Three-dimensional sliding mode guidance law for maneuvering target with prescribed performance and input saturation

Abstract

The three-dimensional sliding mode guidance laws with prescribed performance and saturation actuator are proposed for maneuvering target. The proposed guidance laws can ensure the line of sight (LOS) angle converges according to the prescribed performance with actuator saturation and the convergence rate, the steady state error and the maximum overshoot can be preset in advance. A novel transformed error function is designed by combining the LOS tracking error with the performance constraint function. Then, to further solve the problem of input saturation, the saturation function and auxiliary system are introduced. Additionally, this paper discusses the problem whereby the upper bound of the aggregate uncertainty, including the target information, is unavailable. An adaptive sliding mode guidance law with prescribed performance is presented for this scenario. Experiments comparisons are conducted with other forms of guidance laws. Simulation results show that the guidance laws proposed in this paper achieve effective performance.

Keywords

Prescribed performance input saturation maneuvering target sliding mode guidance law adaptive control

Introduction

The combat capability of missiles can be enhanced by considering a specific impact angle (He and Lin, 2016; Tsalik and Shima, 2016). In the past few decades, various methods have been created in this field. Based on the proportional navigation guidance, Kim et al. (1998) developed a new biased proportional navigation guidance with a supplementary time-varying bias to satisfy the impact angle constraint. Ratnoo and Ghose (2010a, 2010b) proposed an angle-constrained biased proportional navigation guidance law for stationary and non-maneuvering targets. A biased proportional navigation guidance law, considering the seeker’s look angle and acceleration capability limits, was proposed for non maneuvering targets in Park et al. (2017). This guidance law consists of two time-varying biases, and the maximum achievable impact angle under the physical constraints of the missile has been discussed. In Song et al. (1999), an optimal active homing guidance law with impact angle constraint was designed. In Park et al. (2016), an impact angle control guidance law, which considers the impact angle and seeker’s look angle constraints, was proposed based on the optimal control theory with state variable inequality constraint. However, the bias proportional navigation guidance law and the optimal guidance law are mainly aimed at stationary or low speed targets. In recent years, sliding mode control (SMC) theory has been widely developed because of its strong robustness to external disturbance and the uncertainly of the system parameters. A sliding mode guidance law for impact angle control was proposed against a maneuvering target with unknown acceleration (Cho et al., 2015). This guidance law separates the sliding mode surface for the impact angle constraint and the homing constraint, and a virtual controller has been introduced to associate the two surfaces. In order to reduce the convergence time, terminal SMC method in which the system state converges in finite time has been reported (Modirrousta and Khodabandeh, 2017). A nonsingular sliding mode guidance law was proposed to solve the problem of singularities on the sliding mode surface in Kumar et al. (2014). This ensures that, for any initial heading angle, the missile intercepts the target at a desired impact angle without exhibiting any singularity.

Although there has been considerable research on guidance laws with impact angle constraints, most methods restrict the terminal impact angle, which cannot constrain the converge process. In some combat scenarios, the guidance process of missile needs to be constrained. In the problem of a missile intercept a stealth aircraft, the terminal guidance system should converge rapidly due to the decrease of detection distance, and the line of sight (LOS) angle is required to maintain a specific value during the entire terminal guidance phase, as the radar cross section of the target can only be identified at a certain specific angle. In the coordinated operation of multiple missiles, due to the constraints of communication or cooperative detection, the spatial configuration between multiple missiles is required to meet certain constraints. Therefore, the state of the guidance system should converge according to certain prescribed performance criteria.

In this direction, the funnel control approach was proposed (Ilchmann and Ryan, 2008; Ilchmann et al., 2002). This is a continuation of the adaptive high-gain control methodology that replaces the monotonically increasing control gain with a time-varying function. When the output error of the system is close to the funnel boundary, the funnel control admits high values, resulting in a simple nonlinear and time-varying proportional control scheme for classes of nonlinear systems with a known relative degree of one (Hopfe et al., 2010; Ilchmann and Ryan, 2009; Ilchmann et al., 2005, 2010) and, more recently, two (Hackl et al., 2010; Reis and Berger, 2017). In Bu (2018), a funnel non-affine controller for AHVs with prescribed performance was proposed applying neural approximation. In addition, Bechlioulis and Rovithakis (2008) proposed prescribed performance control (PPC) theory. Based on a predesigned performance constraint function, a PPC controller was proposed by transforming the “constrained” system into an equivalent unconstrained system, which can guarantee the system tracking error converges to an arbitrarily small residual set at a rate that is not less than a specified value, and that the maximum overshoot is not less than a small constant. Two robust adaptive PPC controllers (Bechlioulis and Rovithakis, 2009) for single-input/single-output strict feedback nonlinear systems possessing unknown nonlinearities have been proposed. In Bechlioulis and Rovithakis (2010), a control scheme has been designed for multi-input/multi-output affine control nonlinear systems with guaranteed prescribed performance. In Bechlioulis and Rovithakis (2013), adaptive dynamic surface control technology was employed to design a PPC controller, capable of curing the ‘explosion of complexity’ problem, caused by the repeated differentiations of the intermediate control signals. An adaptive SMC method for micro-electro-mechanical system gyroscopes trajectory tracking with prescribed performance has been designed in Lu and Fei (2018). In Bu (2019a), PPC method was combined with adaptive critic design (ACD) method for the first time, and a novel ACD controller satisfying prescribed performance was designed for HV nonaffine systems, which provides a new method to improve tracking performance. A novel envelope-constraint-based neural controller was proposed for air-breathing hypersonic vehicles (AHVs) with unknown non-affine formulations in Bu (2019). The complex design of back-stepping controller is avoided by a model transformation, and a novel adaptive algorithm is designed to reduce the computational costs. In Zhu et al. (2019), a composite controller with prescribed transient and steady-state performances was developed, considering system uncertainty.

In actual control system, the input of the missile’s actuator is limited. If the input exceeds the limit of the actuator, the saturation problem will result in the weakening of guidance system performance or even in system instability. To solve the saturation problem, in Liang et al. (2015), a smooth function and a Nussbaum type function are employed in the dynamic surface control design to deal with the problem of input saturation. In Zhou and Xu (2016), a three-dimensional guidance law with input saturation constraint and autopilot dynamics was developed using lock dynamic surface control. Based on the extended state observer techniques, a three-dimensional integrated guidance law with impact angle constraint was proposed for the bank to turn missile attacking a ground fixed target (Wang et al., 2016). To solve the problem of state constraint and input saturation constraint, this paper presents adaptive sliding mode guidance laws with prescribed performance and input saturation, which adopt a novel transformed error function to ensure the prescribed performance and the auxiliary system to handle input saturation. Besides, SMC method and adaptive control method can be used to solve the problem of uncertain parameters in dynamic model.

The special contributions of this study are:

(1) This paper presents a prescribed performance control method based on SMC theory. A new transformed error function is designed to transform the prescribed performance constraint problem into the bounded problem of sliding mode surface, and the performance of the controller can be enhanced by improving the sliding manifold.

(2) The PPC guidance law designed in this paper can be used to solve the problem of intercepting stealth maneuvering targets by limiting the convergence process of LOS angle. Besides, the ability of communication and cooperative detection of multi missiles can be enhanced by designing the convergence time and steady-state error of LOS angle in advance, and the overshoot can solve the problems of multi missile collision avoidance.

(3) Considering input saturation and unknown target maneuver, an anti-saturation adaptive guidance law is designed by introducing an auxiliary system and adaptive control law, and the closed-loop theory is proved, which makes the designed controller have more practical significance.

The remainder of this paper is organized as follows. In Section 2, the guidance model considering target maneuvering is established. The prescribed performance constraint function and the transformed error function are then designed. In Section 3, the anti-saturation sliding mode guidance law with prescribed performance is introduced. For the unknown upper bound of an external disturbance, an adaptive anti-saturation sliding mode guidance law is derived based on adaptive control theory. Simulation results and analyses are presented in Section 4. Finally, Section 5 summarizes our conclusions.

Preliminaries and problem formulation

Guidance system models with target maneuvering

Figure 1 shows the three-dimensional homing guidance geometry. In this figure, M is the missile and T is the target. $M X_{I} Y_{I} Z_{I}$ is the inertial reference frame. $M X_{M} Y_{M} Z_{M}$ and $T X_{T} Y_{T} Z_{T}$ denote the missile velocity coordinate system and the target velocity coordinate system. R represents the relative distance between the target and the missile. The speed and lateral acceleration of the missile and the target are denoted as $(V_{m}, a_{ym}, a_{zm})$ and $(V_{t}, a_{yt}, a_{zt})$ , respectively. $(θ_{m}, ϕ_{m})$ and $(θ_{t}, ϕ_{t})$ denote the direction of $V_{m}$ and $V_{t}$ with respect to the LOS frame. $θ_{L}$ and $ϕ_{L}$ are the LOS angle. It is assumed that the origin is the missile launch point. The engagement dynamics are given as follows

\overset{\cdot}{R} = (ρ \cos θ_{t} \cos ϕ_{t} - \cos θ_{m} \cos ϕ_{m}) V_{m}

(1)

R {\overset{\cdot}{θ}}_{L} = (ρ \sin θ_{t} - \sin θ_{m}) V_{m}

(2)

R {\overset{\cdot}{ϕ}}_{L} \cos θ_{L} = (ρ \cos θ_{t} \sin ϕ_{t} - \cos θ_{m} \sin ϕ_{m}) V_{m}

(3)

{\overset{\cdot}{θ}}_{m} = \frac{a_{zm}}{V_{m}} - {\overset{\cdot}{ϕ}}_{L} \sin θ_{L} \sin ϕ_{m} - {\overset{\cdot}{θ}}_{L} \cos ϕ_{m}

(4)

\begin{matrix} {\overset{\cdot}{ϕ}}_{m} = \frac{a_{ym}}{V_{m} \cos θ_{m}} + {\overset{\cdot}{ϕ}}_{L} \sin θ_{L} \cos ϕ_{m} \tan θ_{m} \\ - {\overset{\cdot}{θ}}_{L} \sin ϕ_{m} \tan θ_{m} - {\overset{\cdot}{ϕ}}_{L} \cos θ_{L} \end{matrix}

(5)

{\overset{\cdot}{θ}}_{t} = \frac{a_{zt}}{ρ V_{m}} - {\overset{\cdot}{ϕ}}_{L} \sin θ_{L} \sin ϕ_{t} - {\overset{\cdot}{θ}}_{L} \cos ϕ_{t}

(6)

\begin{matrix} {\overset{\cdot}{ϕ}}_{t} = \frac{a_{yt}}{ρ V_{m} \cos θ_{t}} + {\overset{\cdot}{ϕ}}_{L} \sin θ_{L} \cos ϕ_{t} \tan θ_{t} \\ - {\overset{\cdot}{θ}}_{L} \sin ϕ_{t} \tan θ_{t} - {\overset{\cdot}{ϕ}}_{L} \cos θ_{L} \end{matrix}

(7)

where $ρ = \frac{V_{t}}{V_{m}}$ .

Figure 1.

Geometry in three-dimensional space.

The main purpose of this paper is to design sliding mode guidance laws with prescribed performance. In other words, the lateral acceleration of the missile is designed to ensure that the missile intercepts the target at designated angle $θ_{Lf}$ and $ϕ_{Lf}$ , and that the LOS angle tracking error $θ_{L} - θ_{Lf}$ and $ϕ_{L} - ϕ_{Lf}$ converge according to the prescribed performance.

Assumption 1. The signals R, $\overset{\cdot}{R}$ , $θ_{L}$ , $ϕ_{L}$ , ${\overset{\cdot}{θ}}_{L}$ , ${\overset{\cdot}{ϕ}}_{L}$ , $θ_{m}$ and $ϕ_{m}$ are assumed to be measurable. $θ_{m}$ and $ϕ_{m}$ satisfy $θ_{m}, ϕ_{m} \neq \pm (π / 2)$ .

Assumption 2. During the time horizon of the guidance process, $\overset{\cdot}{R} < 0$ , $0 < R < R (0)$ , and $t > 0$ . The missile intercepts the target when $R \neq 0$ , but belongs to the interval $[r_{min}, r_{max}] = [0.1 m, 0.25 m]$

Let $θ_{Lf}$ and $ϕ_{Lf}$ be the desired LOS angle, $x_{1} = θ_{L} - θ_{Lf}$ and $x_{2} = ϕ_{L} - ϕ_{Lf}$ be the LOS angle error.

Differentiating (2)

\begin{matrix} \frac{d (R {\overset{\cdot}{θ}}_{L})}{dt} = \overset{\cdot}{R} {\overset{\cdot}{θ}}_{L} + R {\overset{\cdot\cdot}{θ}}_{L} \\ = (ρ {\overset{\cdot}{θ}}_{t} \cos θ_{t} - {\overset{\cdot}{θ}}_{m} \cos θ_{m}) V_{m} \\ = a_{zt} \cos θ_{t} - a_{zm} \cos θ_{m} - {\overset{\cdot}{ϕ}}_{L}^{2} R \cos θ_{L} \sin θ_{L} - \overset{\cdot}{R} {\overset{\cdot}{θ}}_{L} \end{matrix}

(8)

That is

{\overset{\cdot\cdot}{θ}}_{L} = \frac{\cos θ_{t}}{R} a_{zt} - \frac{\cos θ_{m}}{R} a_{zm} - {\overset{\cdot}{ϕ}}_{L}^{2} \cos θ_{L} \sin θ_{L} - \frac{2 \overset{\cdot}{R} {\overset{\cdot}{θ}}_{L}}{R}

(9)

Similarly, the derivative of (3) is

\begin{matrix} {\overset{\cdot\cdot}{ϕ}}_{L} R \cos θ_{L} = a_{yt} \cos ϕ_{t} - a_{zt} \sin θ_{t} \sin ϕ_{t} + a_{zm} \sin θ_{m} \sin ϕ_{m} \\ - a_{ym} \cos ϕ_{m} + 2 R {\overset{\cdot}{ϕ}}_{L} {\overset{\cdot}{θ}}_{L} \sin θ_{L} - 2 \overset{\cdot}{R} {\overset{\cdot}{ϕ}}_{L} \cos θ_{L} \end{matrix}

(10)

It can be written as

\begin{matrix} {\overset{\cdot\cdot}{ϕ}}_{L} = \frac{\cos ϕ_{t}}{R \cos θ_{L}} a_{yt} - \frac{\sin θ_{t} \sin ϕ_{t}}{R \cos θ_{L}} a_{zt} + \frac{\sin θ_{m} \sin ϕ_{m}}{R \cos θ_{L}} a_{zm} \\ - \frac{\cos ϕ_{m}}{R \cos θ_{L}} a_{ym} + 2 {\overset{\cdot}{ϕ}}_{L} {\overset{\cdot}{θ}}_{L} \tan θ_{L} - \frac{2 \overset{\cdot}{R} {\overset{\cdot}{ϕ}}_{L}}{R} \end{matrix}

(11)

Rearranging dynamic equations (8)–(11), the guidance system can be rewritten as

[\begin{matrix} {\overset{\cdot\cdot}{θ}}_{L} \\ {\overset{\cdot\cdot}{ϕ}}_{L} \end{matrix}] = F + Bu + d

(12)

where

\begin{matrix} F = [\begin{matrix} - {\overset{\cdot}{ϕ}}_{L}^{2} \cos θ_{L} \sin θ_{L} - \frac{2 \overset{\cdot}{R} {\overset{\cdot}{θ}}_{L}}{R} \\ + 2 {\overset{\cdot}{ϕ}}_{L} {\overset{\cdot}{θ}}_{L} \tan θ_{L} - \frac{2 \overset{\cdot}{R} {\overset{\cdot}{ϕ}}_{L}}{R} \end{matrix}], B = [\begin{matrix} - \frac{\cos θ_{m}}{R} & 0 \\ \frac{\sin θ_{m} \sin ϕ_{m}}{R \cos θ_{L}} & - \frac{\cos ϕ_{m}}{R \cos θ_{L}} \end{matrix}], \\ u = [\begin{matrix} a_{zm} \\ a_{ym} \end{matrix}], \end{matrix}

d = [\begin{matrix} d_{1} \\ d_{2} \end{matrix}] = [\begin{matrix} \frac{\cos θ_{t}}{R} a_{zt} \\ \frac{\cos ϕ_{t}}{R \cos θ_{L}} a_{yt} - \frac{\sin θ_{t} \sin ϕ_{t}}{R \cos θ_{L}} a_{zt} \end{matrix}] .

From equation (12), the control input $u$ can be designed only if the variable $B$ is nonsingular during the whole guidance process, which means $R \neq 0$ and the angles $θ_{m}$ and $ϕ_{m}$ are not equal to $\pm (π / 2)$ . With Assumption 1 and Assumption 2, $u$ can be applied to control the system (12).

Assumption 3. The external disturbance $d$ are assumed to be bounded. Let $‖ d ‖_{\infty} = d_{max}$ , where $d_{max}$ is a known positive constant.

Prescribed performance constraint function

In the control process of the nonlinear system, the state of the nonlinear system, such as the convergence rate and the steady state error, should ideally satisfy some requirements in the control process, similar to the analysis of linear systems. Let $ω (t)$ denote the system tracking error, a prescribed performance constraint function $λ (t)$ is designed to ensure the system tracking error $ω (t)$ converges to the prescribed performance.

The prescribed performance constraint function can be designed as

λ (t) = (λ (0) - λ (\infty)) e^{- lt} + λ (\infty)

(13)

where l is a positive constant and $λ (0)$ is the initial value of the prescribed performance constraint function. Let $ω (0)$ denote the initial value of the state tracking error. $λ (0)$ satisfies $0 < | ω (0) | < λ (0)$ , $λ (\infty) < λ (0)$ , $λ (\infty) > 0$ .

The prescribed performance constraint function is shown in Figure 2.

Figure 2.

Prescribed performance constraint function.

The value of l determines the convergence rate of the prescribed performance function: the larger the value of l, the faster the convergence rate. $λ (\infty)$ determines the bound of the prescribed performance constraint function in the steady state. m determines the maximum overshoot.

Similar to the design of collision-avoidance functions in research on spacecraft collision-avoidance control, a transformed error function is designed as follows

h (t) = {\begin{matrix} {(\frac{k_{h}}{(m λ + ω) (λ - ω)})}^{2}, ω (0) > 0 \\ {(\frac{k_{h}}{(λ + ω) (m λ - ω)})}^{2}, ω (0) < 0 \end{matrix}

(14)

The prescribed performance constraint can be expressed as

{\begin{matrix} - m λ (t) < ω (t) < λ (t), ω (0) > 0 \\ - λ (t) < ω (t) < m λ (t), ω (0) < 0 \end{matrix}

(15)

where $ω (t)$ denote the tracking error, $k_{h} > 0$ , and $0 \leq m \leq 1$ .

As $0 < | ω (0) | < λ (0)$ , $h (0)$ exists. The derivative of $h (t)$ in the definition domain can be written as

\overset{\cdot}{h} (t) = {\begin{matrix} \frac{- 2 k_{h}^{2} [(m \overset{\cdot}{λ} + \overset{\cdot}{ω}) (λ - ω) + (m λ + ω) (\overset{\cdot}{λ} - \overset{\cdot}{ω})]}{{[(m λ + ω) (λ - ω)]}^{3}}, ω (0) > 0 \\ \frac{- 2 k_{h}^{2} [(\overset{\cdot}{λ} + \overset{\cdot}{ω}) (m λ - ω) + (λ + ω) (m \overset{\cdot}{λ} - \overset{\cdot}{ω})]}{{[(λ + ω) (m λ - ω)]}^{3}}, ω (0) < 0 \end{matrix}

(16)

Remark 1. To ensure the system tracking error satisfies the prescribed performance (15), as $h (0)$ exists, it is only necessary to ensure that $h (t)$ is bounded in the process of system convergence. According to (14), if $h (t)$ is bounded, $| ω | \neq λ$ and $| ω | \neq m λ$ , which means that the system tracking error $ω$ will not exceed the preset convergence curve.

Therefore, the state constraint problem for the tracking control of nonlinear systems is transformed into the design of controller to ensure that $h (t)$ is bounded in the process of system convergence.

Related assumptions and lemmas

To facilitate the design, the following lemmas and assumptions are stated.

Lemma 1. (Yu et al., 2005). Consider the nonlinear system $\overset{\cdot}{x} = f (x, t), x \in R^{n}$

Suppose that there exists a continuous and positive definite function $V (x)$ such that

\overset{\cdot}{V} (x) \leq - μ V (x) - λ V^{α} (x)

(17)

where $μ$ , $λ > 0$ and $0 < α < 1$ are constants. $x (t_{0}) = x_{0}$ , where $t_{0}$ is the initial time. The system states arrive at the equilibrium point at time T, which satisfies the following inequality

T \leq \frac{1}{μ (1 - α)} \ln \frac{μ V^{1 - α} (x_{0}) + λ}{λ}

(18)

That is, the system states are finite-time convergent.

Lemma 2. (Zhu et al., 2011) Consider the system $\overset{\cdot}{x} = f (x, t), x \in R^{n}$ . Suppose that there exists a Lyapunov function $V (x)$ and scalars $ε \in (0, 1)$ , $c > 0$ , and $0 < δ < \infty$ such that

\overset{\cdot}{V} \leq - c V^{ε} + δ

(19)

Then, the trajectory of this system is practical finite-time stable.

Lemma 3. (Yu et al., 2005) Suppose $b_{1}, b_{2}, . . . b_{n}$ are real number, and q satisfies $0 < q < 1$ , then the following inequality holds

(| b_{1} | + | b_{2} | + . . . + | b_{n} |)^{q} \leq {| b_{1} |}^{q} + {| b_{2} |}^{q} + . . . + {| b_{n} |}^{q}

(20)

Lemma 4. (Wallsgrove and Akella, 2005) For arbitrary number x and nonzero real number y, the following inequality holds

0 \leq | x | (1 - \tanh (| x / y |)) \leq α^{*} | y |

(21)

where $α^{*} = 0.2785$ , and $\tanh (\cdot)$ denote the hyperbolic tangent function.

Guidance law design

Design of sliding mode guidance law with prescribed performance and input saturation

Let $e = [e_{1}, e_{2}]^{T}$ , $\overset{\cdot}{e} = [{\overset{\cdot}{e}}_{1}, {\overset{\cdot}{e}}_{2}]^{T}$ . $e_{1} = θ_{L} - θ_{Lf}$ , $e_{2} = ϕ_{L} - ϕ_{Lf}$ denote the LOS angle tracking error. $h = [\begin{matrix} h_{1} & 0 \\ 0 & h_{2} \end{matrix}]$ denote the prescribed performance constraint function for pitch and yaw channel of the missile. The main objective of this study is to design guidance laws so that the missile can intercept a maneuvering target in three-dimensional at desired LOS angles $θ_{Lf}$ and $ϕ_{Lf}$ with prescribed performance and input saturation.

The sliding mode surface is designed as follows

S = \overset{\cdot}{e} + α_{1} e + α_{2} f (e) + α_{3} he

(22)

f (e) = [\begin{matrix} f (e_{1}) \\ f (e_{2}) \end{matrix}]

(23)

f (e_{i}) = {\begin{matrix} r_{1} e_{i} + r_{2} e_{i}^{2} sign (e_{i}) & | e_{i} | < σ \\ {| e_{i} |}^{r} sign (e_{i}) & otherwise \end{matrix} i = 1, 2

(24)

r_{1} = (2 - r) σ^{r - 1}

(25)

r_{2} = (r - 1) σ^{r - 2}

(26)

where $α_{i} = [\begin{matrix} α_{i 1} & 0 \\ 0 & α_{i 2} \end{matrix}] (i = 1, 2, 3)$ , $0 < r < 1$ and $σ$ is a small positive constant.

The derivative of $S$ can be expressed as

\overset{\cdot}{S} = \overset{\cdot\cdot}{e} + α_{1} \overset{\cdot}{e} + α_{2} \overset{\cdot}{f} (e) + α_{3} \overset{\cdot}{h} e + α_{3} h \overset{\cdot}{e}

(27)

\overset{\cdot}{f} (e) = [\begin{matrix} \overset{\cdot}{f} (e_{1}) \\ \overset{\cdot}{f} (e_{2}) \end{matrix}]

(28)

\overset{\cdot}{f} (e_{i}) = {\begin{matrix} r_{1} {\overset{\cdot}{e}}_{i} + 2 r_{2} e_{i} {\overset{\cdot}{e}}_{i} sign (e_{i}) & | e_{i} | < σ \\ r {| e_{i} |}^{r - 1} {\overset{\cdot}{e}}_{i} & otherwise \end{matrix}

(29)

The design idea presented in this paper is to use the boundedness of $S$ to ensure the transformed error function $h (t)$ is bounded. Based on the terminal SMC, the anti-saturation nonsingular terminal sliding mode guidance law with prescribed performance of a missile intercepting maneuvering targets can be expressed, under Assumptions 1, 2, and 3, using the following equation

u_{0} = sat (u_{1})

(30)

\begin{matrix} u_{1} = B^{- 1} [- F - α_{1} \overset{\cdot}{e} - α_{2} \overset{\cdot}{f} (e) - α_{3} \overset{\cdot}{h} e - α_{3} h \overset{\cdot}{e} - \frac{S}{{‖ S ‖}^{2}} ((he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e})) + k_{η} η \\ - k_{1} S - k_{2} sig (S)^{γ_{1}} - H sign (S) - {\frac{S}{‖ S ‖}}^{2} (k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} + k_{h 1} (\frac{1}{2} (he)^{T} (he)))] \end{matrix}

(31)

\overset{\cdot}{η} = - k_{η_{1}} η + \frac{η}{η^{T} η} S^{T} B (u_{0} - u_{1}) - k_{η_{2}} sig (η)^{γ_{1}}

(32)

where $0 < γ_{1} < 1$ , $sig (η)^{γ_{1}} = {[\begin{matrix} {| η_{1} |}^{γ_{1}} sign (η_{1}) & {| η_{2} |}^{γ_{1}} sign (η_{2}) \end{matrix}]}^{T}$ , $H = [\begin{matrix} H_{1} & 0 \\ 0 & H_{2} \end{matrix}]$ , $k_{η}$ , $k_{1}$ , $k_{2}$ , $k_{h 1}$ , $k_{η_{1}}$ , $k_{η_{2}}$ , $H_{1}$ and $H_{2}$ are positive constants, and $k_{1} > \frac{1}{2} k_{η}$ , $k_{η_{1}} > \frac{1}{2} k_{η}$ , $‖ d ‖_{\infty} < min {H_{1}, H_{2}}$ .

The $sat (u_{1})$ is defined as

sat (u_{1, i}) = {\begin{matrix} u_{max} & u_{max} < u_{1, i} \\ u_{i} & - u_{max} \leq u_{1, i} \leq u_{max} \\ - u_{\max} & u_{1, i} < - u_{max} \end{matrix} (i = 1, 2)

(33)

Theorem 1. Considering the system in (1)–(7), if Assumptions 1, 2, and 3 hold, and the external disturbance $d$ is bounded, then $‖ d ‖_{\infty} = d_{max}$ . If (22) is selected as the sliding mode surface and (30)–(32) is selected as the guidance law, the following conclusions can be drawn:

(i) The sliding mode surface $S$ converge to zero in finite time.

(ii) The LOS tracking error $e$ and the LOS angular rate $\overset{\cdot}{e}$ converge to a small region in finite time.

(iii) In the whole guidance process, the transformed error function $h (t)$ is always bounded, and the LOS tracking error always satisfies the prescribed performance constraint as (15).

Proof. Consider the Lyapunov function candidate to be

V_{1} = \frac{1}{2} S^{T} S + \frac{1}{2} η^{T} η + \frac{1}{2} (he)^{T} (he)

(34)

The time derivative of $V_{1}$ along with (1)–(7) give

\begin{matrix} {\overset{\cdot}{V}}_{1} = S^{T} S + η^{T} η + (he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e}) \\ = S^{T} (F + B u_{0} + d + α_{1} \overset{\cdot}{e} + α_{2} \overset{\cdot}{f} (e) + α_{3} \overset{\cdot}{h} e + α_{3} h \overset{\cdot}{e}) \\ + η^{T} η + (he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e}) \\ = S^{T} d + k_{η} S^{T} η - k_{1} S^{T} S - k_{2} S^{T} sig (S)^{γ_{1}} - d_{max} S^{T} sign (S) \\ - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} \\ - k_{h 1} (\frac{1}{2} (he)^{T} (he)) - k_{η_{1}} η^{T} η - k_{η_{2}} η^{T} sig (η)^{γ_{1}} \end{matrix}

(35)

According to (22)

\begin{matrix} S^{T} η = \sum s_{i} η_{i} \\ = s_{1} η_{1} + s_{2} η_{2} \\ \leq \frac{1}{2} ({s_{1}}^{2} + {η_{1}}^{2}) + \frac{1}{2} ({s_{2}}^{2} + {η_{2}}^{2}) \\ = \frac{1}{2} ({s_{1}}^{2} + {s_{2}}^{2}) + \frac{1}{2} ({η_{1}}^{2} + {η_{2}}^{2}) \\ = \frac{1}{2} S^{T} S + \frac{1}{2} η^{T} η \end{matrix}

(36)

Using Assumption 3, inequality (37) equation reference goes here can be obtained

\begin{matrix} S^{T} d = \sum s_{i} d_{i} \\ \leq | s_{1} | | d_{1} | + | s_{2} | | d_{2} | \\ \leq d_{max} (S^{T} sign (S)) \end{matrix}

(37)

Apply Lemma 3, inequality (38) can be obtained

\begin{matrix} S^{T} sig (S)^{γ_{1}} = {| s_{1} |}^{γ_{1} + 1} + {| s_{2} |}^{γ_{1} + 1} \\ = ({| s_{1} |}^{2})^{\frac{γ_{1} + 1}{2}} + ({| s_{2} |}^{2})^{\frac{γ_{1} + 1}{2}} \\ \geq ({| s_{1} |}^{2} + {| s_{2} |}^{2})^{\frac{γ_{1} + 1}{2}} \\ = (S^{T} S)^{\frac{γ_{1} + 1}{2}} \end{matrix}

(38)

Similar to (38)

η^{T} sig (η)^{γ_{1}} \geq (η^{T} η)^{\frac{γ_{1} + 1}{2}}

(39)

Thus, equation (35) can be written as

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - (k_{1} - \frac{1}{2} k_{η}) S^{T} S - (k_{η_{1}} - \frac{1}{2} k_{η}) η^{T} η - k_{2} (S^{T} S)^{\frac{γ_{1} + 1}{2}} \\ - k_{η_{2}} (η^{T} η)^{\frac{γ_{1} + 1}{2}} \\ - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} - k_{h 1} (\frac{1}{2} (he)^{T} (he)) \\ \leq - ρ_{11} V_{1} - ρ_{12} {V_{1}}^{\frac{γ_{1} + 1}{2}} \end{matrix}

(40)

where $ρ_{11} = min {2 (k_{1} - \frac{1}{2} k_{η}), 2 (k_{η_{1}} - \frac{1}{2} k_{η}), k_{h 1}}$ , $ρ_{12} = min {{k_{2}}^{\frac{γ_{1} + 1}{2}}, {k_{η_{2}}}^{\frac{γ_{1} + 1}{2}}, k_{h 1}}$ . According to Lemma 1, inequality (40) indicates that the sliding manifold $S$ converges to zero in finite time, also $η_{i}$ and $(he)^{T} (he)$ are bounded. Thus, conclusion (i) has been proved.

The stability analysis of the tracking error $e$ and $\overset{\cdot}{e}$ are as follows:

Case 1. If $| e_{i} | < σ$ , this means that $e$ has converged to the region $‖ e ‖ \leq \sqrt{\sum_{i = 1}^{2} {| Δ e_{i} |}^{2}}$ , ${| Δ e |}_{i} = σ$ . As $(he)^{T} (he)$ is bounded, assuming that there exists a certain time point $t_{p}$ that makes $h_{i} (t)$ unbounded in the convergence process of the system, (14) implies that $| e_{i} (t_{p}) | = | λ_{i} (t_{p}) |$ or $| e_{i} (t_{p}) | = | m λ_{i} (t_{p}) |$ . Therefore, at time $t_{p}$ , $(he)^{T} (he)$ is unbounded, which contradicts the previous conclusions. Additionally, when $e_{i} (t) = 0$ , $h_{i} (t) = (\frac{k_{h}}{(m {λ_{i}}^{2})})^{2}$ . Hence, $h_{i} (t)$ is always bounded in the tracking control process, which means that there exists a positive constant $σ_{h}$ satisfying $| h_{i} (t) | \leq σ_{h}$ .

According to (22), it can be obtained

S_{i} = {\overset{\cdot}{e}}_{i} + α_{i 1} e_{i} + α_{i 2} (r_{1} e_{i} + r_{2} e_{i}^{2} sign (e_{i})) + α_{i 3} h_{i} e_{i} = 0

(41)

Then according to (41), inequality (42) can be obtained

\begin{matrix} | {\overset{\cdot}{e}}_{i} | \leq | α_{i 1} e_{i} + α_{i 2} (r_{1} e_{i} + r_{2} e_{i}^{2} sign (e_{i})) + α_{i 3} h_{i} e_{i} | \\ \leq (α_{i 1} + α_{i 3} σ_{h}) σ + α_{i 2} (r_{1} σ + r_{2} σ^{2}) \end{matrix}

(42)

Thus, it can be obtained that $e$ and $\overset{\cdot}{e}$ are convergent for Case 1.

Case 2. If $| e_{i} | \geq σ$ , consider the Lyapunov function candidate as

V_{2} = \frac{1}{2} e_{i}^{2}

(43)

The time derivative of $V_{2}$ along with (1)–(7) give

\begin{matrix} {\overset{\cdot}{V}}_{2} = e_{i} {\overset{\cdot}{e}}_{i} \\ = - e_{i} (α_{i 1} e_{i} + α_{i 2} sig (e_{i})^{γ} + α_{i 3} h_{i} e_{i}) \\ \leq - α_{i 1} e_{i}^{2} - α_{i 2} {| e_{i} |}^{γ + 1} \end{matrix}

(44)

According to Lemma 1, $e$ and $\overset{\cdot}{e}$ can converge to zero in finite time. Thus, it can be obtained that $e$ and $\overset{\cdot}{e}$ are convergent for Case 2.

Synthesizing the analysis of Case 1 and Case 2, conclusion (ii) has been proved.

From the previous analyses, $(he)^{T} (he)$ , $e$ and $\overset{\cdot}{e}$ are bounded in the guidance process. According to the analysis above, $h (t)$ is always bounded, which means that LOS tracking error satisfies the prescribed performance (15). Thus, conclusion (iii) has been proved. This completes the proof of all conclusions in Theorem 1.

Remark 2. In the design process of the controller, the boundedness of the transformed error function $h (t)$ is used to ensure that the LOS tracking error can converge as prescribed performance in the guidance process. According to (14) and (16), if $e (t)$ is too close to $λ (t)$ , then $h (t)$ and $\overset{\cdot}{h} (t)$ may become too large, which requires the controller to ensure better performance. Therefore, the parameter $k_{h}$ in (14) and the parameter $k_{h 1}$ in (31) should not be too large.

Remark 3. In practice, the upper bound of $d$ (including the target maneuvering message $a_{zt}$ , $a_{yt}$ , $θ_{t}$ and $ϕ_{t}$ ) cannot be accurately measured or estimated. To solve this problem, an adaptive anti-saturation controller is designed as follows.

Design of adaptive sliding mode guidance law with prescribed performance and input saturation

An adaptive finite-time anti-saturation sliding mode guidance law is designed in this section to solve the problem of the upper bound of the external disturbance being unknown. The guidance law is designed as (45)–(49), and the external disturbance $d$ satisfies $| d_{1} | \leq d_{M_{1}}$ , $| d_{2} | \leq d_{M_{2}}$ .

It is assumed that $d_{M} = [\begin{matrix} d_{M 1} & 0 \\ 0 & d_{M 2} \end{matrix}]$ . ${\hat{d}}_{M 1}$ and ${\hat{d}}_{M 2}$ are the estimate value of $d_{M 1}$ and $d_{M 2}$ , respectively. ${\hat{d}}_{M} = [\begin{matrix} {\hat{d}}_{M 1} & 0 \\ 0 & {\hat{d}}_{M 2} \end{matrix}]$ . Let ${\tilde{d}}_{M} = d_{M} - {\hat{d}}_{M}$ . Then

u_{2} = sat (u_{3})

(45)

\begin{matrix} u_{3} = B^{- 1} [- F - α_{1} \overset{\cdot}{e} - α_{2} \overset{\cdot}{f} (e) - α_{3} \overset{\cdot}{h} e - α_{3} h \overset{\cdot}{e} \\ - \frac{S}{{‖ S ‖}^{2}} ((he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e})) + k_{η} η - k_{1} S - k_{2} sig (S)^{γ_{1}} \\ - {\hat{d}}_{M} \tanh (\frac{S}{{p_{i}}^{2}}) - \frac{S}{{‖ S ‖}^{2}} (k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} \\ + k_{h 1} (\frac{1}{2} (he)^{T} (he)))] \end{matrix}

(46)

\overset{\cdot}{η} = - k_{η_{1}} η + \frac{η}{η^{T} η} S^{T} B (u_{0} - u_{1}) - k_{η_{2}} sig (η)^{γ_{1}}

(47)

{\overset{\cdot}{\hat{d}}}_{Mi} = γ_{d} | S_{i} |

(48)

\frac{d}{dt} p_{i}^{2} = - α^{*} {\overset{\cdot}{\hat{d}}}_{Mi} p_{i}^{2}

(49)

where $sig (η)^{γ_{1}} = {[\begin{matrix} {| η_{1} |}^{γ_{1}} sign (η_{1}) & {| η_{2} |}^{γ_{1}} sign (η_{2}) \end{matrix}]}^{T}$ , $k_{η}$ , $k_{1}$ , $k_{2}$ , $k_{h 1}$ , $k_{η_{1}}$ , $k_{η_{2}}$ are positive constants, and $k_{1} > \frac{1}{2} k_{η}$ , $k_{η_{1}} > \frac{1}{2} k_{η}$ .

Theorem 2. Considering the system in (1)–(7), if Assumptions 1, 2 and 3 hold, and the external disturbance $d$ is bounded, then if (22) is selected as the sliding mode surface and (45)–(49) is selected as the guidance law, the following conclusions can be drawn:

(i) The sliding manifold $S$ and the adaptive parameter ${\hat{d}}_{M}$ are bounded, and ${\hat{d}}_{Mi}$ has an upper bound of ${\bar{d}}_{Mi}$ , which makes ${\hat{d}}_{Mi} \leq {\bar{d}}_{Mi}$ .

(ii) The sliding manifold $S$ converges to a small region of zero in finite time, and the LOS tracking errors $θ_{L} - θ_{Lf}$ , $ϕ_{L} - ϕ_{Lf}$ and the LOS angular rate ${\overset{\cdot}{θ}}_{L}$ , ${\overset{\cdot}{ϕ}}_{L}$ asymptotically converge to within a small region of zero.

(iii) In the whole guidance process, the LOS tracking error $e$ always satisfies the prescribed performance (15).

Proof: Consider the Lyapunov function candidate to be

V_{3} = \frac{1}{2} S^{T} S + \frac{1}{2} η^{T} η + \frac{1}{2} (he)^{T} (he) + \frac{1}{2 γ_{d}} {\tilde{d}}_{M}^{T} {\tilde{d}}_{M} + \sum p_{i}^{2}

(50)

Taking the time derivative of $V_{3}$ along with (1)–(7) and substituting the controller of (45) and (49) into (50), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{3} = S^{T} \overset{\cdot}{S} + η^{T} \overset{\cdot}{η} + (he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e}) - \sum \frac{1}{γ_{d}} {\tilde{d}}_{Mi} {\overset{\cdot}{\hat{d}}}_{Mi} \\ + \frac{d}{dt} \sum p_{i}^{2} \\ = S^{T} (F + B u_{0} + d + α_{1} \overset{\cdot}{e} + α_{2} \overset{\cdot}{f} (e) + α_{3} \overset{\cdot}{h} e + α_{3} h \overset{\cdot}{e}) \\ + η^{T} η + (he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e}) - \sum \frac{1}{γ_{d}} {\tilde{d}}_{Mi} {\overset{\cdot}{\hat{d}}}_{Mi} + \frac{d}{dt} \sum p_{i}^{2} \\ = k_{η} S^{T} η - k_{1} S^{T} S - k_{2} S^{T} sig (S)^{γ_{1}} - k_{η_{1}} η^{T} η \\ - k_{η_{2}} η^{T} sig (η)^{γ_{1}} + S^{T} d - S^{T} {\hat{d}}_{M} \tanh (\frac{S}{p_{i}^{2}}) \\ - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} - k_{h 1} (\frac{1}{2} (he)^{T} (he)) - \sum {\tilde{d}}_{Mi} | S_{i} | \\ - \sum α^{*} {\hat{d}}_{Mi} p_{i}^{2} \end{matrix}

(51)

According to Lemma 4, inequality (52) can be obtained

\begin{matrix} S^{T} d - S^{T} {\hat{d}}_{M} \tanh (\frac{S}{p_{i}^{2}}) = \sum (S_{i} d_{i} - S_{i} {\hat{d}}_{Mi} \tanh (\frac{S_{i}}{p_{i}^{2}})) \\ = \sum (S_{i} d_{i} - {\hat{d}}_{Mi} p_{i}^{2} \frac{S_{i}}{p_{i}^{2}} \tanh (\frac{S_{i}}{p_{i}^{2}})) \\ \leq \sum (S_{i} d_{i} + {\hat{d}}_{Mi} p_{i}^{2} (α^{*} - \frac{S_{i}}{p_{i}^{2}})) \\ = \sum (S_{i} d_{i} + {\hat{d}}_{Mi} p_{i}^{2} α^{*} - {\hat{d}}_{Mi} S_{i}) \\ \leq \sum (S_{i} {\tilde{d}}_{Mi}_{i} + {\hat{d}}_{Mi} p_{i}^{2} α^{*}) \\ \leq \sum {\tilde{d}}_{Mi} | S_{i} | + \sum α^{*} {\hat{d}}_{Mi} p_{i}^{2} \end{matrix}

(52)

Substituting equation (52) into equation (51) yields

\begin{matrix} {\overset{\cdot}{V}}_{3} \leq k_{η} S^{T} η - k_{1} S^{T} S - k_{2} S^{T} sig (S)^{γ_{1}} - k_{η_{1}} η^{T} η - k_{η_{2}} η^{T} sig (η)^{γ_{1}} \\ - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} - k_{h 1} (\frac{1}{2} (he)^{T} (he)) \\ \leq - (k_{1} - \frac{1}{2} k_{η}) S^{T} S - (k_{η_{1}} - \frac{1}{2} k_{η}) η^{T} η - k_{2} (S^{T} S)^{\frac{γ_{1} + 1}{2}} \\ - k_{η_{2}} (η^{T} η)^{\frac{γ_{1} + 1}{2}} \\ - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} - k_{h 1} (\frac{1}{2} (he)^{T} (he)) \\ \leq 0 \end{matrix}

(53)

According to inequality (53), the sliding manifold $S$ , the auxiliary system parameter $η$ , $he$ , $p_{i}^{2}$ and the adaptive parameter ${\hat{d}}_{M}$ are bounded. This means that there exists a positive constant ${\bar{d}}_{Mi} > 0$ that satisfies ${\hat{d}}_{Mi} \leq {\bar{d}}_{Mi}$ and $({\bar{d}}_{Mi} - {\hat{d}}_{Mi})$ is bounded. Thus, conclusion (i) has been proved.

Consider the Lyapunov function candidate $V_{4}$ to be

\begin{matrix} V_{4} = \frac{1}{2} S^{T} S + \frac{1}{2} η^{T} η + \frac{1}{2} (he)^{T} (he) + \frac{1}{2 γ_{d}} \sum {({\bar{d}}_{Mi} - {\hat{d}}_{Mi})}^{2} \\ + \sum p_{i}^{2} \end{matrix}

(54)

The derivative of $V_{4}$ is

\begin{matrix} {\overset{\cdot}{V}}_{4} = S^{T} \overset{\cdot}{S} + η^{T} \overset{\cdot}{η} + (he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e}) - \frac{1}{γ_{d}} \sum ({\bar{d}}_{Mi} - {\hat{d}}_{Mi}) {\overset{\cdot}{\hat{d}}}_{Mi} \\ - \sum α^{*} {\hat{d}}_{Mi} p_{i}^{2} \\ = S^{T} (F + B u_{0} + d + α_{1} \overset{\cdot}{e} + α_{2} \overset{\cdot}{f} (e) + α_{3} \overset{\cdot}{h} e + α_{3} h \overset{\cdot}{e}) \\ + η^{T} \overset{\cdot}{η} + (he)^{T} (\overset{\cdot}{h} e + h \overset{\cdot}{e}) \\ - \sum ({\bar{d}}_{Mi} - {\hat{d}}_{Mi}) | S_{i} | - \sum α^{*} {\hat{d}}_{Mi} p_{i}^{2} \end{matrix}

(55)

According to Lemma 4, inequality (52) can be rewritten as

\begin{matrix} S^{T} d - S^{T} {\hat{d}}_{M} \tanh (\frac{S}{p_{i}^{2}}) = \sum (S_{i} d_{i} - S_{i} {\hat{d}}_{Mi} \tanh (\frac{S_{i}}{p_{i}^{2}})) \\ = \sum (S_{i} d_{i} - {\hat{d}}_{Mi} p_{i}^{2} \frac{S_{i}}{p_{i}^{2}} \tanh (\frac{S_{i}}{p_{i}^{2}})) \\ \leq \sum (S_{i} {\bar{d}}_{Mi} + {\hat{d}}_{Mi} p_{i}^{2} (α^{*} - \frac{S_{i}}{p_{i}^{2}})) \\ \leq \sum (S_{i} ({\bar{d}}_{Mi} - {\hat{d}}_{Mi}) + {\hat{d}}_{Mi} p_{i}^{2} α^{*}) \\ \leq \sum ({\bar{d}}_{Mi} - {\hat{d}}_{Mi}) | S_{i} | + \sum α^{*} {\hat{d}}_{Mi} p_{i}^{2} \end{matrix}

(56)

Substituting equation (56) into equation (55) yields

\begin{matrix} {\overset{\cdot}{V}}_{4} \leq k_{η} S^{T} η - k_{1} S^{T} S - k_{2} S^{T} sig (S)^{γ_{1}} - k_{η_{1}} η^{T} η \\ - k_{η_{2}} η^{T} sig (η)^{γ_{1}} + \sum ({\bar{d}}_{Mi} - {\hat{d}}_{Mi}) | S_{i} | + \sum α^{*} {\hat{d}}_{Mi} p_{i}^{2} \\ - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} - k_{h 1} (\frac{1}{2} (he)^{T} (he)) \\ - \sum ({\bar{d}}_{Mi} - {\hat{d}}_{Mi}) | S_{i} | - \sum α^{*} {\hat{d}}_{Mi} p_{i}^{2} \\ \leq k_{η} S^{T} η - k_{1} S^{T} S - k_{2} S^{T} sig (S)^{γ_{1}} - k_{η_{1}} η^{T} η - k_{η_{2}} η^{T} sig (η)^{γ_{1}} \\ - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} - k_{h 1} (\frac{1}{2} (he)^{T} (he)) \\ \leq - (k_{1} - \frac{1}{2} k_{η}) S^{T} S - (k_{η_{1}} - \frac{1}{2} k_{η}) η^{T} η - k_{2} (S^{T} S)^{\frac{γ_{1} + 1}{2}} \\ - k_{η_{2}} (η^{T} η)^{\frac{γ_{1} + 1}{2}} - k_{h 1} {(\frac{1}{2} (he)^{T} (he))}^{\frac{γ_{1} + 1}{2}} \\ - k_{h 1} (\frac{1}{2} (he)^{T} (he)) - \frac{1}{2 γ_{d}} \sum {({\bar{d}}_{Mi} - {\hat{d}}_{Mi})}^{2} - \sum p_{i}^{2} \\ - {[\frac{1}{2 γ_{d}} \sum {({\bar{d}}_{Mi} - {\hat{d}}_{Mi})}^{2}]}^{\frac{γ_{1} + 1}{2}} - {[\sum p_{i}^{2}]}^{\frac{γ_{1} + 1}{2}} \\ + \frac{1}{2 γ_{d}} \sum {({\bar{d}}_{Mi} - {\hat{d}}_{Mi})}^{2} + \sum p_{i}^{2} \\ + {[\frac{1}{2 γ_{d}} \sum {({\bar{d}}_{Mi} - {\hat{d}}_{Mi})}^{2}]}^{\frac{γ_{1} + 1}{2}} + {[\sum p_{i}^{2}]}^{\frac{γ_{1} + 1}{2}} \\ \leq - ρ_{21} V_{4} - ρ_{22} V_{4}^{\frac{γ_{1} + 1}{2}} + c_{2} \end{matrix}

(57)

where $ρ_{21} = min {2 (k_{1} - \frac{1}{2} k_{η}), 2 (k_{η_{1}} - \frac{1}{2} k_{η}), k_{h 1}, 1}$ , $ρ_{22} = \min {k_{2}^{\frac{γ_{1} + 1}{2}}, k_{η_{2}}^{\frac{γ_{1} + 1}{2}}, k_{h 1}, 1}$ ,

$c_{2} = \frac{1}{2 γ_{d}} \sum {({\bar{d}}_{M i} - {\hat{d}}_{M i})}^{2} + \sum p_{i}^{2} + {[\frac{1}{2 γ_{d}} \sum {({\bar{d}}_{M i} - {\hat{d}}_{M i})}^{2}]}^{\frac{γ_{1} + 1}{2}} + {[\sum p_{i}^{2}]}^{\frac{γ_{1} + 1}{2}}$ . As $({\bar{d}}_{Mi} - {\hat{d}}_{Mi})$ and $p_{i}^{2}$ are bounded, according to Lemma 2, inequality (57) indicates that the sliding manifold $S$ is practical finite-time stable.

In order to analyze the convergent region of $S$ , inequality (57) can be written in two forms

{\overset{\cdot}{V}}_{4} \leq - (ρ_{21} - \frac{c_{2}}{V_{4}}) V_{4} - ρ_{22} V_{4}^{\frac{γ_{1} + 1}{2}}

(58)

{\overset{\cdot}{V}}_{4} \leq - ρ_{21} V_{4} - (ρ_{22} - \frac{c_{2}}{V_{4}^{\frac{γ_{1} + 1}{2}}}) V_{4}^{\frac{γ_{1} + 1}{2}}

(59)

According to Lemma 1, if $ρ_{21} - \frac{c_{2}}{V_{4}} > 0$ or $ρ_{22} - \frac{c_{2}}{V_{4}^{\frac{γ_{1} + 1}{2}}} > 0$ , that is $V_{4} > \frac{c_{2}}{ρ_{21}}$ or $V_{4} > (\frac{c_{2}}{ρ_{21}})^{\frac{2}{γ_{1} + 1}}$ , $S$ is finite-time convergent. According to (54), the convergent region is $‖ S ‖ \leq Δ_{S} = min {\sqrt{2} (\frac{c_{2}}{ρ_{21}})^{\frac{1}{2}}, \sqrt{2} (\frac{c_{2}}{ρ_{21}})^{\frac{1}{γ_{1} + 1}}}$ .

The stability analysis of the tracking error $e$ and $\overset{\cdot}{e}$ are as follows:

Case 1. If $| e_{i} | < σ$ , this means that $e$ has converged to the region $‖ e ‖ \leq \sqrt{\sum_{i = 1}^{2} {| Δ e_{i} |}^{2}}$ , ${| Δ e |}_{i} = σ$ . As $(he)^{T} (he)$ is bounded, it can be obtained that $h_{i} (t)$ is bounded, which means that there exists a positive constant $σ_{h}$ satisfying $| h_{i} (t) | \leq σ_{h}$ .

According to (22), it can be obtained

S_{i} = {\overset{\cdot}{e}}_{i} + α_{i 1} e_{i} + α_{i 2} (r_{1} e_{i} + r_{2} e_{i}^{2} sign (e_{i})) + α_{i 3} h_{i} e_{i} = Ω_{S}

(60)

where $| Ω_{S} | \leq Δ_{S}$ .

Then according to (60), inequality (61) can be obtained

\begin{matrix} | {\overset{\cdot}{e}}_{i} | \leq α_{i 1} | e_{i} | + α_{i 2} | (r_{1} e_{i} + r_{2} e_{i}^{2} sign (e_{i})) | + α_{i 3} | h_{i} e_{i} | \\ + | Ω_{S} | \leq (α_{i 1} + α_{i 3} σ_{h}) σ + α_{i 2} (r_{1} σ + r_{2} σ^{2}) + Δ_{S} \end{matrix}

(61)

Thus, it can be obtained that $e$ and $\overset{\cdot}{e}$ are convergent for Case 1.

Case 2: If $| e_{i} | \geq σ$ , consider the Lyapunov function candidate as

V_{5} = \frac{1}{2} e_{i}^{2}

(62)

According to (60), the time derivative of $V_{2}$ along with (1)–(7) can be obtained

\begin{matrix} {\overset{\cdot}{V}}_{4} = e_{i} {\overset{\cdot}{e}}_{i} \\ = - e_{i} (α_{i 1} e_{i} + α_{i 2} sig (e_{i})^{γ} + α_{i 3} h_{i} e_{i} - Ω_{S}) \\ \leq - α_{i 1} e_{i}^{2} - α_{i 2} {| e_{i} |}^{γ + 1} + Ω_{S} e_{i} \end{matrix}

(63)

Further, inequality (63) can be rewritten as two forms

{\overset{\cdot}{V}}_{4} \leq - (α_{i 1} - \frac{Ω_{S}}{e_{i}}) e_{i}^{2} - α_{i 2} {| e_{i} |}^{γ + 1}

(64)

{\overset{\cdot}{V}}_{4} \leq - α_{i 1} e_{i}^{2} - (α_{i 2} - \frac{Ω_{S}}{{| e_{i} |}^{γ}}) {| e_{i} |}^{γ + 1}

(65)

According to Lemma 1, if $α_{i 1} - \frac{Ω_{S}}{e_{i}} > 0$ or $α_{i 2} - \frac{Ω_{S}}{{| e_{i} |}^{γ}} > 0$ , $e_{i}$ is finite-time convergent. Thus, the convergent region of $e_{i}$ is $‖ e_{i} ‖ \leq Δ_{e} = min {\frac{Ω_{S}}{α_{i 1}}, (\frac{Ω_{S}}{α_{i 2}})^{\frac{1}{γ}}}$ . As $| h_{i} (t) | \leq σ_{h}$ , inequality (66) can be obtained

\begin{matrix} | {\overset{\cdot}{e}}_{i} | \leq α_{i 1} | e_{i} | + α_{i 2} | {| e_{i} |}^{r} sign (e_{i}) | + α_{i 3} | h_{i} e_{i} | + | Ω_{S} | \\ \leq (α_{i 1} + α_{i 3} σ_{h}) Δ_{e} + α_{i 2} {Δ_{e}}^{γ} + Δ_{S} \end{matrix}

(66)

Thus, it can be obtained that $e$ and $\overset{\cdot}{e}$ are convergent for Case 2.

Therefore, conclusion (ii) has been proved.

As $(he)$ and $e$ are bounded, $h (t)$ is always bounded, which means that LOS tracking error satisfies the prescribed performance (15). Thus, conclusion (iii) has been proved. This completes the proof of all conclusions in Theorem 2.

Remark 4: In the analysis of Theorem 2, the convergence region of $S$ are given qualitatively, but the accurately convergence region cannot be given quantitatively. This is because the system is practical finite-time stable, and $S$ can not converge to zero. But the convergence region of $e$ can be constrained by setting the $λ (\infty)$ in (13) and the convergence rate can be limited by adjusting the l in (13). Furthermore, the maximum overshoot can be designed by the parameter m in (13).

Simulations

In this section, numerical simulations are conducted to verify the performance of the designed guidance laws. To illustrate the effectiveness of the designed guidance laws, two different target maneuvering scenarios are considered:

Case 1: $a_{yt} = a_{zt} = 5 g \cos (π / 2 t) m / s^{2}$

Case 2: $a_{yt} = a_{zt} = 5 g \cos (π t) m / s^{2}$

where $g = 9.8 m / s^{2}$ denotes the acceleration of gravity.

To verify the performance of the designed guidance laws, the results are compared with those from the proportional navigation guidance law (PNGL) and fast nonsingular terminal sliding mode guidance law (FNTSMGL) (Si and Song, 2019).

The PNGL is given as

n_{c} = [\begin{matrix} - N_{1} \overset{\cdot}{R} {\overset{\cdot}{θ}}_{L} \\ - N_{2} \overset{\cdot}{R} {\overset{\cdot}{θ}}_{L} \end{matrix}]

(67)

The fast nonsingular terminal sliding mode surface is given as

S = \overset{\cdot}{e} + α_{1} e + α_{2} f (e) + α_{3} he

(68)

f (e) = [\begin{matrix} f (e_{1}) \\ f (e_{2}) \end{matrix}]

(69)

f (e_{i}) = {\begin{matrix} r_{1} e_{i} + r_{2} e_{i}^{2} sign (e_{i}) & | e_{i} | < σ \\ {| e_{i} |}^{r} sign (e_{i}) & otherwise \end{matrix} i = 1, 2

(70)

r_{1} = (2 - r) σ^{r - 1}

(71)

r_{2} = (r - 1) σ^{r - 2}

(72)

where $α_{i} = [\begin{matrix} α_{i 1} & 0 \\ 0 & α_{i 2} \end{matrix}] (i = 1, 2, 3)$ , $0 < r < 1$ and $σ$ is a small positive constant.

The FNTSMGL is given as

u_{F} = - B^{- 1} (F + α_{1} \overset{\cdot}{e} + α_{2} \overset{\cdot}{f} (e) + k_{1} S + k_{2} sign (S))

(73)

\overset{\cdot}{f} (e) = [\begin{matrix} \overset{\cdot}{f} (e_{1}) \\ \overset{\cdot}{f} (e_{2}) \end{matrix}]

(74)

\overset{\cdot}{f} (e_{i}) = {\begin{matrix} r_{1} {\overset{\cdot}{e}}_{i} + 2 r_{2} e_{i} {\overset{\cdot}{e}}_{i} sign (e_{i}) & | e_{i} | < σ \\ r {| e_{i} |}^{r - 1} {\overset{\cdot}{e}}_{i} & otherwise \end{matrix}

(75)

The two sets of initial scene parameters listed in Table 1 are considered.

Table 1.

Initial scene parameters.

Dataset	$R (0)$ ,m	$θ_{L} (0)$ ,°	$ϕ_{L} (0)$ ,°	$θ_{m} (0)$ , °	$ϕ_{m} (0)$ , °	$θ_{t} (0)$ ,°	$ϕ_{t} (0)$ ,°	$θ_{Lf}$ ,°	$ϕ_{Lf}$ ,°	$V_{m}$ ,m/s	$V_{t}$ ,m/s
1	5000	30	50	20	10	–20	–10	20	45	900	300
2	9000	20	30	10	–10	–10	10	15	35	900	300

The maximum lateral accelerations of the missile provided by the dynamic actuators are limited as

a_{M} = {\begin{matrix} a_{M max} sign (a_{M}) & if | a_{M} | \geq a_{M max} \\ a_{M} & if | a_{M} | < a_{M max} \end{matrix}

(76)

where $a_{M max} = 40 g$ . This means that the maximum lateral acceleration of the missile is $40 g$ .

Simulation results of sliding mode guidance law with prescribed performance

The parameters of guidance law $u_{1}$ are $α_{1} = [\begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}]$ , $α_{2} = [\begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}]$ , $α_{3} = [\begin{matrix} 0.001 & 0 \\ 0 & 0.001 \end{matrix}]$ , $σ = 0.05$ , $r = 0.2$ , $k_{η} = 0.1$ , $k_{1} = 0.34$ , $k_{2} = 0.05$ , $γ_{1} = 0.7$ , $k_{h 1} = 0.01$ , $H = [\begin{matrix} 0.045 & 0 \\ 0 & 0.045 \end{matrix}]$ . The parameters in the prescribed performance constraint function are $λ_{1} (0) = 12$ , $λ_{1} (\infty) = 0.4$ , $l_{1} = 0.8$ , $λ_{2} (0) = 6$ , $λ_{2} (\infty) = 0.4$ , $l_{2} = 0.8$ , $m = 0.4$ , $k_{h} = 0.001$ . The parameters in the PNGL is $N_{1} = 5$ , $N_{2} = 5$ . The parameters in $u_{F}$ are same as $u_{1}$ , $α_{1} = [\begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}]$ , $α_{2} = [\begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}]$ , $σ = 0.05$ , $r = 0.2$ , $k_{1} = 0.34$ , $k_{2} = 0.05$ .

Case 1

When $a_{yt} = a_{zt} = 5 g \cos (π / 2 t) m / s^{2}$ , the target is performing an S-type maneuver. The simulation results under $u_{1}$ are shown in Figure 3. Figure 3(a) shows the relative movement trajectories under three guidance laws and the missile can intercept the target successfully, which verifies the effectiveness of the guidance law presented in this paper. Figure 3(b) shows the LOS angle can converge to the desired angle under guidance laws $u_{1}$ and $u_{f}$ , but the PNGL cannot ensure the convergence of the LOS angle. Figure 3(c) gives the curves of the $θ_{L}$ tracking error for guidance laws $u_{1}$ and $u_{f}$ with the prescribed performance constraint function. The results indicate that the $θ_{L}$ tracking error converges to zero with the prescribed performance under guidance law $u_{1}$ . However, $u_{f}$ cannot guarantee the $θ_{L}$ tracking error converges to the prescribed performance. Similarly, Figure 3(d) describes the curves of the $ϕ_{L}$ tracking error with the prescribed performance constraint function, and $u_{1}$ can ensure the tracking error converge to zero with the prescribed performance. Figure 3(e) shows the acceleration of the missile. The acceleration under $u_{1}$ and $u_{f}$ exhibits saturation at the beginning of the guidance process. This is because the selected sliding mode surface requires a large control input to ensure rapid convergence. Figure 3(f) shows the sliding mode surface under $u_{1}$ and $u_{f}$ . Both surfaces converge to a small region in a short period of time. This is consistent with the previous theoretical analysis.

Figure 3.

Responses under $u_{1}$ for Case 1.

Case 2

For $a_{yt} = a_{zt} = 5 g \cos (π t) m / s^{2}$ , the simulation results under $u_{1}$ are shown in Figure 4. Although the target is maneuvering in a different way, Figure 4(a) shows the missile can still intercept the target successfully under all three guidance laws. Figure 4(b) shows the curves of LOS angle $θ_{L}$ and $ϕ_{L}$ , and the LOS angle converge to the desired angle under $u_{1}$ and $u_{f}$ . The $θ_{L}$ tracking error with the prescribed performance is presented in Figure 4(c). Similar to case 1, $u_{1}$ can guarantee the $θ_{L}$ tracking error converges to the prescribed performance. Figure 4(d) gives the $ϕ_{L}$ tracking error with the prescribed performance constraint function. Figure 4(e) describes the curves of the missile acceleration under the three guidance laws. Compared with case 1, the acceleration of the missile changes more frequently as the target maneuvers more frequently. As shown in Figure 4(f), the sliding mode surface under $u_{1}$ and $u_{f}$ converges to a small region within a short period of time.

Figure 4.

Responses under $u_{1}$ for Case 2.

From the simulation results of Figures 3 and 4, it can be concluded that the guidance law $u_{1}$ designed in this paper performs well when the target is maneuvering in different ways. Compared with previously designed sliding mode guidance laws, $u_{1}$ can ensure the LOS angle tracking error converges according to the prescribed performance in the case of missile acceleration saturation. This agrees with the analysis of the dynamic process of the guidance system. However, guidance law $u_{1}$ requires the upper bound of the maneuvering information of the target to be known in advance. Considering that the parameters related to the target cannot be easily obtained in actual guidance systems, the next section describes simulations using the adaptive sliding mode guidance law with prescribed performance.

Simulation results of adaptive sliding mode guidance law with prescribed performance

Similar to 4.1, the PNGL and FNTSMGL $u_{f}$ are compared with guidance law $u_{3}$ . To separate this experiment from the simulation of $u_{1}$ , dataset 2 in Table 1 is used as the initial scene parameters for the simulation. The parameters in guidance law $u_{3}$ are $α_{1} = [\begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}]$ , $α_{2} = [\begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}]$ , $α_{3} = [\begin{matrix} 0.001 & 0 \\ 0 & 0.001 \end{matrix}]$ , $σ = 0.05$ , $r = 0.2$ , $k_{η} = 0.1$ , $k_{1} = 0.65$ , $k_{2} = 0.05$ , $γ_{1} = 0.7$ , $k_{h 1} = 0.01$ . The adaptive gain $γ_{d} = 0.02$ . The parameters in the prescribed performance constraint function are $λ_{1} (0) = 6$ , $λ_{1} (\infty) = 0.4$ , $l_{1} = 0.8$ , $λ_{2} (0) = 6$ , $λ_{2} (\infty) = 0.4$ , $l_{2} = 0.8$ , $m = 0.4$ , $k_{h} = 0.001$ . The parameters in the PNGL is $N_{1} = 5$ , $N_{2} = 5$ . The parameters in $u_{F}$ are $α_{1} = [\begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}]$ , $α_{2} = [\begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}]$ , $σ = 0.05$ , $r = 0.2$ , $k_{1} = 0.6$ , $k_{2} = 0.05$ .

Case 1

When $a_{yt} = a_{zt} = 5 g \cos (π / 2 t) m / s^{2}$ , the simulation results under $u_{3}$ are shown in Figure 5. Figure 5(a) shows the relative movement trajectories under three guidance laws. The missile can intercept the target successfully under $u_{3}$ . Figure 5(b) presents the curves of $θ_{L}$ and $ϕ_{L}$ . The LOS angle can converge to the desired angle under both $u_{3}$ and $u_{f}$ . Figure 5(c) gives the curves of the $θ_{L}$ tracking error with the prescribed performance constraint function under $u_{3}$ and $u_{f}$ . It is obvious that the $u_{3}$ can ensure the $θ_{L}$ tracking error converge to zero with the prescribed performance. However, $u_{f}$ cannot guarantee the $θ_{L}$ tracking error converges to the prescribed performance. Similarly, Figure 5(d) gives the curves of the $ϕ_{L}$ tracking error with the prescribed performance constraint function. From Figure 5(e), the missile acceleration under $u_{3}$ and $u_{f}$ exhibits saturation at the beginning of the process, as the selected sliding manifold requires a large control force to ensure rapid convergence. The curves of the sliding mode surface are presented in Figure 5(f). The results indicate that the sliding mode surface under $u_{3}$ and $u_{f}$ converge to a small region in a short period of time. This is consistent with the previous theoretical analysis.

Figure 5.

Responses under $u_{3}$ for Case 1.

Case 2

For $a_{yt} = a_{zt} = 5 g \cos (π t) m / s^{2}$ , the simulation results under $u_{3}$ are shown in Figure 6. Figure 6(a) shows that the missile can still intercept the target successfully under all three guidance laws. Figure 6(b) indicates that the LOS angle $θ_{L}$ and $ϕ_{L}$ converges to the desired angle under $u_{3}$ and $u_{f}$ . As shown in Figure 6(c) and Figure 6(d), $u_{3}$ can still ensure the $θ_{L}$ and $ϕ_{L}$ tracking error converges to zero to satisfy the prescribed performance. Figure 6(e) describes the acceleration of the missile. Figure 6(f) shows that the sliding mode surface under $u_{3}$ and $u_{f}$ converges to a small region in a short period of time.

Figure 6.

Responses under $u_{3}$ for Case 2.

Figures 5 and 6 indicate that the adaptive guidance law $u_{2}$ can guarantee the effectiveness of the guidance system when the upper bound of the target maneuver is unknown. Note that the prescribed performance constraint function can, in theory, be designed arbitrarily. However, as the control force in actual system is limited, the parameters of the prescribed performance constraint function cannot be selected arbitrarily. For example, if the parameter l in the prescribed performance function is too large, the system will converge faster, requiring a larger control force. This may lead to the failure of the prescribed performance constraint function. Therefore, the parameters should be selected according to the requirements of the actual system

In conclusion, Figure 3 –Figure 6 indicate that the guidance law designed in this paper performs well when the target is maneuvering in different ways. Furthermore, $u_{1}$ and $u_{3}$ can ensure the LOS angle $θ_{L}$ and $ϕ_{L}$ converge according to the prescribed performance.

Conclusion

In this paper, a novel three-dimensional sliding mode guidance law with prescribed performance and input saturation is designed for maneuvering target, capable of allowing attributes such as maximum overshoot, convergence rate and steady state error to be designed in advance. Visualizing the prescribed performance characteristics as tracking error constraints, the key concept in this approach is to transform the prescribed performance constraint problem into that of the boundedness of the transformed error function. The auxiliary system is introduced to deal with the problem of input saturation. Besides, the guidance laws designed in this paper can deal with the problem of target acceleration under known and unknown bounds. A strict theoretical proof was presented, and simulation results verified the effectiveness and superiority of the guidance laws. The proposed prescribed performance controller is more robustness and simpler, and it can also be conveniently used in other nonlinear control systems, such as spacecraft attitude control, robot control and so on. However, the PPC guidance law proposed in this paper relies on the sign of initial LOS error, which is complex for engineering practice. In the future work, the guidance law with improved PPC method will be considered.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Aeronautical Science Foundation of China (20170112012, 20180112013), the National Natural Science Foundation of China (61703126, 6191101340), and the Major Program of Natural Science Foundation of China (number 61690210).

ORCID iDs

Meng-chen Ma

Li-Guo Tan

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