J. Cent. South Univ. (2016) 23: 859-868

DOI: 10.1007/s11771-016-3133-x

Differential geometric guidance command with finite time convergence using extended state observer

MA Yi-wei(麻毅威)^{1, 2}, ZHANG Wei-hua(张为华)¹

1. College of Aerospace Science and Engineering, National University of Defense Technology,

Changsha 410073, China;

2. Unit 61541 of PLA, Beijing 100094, China

 Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: For improving the performance of differential geometric guidance command (DGGC), a new formation of this guidance law is proposed, which can guarantee the finite time convergence (FTC) of the line of sight (LOS) rate to zero or its neighborhood against maneuvering targets in three-dimensional (3D) space. The extended state observer (ESO) is employed to estimate the target acceleration, which makes the new DGGC more applicable to practical interception scenarios. Finally, the effectiveness of this newly proposed guidance command is demonstrated by the numerical simulation results.

Key words: missile guidance; differential geometric guidance command (DGGC); finite time convergence (FTC); extended state observer (ESO); maneuvering targets

1 Introduction

Due to the simplification and feasibility, the proportional navigation (PN) has been widely used in engineering practices of missile guidance community since its appearance in late 1940 s [1]. PN has been proved to be capable of intercepting non-maneuvering targets. However, the performance of PN will be greatly degraded against highly maneuvering targets. Then, researchers began to study new guidance laws which aim at handling modern agile targets, based on modern control theories such as optimal control and H-∞ control.

Differential geometric guidance command (DGGC) is a novel guidance law proposed recently, which aims to improve the performance of the endoatmospheric interception of maneuvering targets. KUO and CHIOU [2-4] firstly studied the three-dimensional motion of the missile and target in the arc-length system with the help of classical differential geometry theory, and proposed DGGC in the arc-length system. LI et al [5-6] transformed DGGC from the arc-length system to the time domain, and studied the application of DGGC in the scenario of intercepting a tactical ballistic missile. The initial capture condition of this case was also investigated. LI et al [7-9] analyzed DGGC by utilizing the relative kinematic equations based on the line of sight (LOS) rotation reference system, and proposed a new formation of DGGC which was much simpler. Actually, the main contribution of DGGC is that the DGGC provides a new direction of command acceleration which has better performance of controlling LOS rate compared with traditional 3D PPN. This property of DGGC is able to improve its performance by combining some other guidance strategies with it. Ye et al [10] presented a novel DGGC combined with the sliding mode control theory. however, only the asymptotic convergence of the LOS rate is guaranteed and the upper bound of the target acceleration is required for the construction of the guidance command. For other types of differential geometric guidance laws, ARIFF et al [11] proposed a differential geometric guidance law by using the involute of the target’s trajectory. WHITE et al [12] presented another differential geometric guidance law which was thought to be better than the classic PN.

It is obvious that the finite time convergence of the LOS rate is more important for the modern guidance laws against maneuvering targets. Therefore, a lot of researchers have been devoted to this sector. HAIMO [13] firstly proposed the finite time control law in 1986. GURFIL et al [14] discussed the finite time stability issue of the guidance system. WU et al [15] gave a design of the finite time guidance law to intercept the fixed targets based on the nonlinear three-dimensional relative kinematics for the missile and target. ZHOU et al [16] proposed the finite time convergence guidance law on the purpose of intercepting maneuvering targets, and it could guarantee the convergence of LOS rate in finite time, but the upper bound of target acceleration needed to be initially known. WANG and WANG [17] proposed a partial integrated sliding mode guidance law, while, it was quite complex and some high order variable was used in the guidance command which was difficult to measure and had to be estimated by using a sliding mode observer.

Target acceleration is the main factor which causes the missile to miss the target. Recently, it becomes popular to use the extended state observer (ESO) to estimate the target acceleration during the guidance process. The effectiveness of ESO has been proved. YAO and WANG [18] designed an ESO to estimate the target acceleration in real time. ZHU et al [19] proposed a sliding mode guidance law with ESO, which could guarantee the convergence of states of the guidance system to the designed sliding surface in finite time. Based on ZHU’s research, XIONG et al [20] explored a guidance law against maneuvering targets with intercept angle constraint, and the application range of ESO was extended.

In this work, the DGGC with FTC using ESO (DGGC-FE) is presented in order to overcome the shortage of previous work. Firstly, the finite time convergence guidance law (FTCG) is directly constructed in the engagement plane and combined with DGGC in 3D space, which is different from and simpler than the previous work by ZHOU et al [16] that realizes FTCG in 3D space by constructing two 2D FTCGs in the pitch and yaw planes of the missile and taking their coupling effect into account. Secondly, the ESO is used to estimate the target acceleration vertical to the line of sight directly in three-dimensional space.

2 Dynamic equations and form of guidance law

Firstly, the differential geometry theory will be introduced. The flight trajectories of the endoatmospheric missile and target can be approximately treated as continuous smooth space curves approximately. Let t, n, and b be the tangential, normal, and binormal unit vectors of a space curve, respectively, as shown in Fig. 1.

Frenet-Serret formula [3] is essential to describe the motions of space curves, which is given as

                                  (1)

where κ is the curvature; τ is the torsion; ds represents the derivative with respect to the trajectory of the space curve, viz.,

Fig. 1 Space curve and Frenet frame

ds/dt=V                                           (2)

The rotating line of sight (LOS) kinematic equation set is written as

                                (3)

where e_r is the unit vector along LOS, and e_ω is the unit vector along the LOS angular velocity, e_θ=e_ω×e_r is the unit normal vector of LOS; e_r, e_θ, and e_ω form the bases of the rotating LOS coordinate system; e_r and e_θ constitute the instantaneous rotation plane of LOS (IRPL); ω_s is the instantaneous LOS rate; IRPL may rotate around e_r in 3D space, and Ω_s is the IRPL rotation rate.

The relative dynamic equation set is written as

                             (4)

where r is the relative distance; a represents the acceleration; subscripts t, m represent the quantities belong to the target and missile, respectively; subscripts r, θ, and ω represent projections along e_r, e_θ, and e_ω, respectively.

LI et al [7-9] deduced a simple expression of DGGC in the time domain by

                             (5)

where a_mθ represents the desired commanded acceleration of the guidance law vertical to LOS; n_m is the designed direction of the commanded acceleration, which is also the normal direction of the missile trajectory.

It is shown by system (4) that the first two equations determine the relative motion in the instantaneous rotation plane of LOS (IRPL) between the missile and target, and the third equation determines the rotation of IRPL. Since the rotation of IRPL does not affect the final interception, the main problem of guidance lies in the countermeasure between the missile and target in the IRPL. According to the characteristics of relative motion and the viewpoint of system control theory, a_mθ can be selected as the control variable, ω_s can be selected as the state variable. In order to make the relative motion of missile and target close to parallel approaching, an effective control variable a_mθ should be obtained, which will restrain ω_s within certain realms, or even better, dropping to 0.

Substituting Eq. (5) into Eq. (4), the following equations can be obtained:

                    (6)

And a solution of n_m is described as follows [9].

For the well-definition of Eq. (5), suppose

                                      (7)

where γ<1 is a constant. Simultaneously, n_m needs to meet the following two constraints:

                                   (8)

where t_m is the direction of the missile velocity, which is also the tangent direction of the missile trajectory. By simultaneously solving Eqs. (7) and (8), the following equation can be obtained:

 (9)

Substituting Eqs. (7) and (9) into Eq. (5), the following equation can be obtained:

                   (10)

Hence, an expression of DGGC in three- dimensional case is obtained.

However, according to Eq. (10), a_mθ remains unknown, thus a_DGGC cannot be used directly as a command. Some scholars designed a_mθ by using the PN concept, which cannot guarantee the finite time convergence of LOS rate. In addition, they were ineffective for intercepting the high maneuvering targets. In this work, in order to improve the interception of the high maneuvering targets, a_mθ will be designed by using a new theory as below.

3 DGGC with FTC

Traditional robust control methods are mainly based on Lyapunov theorems of asymptotic stability or exponential stability. The theoretical results only indicated that the state of system would converge to zero or its small neighborhood as time approaches infinity. Finite time control is able to guide the states of system to zero in a finite time, which definitely brings stronger robustness and better performance to the control system.

3.1 Brief introduction of nonlinear control system

Firstly, some basic concepts of finite time stability theory for nonlinear systems will be introduced here [16].

Definition 1: Consider a system in the form of

                    (11)

where  is continuous on  and U₀ is an open neighborhood of the origin x=0. The state of the system is said to converge to its local equilibrium x=0 in finite time if, for any given initial time t₀ and initial state  there exists a settling time T≥0, which is dependent on x₀, such that every solution of the system (11),  satisfies

            (12)

moreover, if the system (local) equilibrium x=0 is Lyapunov stable with finite time convergence in a neighborhood of the origin  then the system equilibrium is called finite time stable. If  then the origin is a global finite time stable equilibrium.

Lemma 1: Consider the nonlinear system described by Eq. (11). Suppose that there is a C¹ (continuously differentiable) function V(x,t) defined in a neighborhood  of the origin, and that there are real numbers α>0 and 0<λ<1, such that V(x,t) is positive-definite on  and that  on  Then, the zero solution of system (11) is finite-time stable.

3.2 Guidance command

ZHOU et al [[16]] discussed the missile-target relative motion in three-dimensional space. Since the three relative kinematic equations were not decoupled, the guidance law design was conducted in the horizontal and vertical planes of LOS, respectively, which made the guidance law complex and expensive in terms of the computational demand. Differential geometry guidance model introduced earlier is used here. Then, the design of guidance law becomes much simpler since the relative motion between the missile and target in IRPL is decoupled with the rotation of IRPL.

The second equation of Eq. (6) could be rewritten as

                   (13)

where a_mθ is the control variable; a_tθ is taken as the uncertainty and disturbance. Suppose the initial time of terminal guidance t₀=0, and initial states r(0),  and ω_s(0). Meanwhile, in the terminal guidance process we have

                       (14)

Next, the theory introduced in Section 3.1 will be used to design the control variable of the system (13). then, a finite time convergence guidance command can be obtained. Firstly, we introduce the following theorem, which is an extension of Theorem 1 [16] in 3D space.

Theorem 1: Consider the guidance system (13). If there exists a control a_mθ such that the system state satisfies

                  (15)

where   and  then ω_s converges to zero in finite time. Furthermore, the convergence rate increases as the value of β is increased or η decreases.

Proof: Choose a continuously differentiable positive-definite function as

                                    (16)

Take a derivative of V₁ with respect to time, and according to Eqs. (14) and (15), the following equation can be obtained.

               (17)

By Lemma 1, ω_s converges to zero in finite time t_r1, and the settling time is given by

                                (18)

It is revealed in Eq. (18) that the convergence rate increases as the value of β increases. Moreover, in practice, the absolute value of the initial LOS angular rate |ω_s(0)| must be much less than 1 rad/s, and so the convergence rate increases as the value of η increases.

Substituting Eq. (13) into Eq. (15) gives

 (19)

Thus, the guidance law can be obtained as

                           (20)

Where   and |x(0)|<<1, which makes ω_s converge to zero in finite time. Furthermore, the convergence rate increases as the value of β increases or η decreases.

According to Eq. (20) it is obvious that, if -1<η<1, there exists a singularity at ω_s=0. Hence, a reasonable range of η is 0≤η<1.

Equation (20) involves a signum function, which indicates that the control variable may switch during the guidance process. In a practical system, the occurrence of a switching cannot be completely instantaneous. The delay of the switching induces the chattering effect. To remove the chattering effect, we may smooth the signum function usually by replacing it with a saturation function sat_δ(x) which is expressed as

                         (21)

where d>0 is a designed constant. Then, the guidance law can be rewritten as

(22)

Substituting Eq.(22) into Eq.(10), the differential geometric guidance command with finite time convergence (DGGC-F) is obtained:

    (23)

Since a_tθ is not completely known to the guidance law designer, it is usually replaced by the upper bound d for sliding mode guidance laws [16], which satisfies

                                    (24)

Then, the guidance command of DGGC-F becomes

       (25)

It can be proved that the finite time convergence of the LOS rate can also be guaranteed by using DGGC-F of Eq. (25). For the length of the paper, the proof is omitted here.

4 DGGC with FTC using ESO

Although DGGC-F of Eqs. (23) and (25) can guarantee the finite time convergence of the system states, the target acceleration a_tθ or its upper bound d is difficult to obtain. Therefore, the guidance performance of DGGC-F may be degraded when an inaccurate a_tθ or d is used. In the following section, we will use ESO to estimate a_tθ.

The extended state observer (ESO) is a nonlinear state observer designed to remove the requirement of a precise model of the plant by rejecting the unmodeled dynamics. This observer uses a simple canonical form, and the unmodeled dynamics is included in the disturbance which has to be estimated.

Let ω_s=x₁ in Eq. (13), and expanding the term with a_tθ as an one-order state:

                                (26)

Let  g(x₂) is an unknown quantity here. Then, constructing the system as

                    (27)

Based on state filtering theory, Eq. (27) corresponds to the ESO as

                   (28)

where z₁ and z₂ are observed values of x₁ and x₂, respectively. Replacing x₂ by its observed value z₂, and substituting it into Eq. (26), the following equation can be obtained:

                              (29)

In addition, α₁, β₀₁, β₀₂ and d₁ in Eq. (28) are the ESO parameters. The function fal(·) is defined as

                   (30)

Substituting Eq. (29) into Eq. (23), we now obtain the expression of DGGC-FE as

 (31)

where N=const.>2, β=const.>0, -1<η=const.<1.

According to Eq. (31), for the guidance command of DGGC-FE, all the variables and parameters concerned can be easily obtained in practical interception scenarios. Then, DGGC-FE can be realized in practical interception scenarios. Besides, according to the above deduction and analysis of DGGC-FE, the nonlinear dynamics of the three-dimensional interception situation has been taken into full account, which is a major progress compared with Zhou’s method [16].

5 Simulation results

In this section, numerical simulations are performed in the scenario of intercepting an endoatmospheric maneuvering target. The performances of DGGC-F and DGGC-FE are analyzed and validated, while PPN is used as a benchmark guidance law, whose expression is

                               (32)

where  is the LOS angular velocity, and V_m the missile velocity. The navigation constant is selected to be 4, and the sampling period is T=0.01 s. The dynamic lag of the missile is not considered, for the accordance of the analysis in above sections.

The initial sates of missile and target in the Launch Inertial Coordinate System are listed in Table 1.

Table 1 Initial states of missile and target

For both of DGGC-F and DGGC-FE, N=3, and γ=0.7 are adopted. The saturation function of Eq. (21) is used with d=0.15 (°)/s.

The initial Frenet Frame of the target is selected as t_t0=[-0.5985;0;0.8012], n_t0=[0;1;0], and b_t0=[0.8012;0;- 05985]. The curvature of target is selected as  and the torsion τ_t=0.07. The profile of the target acceleration is shown in Fig. 2.

5.1 Performance demonstration of DGGC-F

Firstly, we demonstrate the performance of DGGC-F against the maneuvering target. For DGGC-F, d=30 is used. For the finite time convergence term of the guidance laws, β=5, 20, 40, and η=0.1, 0.5, 0.9 are adopted. The simulation results are shown in the following figures.

Fig. 2 Profile of target acceleration in target velocity reference coordinate system

Fig. 3 Three-dimensional trajectories of PPN and DGGC-F with η=0.1

Fig. 4 Commanded accelerations of PPN and DGGC-F with η=0.1

The simulation results of PPN and DGGC-F with β=5, 20, 40 when η=0.1 are shown in Figs. 3-6. The 3D trajectories are shown in Fig. 3. It can be seen that theflight trajectory of PPN is quite different from those of DGGC-F. The trajectory of PPN is curvy during the whole engagement, while the trajectories of DGGC-F are quite curvy in the beginning and then become straight in the latter half of the guidance process.

Fig. 5 Three-dimensional LOS rates of PPN and DGGC-F with η=0.1

Fig. 6 ZEMs of PPN and DGGC-F with η=0.1

The commanded accelerations in this case are shown in Fig. 4. We can see that, |a_m| of PPN is firstly the smallest. However, it becomes the largest in the end. For DGGC-F, in the beginning the commanded accelerations are very large, since large d and β are applied. However, the commanded accelerations converge to the constant states quickly and keep constant after that. as β increases, the initial value of |a_m| of DGGC-F increases, and so is the convergence rate.

The 3D LOS rates in this case are shown in Fig. 5. ω_s of PPN is always the highest and goes up to a very large value in the end, while the LOS rates of DGGC-F all converge to the boundary layer rapidly. For DGGC-F, as β increases, the convergence rate increases.

The ZEMs in this case are shown in Fig. 6. From the figure, it indicates that the convergence rate of ZEM of PPN is the slowest one. For DGGC-F, as β increases, the convergence rate increases.

According to the simulation results, a large β results in a large convergence rate, but also a large initial commanded acceleration. Therefore, a small β may be good enough for the guidance command.

The simulation results of PPN and DGGC-F with η=0.1, 0.5, 0.9 when β=5 are shown in Figs. 7-10.

Fig. 7 Three-dimensional trajectories of PPN and DGGC-F with β=5

Fig. 8 Commanded accelerations of PPN and DGGC-F with β=5

Fig. 9 Three-dimensional LOS rates of PPN and DGGC-F with β=5

Fig. 10 ZEMs of PPN and DGGC-F with β=5

The 3D flight trajectories in this case are shown in Fig. 7 which is similar as Fig. 3. The commanded accelerations in this case are shown in Fig. 8. As η increases, the initial value of |a_m| of DGGC-F decreases. However, when η is big, such as 0.5 and 0.9, the different between the corresponding commanded acceleration curves are not so clear. |a_m| of PPN is also firstly the smallest and then becomes the largest in the end of the guidance process.

The LOS rates in this case are shown in Fig. 9. From the figure, as η increases, the convergence rate of LOS rate decreases, but the differences among the curves are not so obvious. The ZEMs are shown in Fig. 10, where the curves have similar change tendencies as those in Fig. 9.

According to the above simulation results, the influence of η on the performance of the guidance command is not as strong as that of β.From the above two cases of simulations, the performance of the variable structure term, i.e.,  is demonstrated. In the following simulations, β=5 and η=0.1 is adopted.

5.2 Demonstration of estimation performance of ESO

Then, we demonstrate the estimation performance of ESO of DGGC-FE. For ESO, the parameters are selected as α₁=0.2, 0.4, 0.6, 0.8, β₀₁=20, 40, 60, 80, β₀₂=50, 100, 150, 200, d₁=0.01, 0.05, 0.1, 0.2. The results are shown in the following figures.

Figure 11 shows the estimation of the target acceleration a_tθ of ESO with α₁=0.2, 0.4, 0.6, 0.8 when β₀₁=40, β₀₂=100, and d₁=0.1. It can be seen that, as α₁ increases, the estimation error increases, and the curve of estimation becomes smoother. According to the simulation result, α₁ should not be large.

Figure 12 shows the estimation of the target acceleration a_tθ of ESO with β₀₁=20, 40, 60, 80 when α₁=0.2, β₀₂=100, and d₁=0.1. From the figure, when β₀₁=20, there is a small vibration in the beginning of the estimation, and there exists a small over estimation around 4s. As β₀₁ increases, the curve becomes smoother, but the estimation error increases. Therefore, β₀₁ can neither be too small nor too large.

Fig. 11 Estimation of a_tθ of ESO with α₁=0.2, 0.4, 0.6, 0.8 when β₀₁=40, β₀₂=100, and d₁=0.1

Fig. 12 Estimation of a_tθ of ESO with β₀₁=20, 40, 60, 80 when α₁=0.2, β₀₂=100, and d₁=0.1

Fig. 13 Estimation of a_tθ of ESO with β₀₂=50, 100, 150, 200, when α₁=0.2, β₀₁=40, and d₁=0.1

Figure 13 shows the estimation of the target acceleration a_tθ of ESO with β₀₂=50, 100, 150, 200, when α₁=0.2, β₀₁=40, and d₁=0.1. According to the figure, the effect of β₀₂ on the performance of ESO is approximately the adverse effect of β₀₁ on the performance of ESO.

Figure 14 shows the estimation of the target acceleration a_tθ of ESO with d₁=0.01, 0.05, 0.1°, 0.2° when α₁=0.2, β₀₁=40, and β₀₂=100. From the figure we can see that, when d₁=0.01, there exist large vibrations in the beginning of the estimation and around 4s. As d₁ increases, the vibration is attenuated, and the curve becomes smoother; however, the estimation error is also increased. Therefore, d₁ should not be too small.

Fig. 14 Estimation of a_tθ of ESO with d₁=0.01, 0.05, 0.1°, 0.2° when α₁=0.2, β₀₁=40, and β₀₂=100

5.3 Comparison between DGGC-F and DGGC-FE

Finally, we will compare the performance of DGGC-F with that of DGGC-FE in the following figures. For both guidance laws, N = 3, β=5, and η=0.1 are used. For DGGC-F, d=30 is selected, while for DGGC-FE, α₁=0.2, β₀₁=40, β₀₂=100, and d₁=0.1 are adopted. The simulation results are shown in the following figures.

The 3D flight trajectories of the three guidance laws are shown in Fig. 15. It can be seen that the trajectories of DGGC-F and DGGC-FE are much curvier than that of PPN in the beginning of the guidance, which means large commanded accelerations are used in the beginning, as shown in Fig. 16. From Fig. 16, |a_m| of PPN will rise to a very large value in the end of guidance. |a_m| of DGGC-F is initially larger than that of DGGC-FE, and it converges to the constant state rapidly. However, the distribution of |a_m| of DGGC-FE is better than that of DGGC-F.

The LOS rates of three guidance laws are shown in Fig. 17. It can be seen that, ω_s of PPN cannot converge to a stable state, and it will rise to a very large value in the end of the guidance process. The convergence speed of ω_s of DGGC-F is the most rapid one. The LOS rate of DGGC-FE can converge to the constant stable state which is in the boundary layer in finite time, although the convergence speed is not as fast as that of DGGC-F.

Fig. 15 Three-dimensional trajectories of three guidance laws

Fig. 16 Commanded accelerations of three guidance laws

Fig. 17 Three-dimensional LOS rates of three guidance laws

The ZEMs of three guidance laws are shown in Fig. 18. From the figure, the ZEM of DGGC-F converges to zero with the largest speed, while the convergence speed of ZEM of PPN is the slowest one. The convergence speed of ZEM of DGGC-FE is in the middle.

According to the simulation results, both DGGC-F and DGGC-FE can guarantee the finite time convergence of the LOS rate. However, DGGC-FE is better than DGGC-F, since the overload distribution of DGGC-FE is better. For DGGC-F, if a quite large d is adopted, the guidance command may exceed the physical limitation of the missile. While for DGGC-FE, since the target acceleration a_tθ can be estimated with an acceptable precision, the guidance command cannot be too strong, and the physical limitation problem can be avoided.

Fig. 18 ZEMs of three guidance laws

6 Conclusions and future work

1) DGGC is effective in the endoatmospheric interception scenario, and it can be integrated with other guidance approaches to improve the performance of the airborne missiles.

2) The finite time control theory can be integrated with DGGC to improve the controllability of the LOS rate. In this way, a new guidance law DGGC-F is obtained. However, for DGGC-F, the target acceleration or its upper bound must be initial known, which limits the application of this guidance law.

3) ESO is effective in estimating the target acceleration vertical to LOS when it is employed in the guidance command, and the newly proposed DGGC-FE is robust and easy for implementation in practical interception scenarios, while the finite time convergence of the LOS rate can be guaranteed.

4) However, in this paper only the deterministic problem is discussed. In the future, the statistic problem may need to be explored. The impacts on the guidance performance of DGGC-F and DGGC-FE caused by the measurement errors, missile dynamic lags, and other factors may need to be studied further.

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(Edited by DENG Lü-xiang)

Received date: 2015-06-18; Accepted date: 2016-01-21

Corresponding author: MA Yi-wei, PhD candidate; Tel: +86-15974106634; E-mail: mayiweidyx@163.com