{k}) . \end{matrix}

(20)

From (12), (13), (14), (15), and (19), we have

{\begin{matrix} x_{2 c} = P_{2} (z_{1}, x_{1}, {\hat{D}}_{1}), \\ x_{ic} = P_{i} (z_{1}, . . ., z_{i - 1}, x_{1}, . . ., x_{i - 1}, {\hat{D}}_{1}, . . ., {\hat{D}}_{i - 1}, y_{2}, . . ., y_{i - 1}), i = 3, 4, 5 \\ x_{6 c} = P_{6} (z_{1}, . . ., z_{4}, z_{5}, x_{1}, . . ., x_{5}, {\hat{D}}_{1}, . . ., {\hat{D}}_{4}, y_{2}, . . ., y_{5}), \end{matrix}

(21)

and

{\begin{matrix} {\overset{\cdot}{x}}_{2 c} = S_{2} (z_{1}, x_{1}, x_{2}, {\hat{D}}_{1}), \\ {\overset{\cdot}{x}}_{ic} = S_{i} (z_{1}, . . ., z_{i - 1}, x_{1}, . . ., x_{i}, {\hat{D}}_{1}, . . ., {\hat{D}}_{i - 1}, y_{2}, . . ., y_{i - 1}), i = 3, 4, \\ δ_{i} - {\overset{\cdot}{x}}_{ic} = S_{i} (z_{1}, . . ., z_{i - 1}, x_{1}, . . ., x_{i}, {\hat{D}}_{1}, . . ., {\hat{D}}_{4}, y_{2}, . . ., y_{i - 1}), i = 5, 6, \end{matrix}

(22)

where $P_{k} (\cdot), S_{k} (\cdot) (k = 2, . . ., 6)$ are continuous function of $(\cdot)$ . Define a compact set $Ω : = {L_{1} + L_{5} + L_{6} + \frac{1}{2} \sum_{i = 2}^{4} z_{i}^{T} z_{i} + \sum_{j = 1}^{4} \frac{1}{2 γ_{j}} {\tilde{D}}_{j}^{2} + \frac{1}{2} \sum_{k = 2}^{6} y_{k}^{T} y_{k} \leq ϱ}$ , where $ϱ > 0$ . Therefore, there exist five positive constants $E_{k, max} (k = 2, . . ., 6)$ to make the following inequalities hold on the compact $Ω$ .

{\begin{matrix} ‖ {\overset{\cdot}{x}}_{kc} ‖ \leq E_{k, max}, k = 2, 3, 4, \\ ‖ δ_{k} - {\overset{\cdot}{x}}_{kc} ‖ \leq E_{k, max}, k = 5, 6 . \end{matrix}

(23)

According to the young’s inequality, the following inequalities hold

\begin{matrix} z_{i}^{T} ϕ_{i} y_{i + 1} \leq z_{i}^{T} ϕ_{i}^{2} z_{i} + \frac{1}{4} ‖ y_{i + 1} ‖^{2}, i = 1, 5, \\ z_{i}^{T} g_{i} y_{i + 1} \leq z_{i}^{T} g_{i}^{2} z_{i} + \frac{1}{4} ‖ y_{i + 1} ‖^{2}, i = 2, 3, 4 . \end{matrix}

(24)

The following inequalities also hold by the young’s inequality

\begin{matrix} - y_{k}^{T} {\overset{\cdot}{x}}_{kc} \leq \frac{1}{4} + E_{k, max}^{2} ‖ y_{k} ‖^{2}, k = 2, 3, 4, \\ y_{k}^{T} (δ_{k} - {\overset{\cdot}{x}}_{kc}) \leq \frac{1}{4} + E_{k, max}^{2} ‖ y_{k} ‖^{2}, k = 5, 6 . \end{matrix}

(25)

Substituting inequalities (23), (24), and (25) into (20), one has

\begin{matrix} \overset{\cdot}{V} \leq - z_{1}^{T} ϕ_{1} K_{1} z_{1} - \sum_{i = 2}^{4} z_{i}^{T} (K_{i} - g_{i}^{2}) z_{i} - z_{5}^{T} ϕ_{5} K_{5} z_{5} - z_{6}^{T} ϕ_{6} K_{6} z_{6} \\ - \sum_{j = 1}^{4} \frac{μ_{j}}{2 γ_{j}} {\tilde{D}}_{j}^{2} - \sum_{k = 2}^{6} y_{k}^{T} (Γ_{k}^{- 1} - \frac{1}{4} I - E_{k, max}^{2} I) y_{k} + C_{T}, \end{matrix}

(26)

on the compact $Ω$ , where $C_{T} = \sum_{j = 1}^{4} \frac{μ_{j}}{2 γ_{j}} D_{j}^{2} + \frac{9}{4}$ is an positive constant, $I$ is the unit matrix of suitable dimension. Then, according to Lemma 2, (16) and (26), $\overset{\cdot}{V}$ satisfies

\overset{\cdot}{V} \leq - K_{v} V + C_{T},

(27)

where $K_{v} = min {k_{1 i}, 2 (k_{2 i} - g_{2 i}^{2}), 2 (k_{3 j} - g_{3 j}^{2}), 2 (k_{4 j} - g_{4 j}^{2}), k_{5 j}, k_{6 j}, 2 (τ_{m_{1} i}^{- 1} - \frac{1}{4} - E_{m_{1}, max}^{2}), 2 (τ_{m_{2} j}^{- 1} - \frac{1}{4} - E_{m_{2}, max}^{2}), μ_{m_{3}}}$ $(i = 1, 2, j = 1, 2, 3, m_{1} = 2, 3, m_{2} = 4, 5, 6, m_{3} = 1, . . ., 4)$ . Let $K_{v} > C_{T} / ϱ$ , then $\overset{\cdot}{V} < 0$ , on $V = ϱ$ , that is, if $V (0) \leq ϱ$ , the $V (t) \leq ϱ$ always holds on $t \geq 0$ . Thus, it can be deduced that the compact set $Ω$ is an invariant set.

According to Lemma 2, the inequality $\frac{1}{2} z_{m}^{T} z_{m} \leq L_{m} (z_{m}, x_{md}) (m = 1, 5, 6)$ holds, then combining with (16), we get

\sum_{i = 1}^{6} \frac{1}{2} z_{i}^{T} z_{i} + \sum_{j = 1}^{4} \frac{1}{2 γ_{j}} {\tilde{D}}_{j}^{2} + \sum_{k = 2}^{6} \frac{1}{2} y_{k}^{T} y_{k} \leq V (t) .

(28)

Thus, the boundedness of $V$ implies that $z_{i} (i = 1, . . ., 6)$ , ${\tilde{D}}_{j} (j = 1, . . ., 4)$ , and $y_{k} (k = 2, . . ., 6)$ are bounded, and since $D_{j}$ is a positive constant, we know that ${\hat{D}}_{j}$ is bounded. Besides, it can be known that $x_{kc}$ is bounded on the compact set $Ω$ from (21), and consequently leads to $x_{kd}$ is also bounded on the compact set $Ω$ . Then, from (11), the state $x_{i}$ is also bounded. Furthermore, it can be obtained from (12) that the virtual control law $v$ is also bounded since $v$ is continuous function of $(z_{5}, z_{6}, y_{5}, y_{6}, x_{5 c}, x_{6 c})$ on the compact set $Ω$ . Consequently, all closed-loop signals are bounded, and tracking errors $z_{i}$ are uniformly ultimately bounded. Meanwhile, the $L_{m} (z_{m}, x_{md}) (m = 1, 5, 6)$ are also bounded on the compact set $Ω$ , then the set (4) holds from Lemma 1.

This ends the proof.

Remark 2. It is obvious that the value of $L_{1}$ will become infinite when the state $x_{1}$ approach to the boundary of FOV constraint according to Lemma 1. In other words, given that $- k_{a 1} < x_{1 i} (0) < k_{b 1}, (i = 1, 2)$ the limitation $- k_{a 1} < x_{1 i} (t) < k_{b 1}$ always holds for $t \geq 0$ , which means that the FOV constraint will never violated. Similarly, it is easy to infer that states $x_{5}$ and $x_{6}$ satisfy the ${x_{5} \in R^{3} | - k_{a 5} < x_{5 j} < k_{b 5}}$ and ${x_{6} \in R^{3} | - k_{a 6} < x_{6 j} < k_{b 6}} (j = 1, 2, 3)$ . Therefore the limitations of input magnitude and rate for IGC system are also never violated.

Remark 3. From the proof of Theorem 1 and (27), it can be seen that tracking errors $z_{i} (i = 1, . . ., 6)$ converge to a small neighborhood of zero, which can be regulated to arbitrarily small by increasing the $K_{i}$ , $μ_{j}$ and decreasing $Γ_{k} (i = 1, . . ., 6, j = 1, . . ., 4, k = 2, . . ., 6)$ . Furthermore, if $μ_{j} (j = 1, . . ., 4)$ are required to be very large, then large $γ_{j}$ are needed to avoid the increase in $C_{T}$ . Similarly, other closed-loop signals are bounded and can be made arbitrarily small by the same principle of parameter selection.

Numerical simulation result

In this section, the effectiveness and robustness of the proposed adaptive FOV-constrained IGC scheme with input saturation are verified by numerical simulations. For the vehicle, the initial position and velocity are given by $X_{m} = [0, 200, 0]^{T} m$ and $v_{m} (0) = 400 m / s$ . The initial pitch, yaw and roll angles, pitch, yaw and roll rates, the flight path, and heading angles are set as $ϑ (0) = - 0.1 rad$ , $ψ (0) = 0 rad$ , $γ (0) = 0 rad$ , $ω_{x} (0) = 0.1 rad / s$ , $ω_{y} (0) = 0.1 rad / s$ , $ω_{z} (0) = 0.1 rad / s$ , $θ (0) = 0.01 rad$ , $ψ_{v} (0) = - 0.01 rad$ .

For the moving target, the initial position, velocity, and acceleration vectors are given by $X_{t} = [3000, 0, 0]^{T} m$ , $v_{t} (0) = [40, 40, 20]^{T} m / s$ , and $a_{t} = [4, 3, 3]^{T} m / s^{2}$ .

The parameters of input saturation are set as $| δ_{i} | \leq 15 \deg$ and $| {\overset{\cdot}{δ}}_{i} | \leq 35 \deg / s$ $(i = x, y, z)$ with consideration of physical constraints. The designed parameters for the proposed IGC algorithm are selected as $K_{1} = diag {0.1, 0.2}, K_{2} = diag {3, 5}, K_{3} = diag {10, 1, 2}, K_{4} = diag {2, 1, 1},$ $K_{5} = diag {10, 20, 10}, K_{6} = diag {10, 60, 80},$ $μ_{j} = 0.1, γ_{j} = 1 (j = 1, 2, 3, 4)$ , $Γ_{2} = diag {0.1, 0.1}, Γ_{3} = diag {0.1, 0.1}, Γ_{4} = diag {0.1, 0.1, 0.1},$ $Γ_{5} = diag {0.01, 0.001, 0.01}, Γ_{6} = diag {0.001, 0.001, 0.001}$ . The nominal parameters of the IGC system are given from Hou et al.² Then, there are three simulations to illustrate the performance of the proposed IGC law.

Case 1. To demonstrate the feasibility of the proposed law under FOV constraint with different boundaries, simulations are carried out considering FOV angles with symmetric limitation boundaries $(- 3, 3) \deg$ and $(- 5, 5) \deg$ , and asymmetric limitation boundaries $(- 3, 4) \deg$ , $(- 3, 5) \deg$ , and $(- 4, 5) \deg$ .

Case 2. To validate the superiority of the proposed IGC law (marked as AIGC ) in coping with the FOV constraint and input saturation, the traditional IGC law (marked as QIGC) based on quadratic Lyapunov function and BLF-based IGC (marked as BIGC) designed by Guo et al.¹⁴ are employed for comparison, where the maximum FOV angle is chosen as 5°. The QIGC scheme can be obtained by rewriting $x_{2 c}$ and $\overset{\cdot}{{\hat{D}}_{1}}$ of BIGC as

\begin{matrix} x_{2 c} = - K_{1} s_{1} - {\hat{D}}_{1} λ_{1} s_{1}, \\ {\overset{\cdot}{\hat{D}}}_{1} = - μ_{1} {\hat{D}}_{1} + γ_{1} z_{1}^{T} z_{1} . \end{matrix}

(29)

Case 3. To verify the robustness performance of the AIGC scheme without loss of generality, simulation cases are carried out for perturbations to atmospheric parameters, structural parameters, and aerodynamic coefficients given in Table 1.

$Δ i (i = ρ, m, J, M, c)$ represent the perturbations of corresponding parameters and the FOV constraints are set as 5°.

Table 1.

Perturbations of corresponding parameters.

Biased parameter	Bias value
Air density	$Δ ρ \pm 5 %$
Mass	$Δ m \pm 5 %$
Moment of inertial	$Δ J \pm 5 %$
Moment coefficient	$Δ M \pm 20 %$
Force coefficient	$Δ c \pm 20 %$

The simulation results of three cases are shown in Figures 2 to 4. It can be seen from Figure 2 that targets are intercepted successfully with acceptable miss distances and the body-LOS angles remain within the FOV constraint boundaries despite asymmetric constraint boundaries chosen for the FOV angle. Meanwhile, the curves of $δ_{i} (i = x, y, z)$ and ${\overset{\cdot}{δ}}_{i}$ are always under the required constraint, which indicates that input magnitude and rate under the proposed IGC method have not exceeded the maximum capability of actuator behavior.

Figure 2.

Simulation results under Case 1: (a) curves of the 3D trajectories, (b) elevation angle of body-LOS, (c) azimuth angle of body-LOS, (d) aileron deflection, (e) rudder deflection, (f) elevator deflection, (g) rate of aileron deflection, (h) rate of rudder deflection, and (i) rate of elevator deflection.

Figure 3.

Simulation results under Case 2: (a) curves of the 3D trajectories, (b) elevation angle of body-LOS, (c) azimuth angle of body-LOS, (d) aileron deflection, (e) rudder deflection, (f) elevator deflection, (g) rate of aileron deflection, (h) rate of rudder deflection, and (i) rate of elevator deflection.

Figure 4.

Simulation results under Case 3: (a) curves of miss distance, (b) elevation angle of body-LOS, (c) azimuth angle of body-LOS, (d) aileron deflection, (e) rudder deflection, (f) elevator deflection, (g) rate of aileron deflection, (h) rate of rudder deflection, and (i) rate of elevator deflection.

From Figure 3, it can be seen that body-LOS angles, $δ_{y}$ , $δ_{z}$ , and ${\overset{\cdot}{δ}}_{z}$ all exceed the required boundaries in QIGC law; although the maximum of $ε_{b}$ , $η_{b}$ are always within the boundaries of the seeker’s FOV in BIGC laws, $δ_{z}$ and ${\overset{\cdot}{δ}}_{z}$ of BIGC both exceed the required boundaries. In contrast, the simulation results confirm that the AIGC law can ensure not only the constraint of body-LOS angles but also the limit of the actuator input and its rate. Therefore, the efficiency and advantage of the AIGC scheme are validated by comparison. When the biased parameters change with bias values listed in Table 1, the corresponding simulation results are given in Figure 4, where the blue line represents the result using the normal values, light blue lines correspond to the results using the bias values, purple and yellow dash-dot lines represent upper and lower bounds of the input saturation, respectively. From Figure 4, it is indicated that body-LOS angles as well as the input and input rate of the actuator of the IGC system with proposed IGC Law are always under the required limitation boundaries even if the existence of the complex aerodynamic and structural parameters uncertainty.

In summary, the control objective is achieved by the proposed IGC scheme. Therefore, the proposed law can effectively address the issue of IGC with FOV constraint and input saturation.

Conclusion

This paper proposes a novel IGC scheme for the flight vehicle to intercept the maneuvering target considering asymmetric FOV constraint, input saturation, and uncertainties. By combining the model of the IGC system with strap-down seeker and the dual-integral control law, a new 3D IGC model is established. Based on AIBLF method and DSC approach, the proposed IGC law provides a novel solution from point of state constraint to solve the issue of the asymmetric FOV constraint and actuator input saturation for the IGC system. Besides, adaptive control laws are introduced to compensate for model uncertainties. By comparison, the designed IGC scheme not only guarantees the FOV angle constraint with asymmetric limitation boundaries but also ensures safe actuator operation. Finally, simulation results illustrate the superiority and effectiveness of the proposed IGC scheme.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Dan Feng

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