Schoolof Astronautics,Northwestern Polytechnical University,Xi’an 710072,China

Design ofa robustguidance law via active disturbance rejection control

Yanbo Yuan?and Ke Zhang

Schoolof Astronautics,Northwestern Polytechnical University,Xi’an 710072,China

Focusing on the three-dimensionalguidance problem in case oftarget maneuvers and response delay of the autopilot,the missile guidance law utilizing active disturbance rejection control (ADRC)is proposed.Based on the nonlinear three-dimensional missile target engagement kinematics,the guidance modelis established.The target acceleration is treated as a disturbance and the dynamics of the autopilot is considered by using a fi rst-order model.A nonlinear continuous robust guidance law is designed by using a cascaded structure ADRC controller.In this method the disturbance is estimated by using the extended state observer (ESO)and compensated during each sampling period.Simulation results show thatthe proposed cascaded loop structure is a viable solution to the guidance law design and has strong robustness with respectto targetmaneuvers and response delay ofthe autopilot.

three-dimensionalguidance,active disturbance rejection control,targetmaneuver,autopilotresponse delay.

1.Introduction

Featuring easy implementation,the proportional navigation(PN)is the most popular and widely used guidance law,butithas its drawbacks.In some realistic interception scenarios,for example,a missile with limited maneuverability pursuing a target with a large maneuver capability, a signi fi cant miss distance may occur using the PN guidance law.Furthermore,the controlsystem inevitably has its lag time which can result in a profound impact on the guidance accuracy[1].To resolve these problems,modern guidance laws are formulated.

During the lastdecade,various practicalproblems have been considered for real implementation.Guidance laws have been extended to the three-dimensional engagement case for highly maneuverable targets[2]and have been applied to missile systems considering the induced drag and the internal dynamics of the control loop[3].New guidance laws using nonlinear control theories such as the Lyapunov function method[4],the optimalcontrol theory [5],the nonlinear H∞controltheory[6-7],and the adaptive sliding-mode control theory[8-11]have also been developed.As to the robustness in missile guidance law design,the guidance law proposed in[12]and[13]took into accountthe lag of the autopilot.Howerver,robust control techniques considering both autopilot lag and target maneuvers are still rare in the literature.

The active disturbance rejection control(ADRC)is a control method which was originally proposed by Han in [14].Itis based on an idea thatin orderto controla process, the user does not need to have an explicit mathematical model of the plant.By using a state observer,both internaland externaldisturbances are estimated and rejected in each sampling period.The close-loop system stability and transient performance analysis of the ADRC-based controller were presented in[15-19].Studies on using ADRC to design fl ight control and guidance systems were performed in[20-26].Specially for the guidance problem of maneuvering targets,it is proven by[27-29]that the target’s acceleration can be estimated by the ADRC controller,which is favorable for improving the robustness of the guidance law.

In this paper,a new three-dimensional guidance law based on the ADRC theory is proposed.Combined with the back-stepping method,a cascaded-loop structure is adopted in the presented guidance law design considering target maneuvers and autopilot dynamics,and then controllers are designed respectively according to the ADRC theory.It is shown that the proposed guidance law has strong robustness with respectto targetmaneuvers and response delay of the autopilot.Moreover,the novel design method featuring cascaded-loop ADRC controllers presented in this paper is a viable solution to the guidance law design and is valuable to engineering applications.

The paper is organized as follows.In Section 2,thethree-dimensional missile-target engagement problem is formulated.A new guidance law design via the ADRC theory is proposed in Section 3.Numericalsimulation results are shown in Section 4 and conclusions are reported in Section 5.

2.Three-dimensionalguidance model

Consider a three-dimensional air-to-air engagement as shown in Fig.1,where a missile is attempting to intercept a moving target.

Fig.1 Missile target engagement geometry

The missile and the target are both assumed as point masses.The dynamic equations[5]are given:

where r denotes the relative distance between the missile and the target,θandψare the verticaland horizontalcomponents of the line of sight(LOS)angle,(atr,atθ,atψ) and(amr,amθ,amψ)are the components of target and missile acceleration on the three axes(er,eθ,eψ)of the spherical coordinate system,respectively.Note from(1)-(3)thatthe targetacceleration can be treated as uncertainties of the system.To achieve the goal of guidance,it is required to design amθand amψto make the LOS rate˙θ and˙ψapproach to zero according to the parallelapproaching method.

Considering the fi rst-orderdynamic characteristic ofthe missile autopilot,we get

Thus,the guidance problem considering the response delay of the autopilotcan be described as:design the missile accelerationforsystems(1)-(5)to make the LOS rateθ˙andψ˙approach to zero.

3.Cascaded guidance law design via ADRC theory

Generally,the guidance law can be designed in two planes separately:the longitudinal plane and the lateral plane. Taking the longitudinal plane as an example,the detailed design process is presented.The guidance law in the lateral plane can be derived similarly.

To design the guidance law in the longitudinal plane, study the system consisting of(2)and(4).Letus introduce the following de fi nitions:

Then,we can geta cascaded system as follows:

According to the methodology of back-stepping,the system can be divided into two cascaded loops which are designed separately.For system(11)with respect to state variable x1,regarding x2as a control variable,design an outer loop controller ADRC1 to guarantee that x1converges at zero under the existence of uncertaintyω1(t) caused by target unknown maneuvers,and its output is pseudo controlvariable x2c.For system(12)with respect to state variable x2,regarding u as a control variable,design an inner loop controller ADRC2 to guarantee that x2can track x2cin realtime,thus the dynamic response delay of the autopilot can be compensated to improve the guidance accuracy.The cascaded loop design concept is shown in Fig.2.

Fig.2 Illustration of cascaded loop concept

With regard to the ADRC controller,a typical ADRC controller consists of three parts:a nonlinear tracking differentiator(TD),an extended state observer(ESO),and anonlinearstate errorfeedback(NLSEF).The TDis used to arrange the idealtransientprocess of the system.The ESO could estimate all the disturbances from the system output,and then the ADRC compensates for the disturbance according to estimated values.The NLSEF is used to get the control input of the system[10].Considering systems (11)-(13),the structure of the cascaded ADRC system is shown in Fig.3.Itis clear thatthe ADRC approach makes an effortto compensate for the unknown dynamics and externaldisturbances in the time domain.

Fig.3 Illustration of cascaded ADRC system

3.1 Design of ADRC1

3.1.1 Arranging the transientprocess by TD

The purpose of the controller ADRC1 is to guarantee that the fi rst-order system about x1converges at zero and the differential information of the expected signal is not required,so we do not need to use the common formation of a TD.The following nonlinear function is employed to arrange the transientprocess,which can generate the tracking signalof the transientprocess:

where h is the sampling step,r0and h0are adjustable parameters.The magnitude of r0determines the tracking speed and h0determines the tracking accuracy.The function of f al(·,·,·)is given by

where 0＜α≤1.

3.1.2 Estimating state and totaldisturbance by ESO

Suppose thatthe relative distance r,relative speed˙r,LOS anglesθandψare all measurable parameters,and regard the uncertaintyω1caused by targetmaneuveracceleration as a disturbance.A three-order ESO must be designed to estimate the values ofθ,˙θ,andω1.

Speci fi cally,add an additional stateand regard x2as an external input variable,we get a system as follows:

Then,as long as˙ω1(t)is bounded,a three-order ESO can be constructed as[30]

whereβ01,β02,β03andδare alladjustable parameters.

When adopting the Euler integral method,the discrete realization of the ESO in(17)is given as

The state variables of the ESO,z1and z2,can track the variablesθand˙θ;at the same time,the estimated value of the disturbanceω1can be given by z3.Thus,the targetmaneuver acceleration in the longitudinalplane can be estimated as

3.1.3 Generating controlvariable and disturbance compensation by NLSEF

As is stated above,the transient process of the expected signalis obtained by v1,and the state variables and disturbance of the controlled object are estimated by the ESO. To control x1,the following error is introduced:

The NLSEF is achieved by the following nonlinearfunction:

whereβ1＞0,δ1＞0,and they are adjustable parameters. Compared with the linear feedback control law,this nonlinear feedback controllaw has better performance on disturbance rejection and controlsaturation avoidance[10].

After adding the disturbance compensation term,the practicalcontrollaw of the outer loop ADRC1 is given as

3.2 Design of ADRC2

For the inner loop dynamic system with respect to atθ, the response time of the autopilotτis a known constant, and the system(12)contains no other uncertainties.Consequently,the TD and ESO of ADRC2 can be eliminated to simplify the design process.To design ADRC2,we only need to design an NLSEF controllerwhich tracks the command signal x2cof the outer loop.

De fi ne the following error:

Similarly,the NLSEF of ADRC2 can be designed as follows:

whereβ2＞0,a＞0,δ2＞0,and they are adjustable parameters.

The practical control law of the inner loop ADRC2 is given as

Hence,the guidance law in the longitudinalplane can be given by(14),(18),(22)and(25).

For the guidance law in the lateral plane,we can use the system modelof(3)and(5),and the guidance law can be derived similar to the cascaded-loop ADRC controller design process presented above.

4.Numericalsimulation

The initialconditions of the simulation are setas follows:

(i)The initial speed of the missile is 800 m/s,and the speed inclination angles are 15?and 5?in the longitudinal plane and the lateralplane,respectively.

(ii)The initial speed of the target is 400 m/s,and the speed inclination angles are-10?and 5?in the longitudinalplane and the lateralplane,respectively.

(iii)The initial position of the missile is[0,0,0]T,and the initialposition of the targetis[5 000,4 000,200]T.

(iv)The accelerations of the target in the longitudinal plane and the lateralplane are+2g and-2g,respectively.

(v)The maximum available acceleration of the missile is 20g,and the time constantof the autopilotisτ=0.2 s.

(vi)The ADRC parameters are set as follows:h= 0.01 s,r0=0.1,h0=0.01,β01=100,β02=300, β03=1 000,δ=0.01,β1=0.5,δ1=0.1,β2=20, a=0.4,δ2=0.1.

According to the common scope of the LOS rate during terminalguidance,the initialvalue ofthe transientprocess is setas v1(0)=0.01.The ADRC parameters in the lateral plane are completely the same as those in the longitudinal plane.

Simulation results using the cascaded ADRC guidance law are shown in Figs.4-7.

Fig.4 Three-dimensional trajectory

Fig.5 LOS rate

Fig.6 Guidance command

Fig.7 Target maneuver acceleration and its estimated value

Fig.4 shows the three-dimensional trajectories of the missile and the targetfrom which we can see that the trajectory of the missile is relatively smooth.The terminal miss distance is 0.172 m,which demonstrates thatthe missile can intercept the maneuvering target with high accuracy using the proposed guidance law.Fig.5 indicates that the LOS rate is controlled well.During the guidance process,it converges to zero with a fast speed which guarantees the guidance purpose.At the end of the trajectory, due to the extremely small relative distance,the LOS rate turns to diverge.Fig.6 shows the guidance command.At the beginning,a small period of saturation can be found in the fi gure.This is because the initial LOS angle is relatively large.Fig.7 shows the estimated values ofthe target maneuver acceleration using ESO in real time.The estimated values deviate from the realones before 4 s,butafter 4 s,the estimated values begin to converge,which demonstrates thatthe effectof targetmaneuvers on the guidance accuracy can be compensated by the designed ESO.

In addition,simulations under different conditions are also conducted.Taking the target maneuver acceleration as a disturbance and adding some perturbations,the simulation results are shown in Table 1.

Table 1 Simulation results under different perturbations

Table 1 demonstrates that under different target maneuveraccelerations and differentdisturbance perturbation conditions,the proposed guidance law yields reasonably satisfactory guidance performances.

5.Conclusions

A systematic approach to design a new guidance law is proposed using active disturbance rejection control.The guidance command is derived based on the nonlinearthreedimensional engagement kinematics.The target acceleration is treated as the disturbance and the dynamics of the autopilot is considered using a fi rst-order model.Supposing the relative distance,closing velocity and LOS angle are all known information,a nonlinear continuous robust guidance law is designed using a cascaded structure ADRC controller.The proposed cascaded control loop structure effectively rejects the externaldisturbance through the incorporation of ADRC,which compensates for the disturbance based on the estimation of ESO.Simulation results demonstrate thatthe proposed cascaded loop structure is a viable solution to the guidance law design and has strong robustness with respect to target maneuvers and response delays of the autopilot.

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Biographies

Yanbo Yuan was born in 1981.He is a Ph.D. candidate in armament science and technology at Northwestern Polytechnical University,Xi’an, China.He received his B.S.degree in aerospace engineering from National University of Defense Technology in 2004,and M.S.degree in spacecraftdesign from Harbin Institute of Technology in 2007.His current research interests focus on missile guidance and controlsystem design.

E-mail:runble@163.com

Ke Zhang was born in 1968.He is a professor in the Schoolof Astronautics,Northwestern Polytechnical University,Xi’an,China.He received his Ph.D.degree for the Schoolof Astronautics at Northwestern Polytechnical University.He is currently a member of“remote control,telemetry and remote sensing”professional committee of the Chinese Institute of Electronics.He has authored three college textbooks and academic monographs.His current research interests lie in the fi eld of guidance law design,imaging guidance technology,visualsimulation technology,and advanced control theory and applications.He has contributed more than 70 journalarticles to professionaljournals.

E-mail:zhangke@mail.nwpu.edu.cn

10.1109/JSEE.2015.00041

Manuscriptreceived April01,2014.

*Corresponding author.

This work was supported by the Aviation Science Foundation (2013ZC12004).