Roll-pitch-yaw autopilot design for nonlinear time-varying missile using partial state observer based global fast terminal sliding mode control

Ahmed Awad,Wang Haoping

Sino-French International Joint Laboratory of Automatic Control and Signal Processing (LaFCAS),School of Automation,Nanjing University of Scienceamp;Technology,Nanjing 210094,China

Roll-pitch-yaw autopilot design for nonlinear time-varying missile using partial state observer based global fast terminal sliding mode control

Ahmed Awad,Wang Haoping*

Sino-French International Joint Laboratory of Automatic Control and Signal Processing (LaFCAS),School of Automation,Nanjing University of Scienceamp;Technology,Nanjing 210094,China

The acceleration autopilot design for skid-to-turn (STT) missile faces a great challenge owing to coupling effect among planes,variation of missile velocity and its parameters,in existence of a complete state vector,and nonlinear aerodynamics.Moreover,the autopilot should be designed for the entire flight envelope where fast variations exist.In this paper,a design of integrated roll-pitch-yaw autopilot based on global fast terminal sliding mode control (GFTSMC) with a partial state nonlinear observer (PSNLO) for STT nonlinear time-varying missile model,is employed to address these issues.GFTSMC with a novel sliding surface is proposed to nullify the integral error and the singularity problem without application of the sign function.The proposed autopilot consisting of two-loop structure,controls STT maneuver and stabilizes the rolling with a PSNLO in order to estimate the immeasurable states as an output while its inputs are missile measurable states and control signals.The missile model considers the velocity variation,gravity effect and parameters' variation.Furthermore,the environmental conditions' dynamics are modeled.PSNLO stability and the closed loop system stability are studied.Finally,numerical simulation is established to evaluate the proposed autopilot performance and to compare it with existing approaches in the literature.

1.Introduction

The acceleration autopilot design for skid-to-turn missile is still considered as one of the most attractive topics for control engineers due to its enormous nonlinear dynamics,the coupling effect between channels,and its rapid parameters' variation.1The most significant variation of missile parameters is its velocity which changes rapidly as a result of subjecting the missile to a sudden acceleration during boosting phase and deceleration during gliding phase due to aerodynamic drag.2

Researches in this field have been started since 1944.One of the commonly used autopilots was the three-loop autopilot topology.3The conventional and linear quadratic regulators based on the linearization of model dynamics around fixed operating points were used.This technique is so-called''gain schedulingquot;.In Ref.4,a classical gain-scheduling design was introduced.In the 1990s,extensions of these techniques had brought many developments,like guaranteed stability margins and performance levels.5,6The authors in Ref.7presented a linear quadratic Gaussian with loop transfer recovery technique to design gain scheduling autopilot.A gain scheduling based autopilot in the presence of hidden coupling terms is illustrated in Ref.8The combined optimal/classical approach was applied to design the optimal controller in Ref.9as well.Also,robustness issues were introduced with suitable extensions of H∞techniques.10,11Clearly,the gain scheduling approach shows a good performance during the entire envelope,but the global stability is guaranteed only in the case of slow variation of both the states and the missile parameters.

The development of linear parameter-varying (LPV) and quasi-LPV approaches in the last two decades had pushed the researchers towards a new systematic and strict methodology.For example,an acceleration autopilot design using the LPV reference model was presented for portable missile.12Unfortunately,the disadvantage of these approaches,especially for quasi-LPV autopilot,is the increment of conservative level.13Also,most of these approaches except quasi-LPV approach,were still based on the linearization of dynamics around operating points.Other drawbacks of LPV were the difficulty in parameter variation recognition and the demand of an additional filter for parameter estimation as well.14

The requirements of high maneuverability and the development of nonlinear control methods pushed the research towards new control design approaches that consider essential nonlinear dynamics.This led to the first generation of nonlinear autopilots which were based on both the inversion of dynamics15and the feedback linearization techniques.16New approaches were introduced in the last decades based on recent control techniques,such as Lyapunov stabilization techniques17,sliding mode control(SMC)18,19,adaptive SMC20,adaptive block dynamic surface control21,l1adaptive control22,simple adaptive control algorithm23,adaptivefussy sliding mode control24,robust hybrid control25,immersion and invariance control26,backstepping control27,state-dependent Riccati equation (SDRE) approach28,adaptive SDRE with neural networks29,and fuzzy control.30The approaches introduced in the nonlinear and/or adaptive context failed when massive unstructured dynamics existed.Moreover,the strict requirements needed for the response speed cannot often be achieved due to adaptation laws.Therefore,robust nonlinear approaches based on the geometric theory as in Ref.31and the extensive use of Lyapunov direct criterion as in Ref.32were presented,demonstrating good performance at a high angle of attack.It should be mentioned that the approaches based on these methods have been only applied to simple single-input/single-output cases,disregarding the coupling and nonlinearities occurring between pitch and yaw planes.

A few works paid attention to presenting the integrated autopilot to overcome the coupling effect between channels.For example,a robust backstepping approach has been applied to multi-input/multi-output (MIMO) model to achieve both bank-to-turn and skid-to-turn(STT)maneuvers1,a threeaxis autopilot design using the classical three-loop autopilot approach33,and an acceleration autopilot based on a linear robust control scheme was presented to control roll,pitch,and yaw channels in an integrated way.34These works showed a good performance while considering the missile velocity constant,and the controller design is still based on linearization except in Ref.1.Ref.35presented a sliding mode based integrated attitude control scheme considering velocity change.It showed good results,but the acceleration control was not presented.In Ref.2,a sliding mode based roll-pitch-yaw integrated attitude and acceleration autopilot for a time-varying velocity STT missile was proposed.It showed a good performance,but a complete state vector is essential and the velocity variation was considered as a function of time.Likewise,gravity effect,missile parameters' variation,chattering phenomenon,and environmental dynamics were neglected.

Thus,to achieve a good tracking performance of the integrated acceleration autopilot in presence of the above referred neglected factors without seeking for chattering elimination or complete state vector feedback,an integrated roll-pitch-yaw autopilot using a partial state nonlinear observer (PSNLO) based global fast terminal sliding mode control (GFTSMC) approach is proposed for a skid-to-turn nonlinear timevarying (STTNTV) missile model.The missile model has taken into account the coupling effect,gravity effect,missile parameters' variation,environmental conditions' dynamics,and nonlinear aerodynamics.In a similar manner,the missile velocity and its height have been considered as a function of its states.GFTSMC with a suggested sliding surface is provided to avoid the chattering phenomena of SMC,the singularity problem of normal GFTSMC,and the demand for a relation between the second derivative of system states and its inputs.These sliding mode surfaces are suggested and used in the integrated twoloop autopilot structure to nullify the integral error.PSNLO is presented to estimate the immeasurable states(angle of attack α and side slip angle β)used to feedback the autopilot.The stability of closed loop system including PSNLO stability is discussed.

The remaining paper is organized as follows:In Section 2,system description and modeling are provided.In Section 3,integrated roll-pitch-yaw autopilot design,PSNLO design and its stability analysis,and the closed loop stability analysis are presented.The integrated autopilot design includes outerloop controller design based on GFTSMC with acceleration dynamics modeling and inner-loop controller design based on GFTSMC.In Section 4,numerical simulations are presented,and Section 5 is devoted to summary and concluding remarks.

2.System description and modeling

The STTNTV missile model is aerodynamically controlled via canard fins,and it has an axis-symmetric and cruciform shape.Thus,the next general assumptions can be considered:

(1)The moments of inertia Iyy(t)and Izz(t)are identical and products of inertia moments can be discarded.

(2)For short-range missiles,the Earth has been considered flat and non-rotating.

2.1.Mathe matical modeling

The missile states,and its dynamic parameters measured in different coordinates;the orientation of the commonly used coordinate systems,and the related angles between them are depicted in Fig.1.Missile motion in space is described by means of the following differential equations.Solution of these equations gives missile linear velocity components(u,v,w)in(Xb,Yb,Zb)axes of body coordinate system (BCS),respectively,missile angular rates(p,q,r)around(Xb,Yb,Zb)axes,respectively,the Euler angles(φ,θ),(α,β),and the missile height h in Ziaxis of inertial coordinate system (ICS).As mentioned in Refs.36–38,the differential equations,the missile velocity Vm,dynamic pressure Q(h,Vm),acceleration components(axb,ayb,azb)applied on the missile in(Xb,Yb,Zb),respectively,and force components(Fxb,Fyb,Fzb)applied on the missile in(Xb,Yb,Zb),respectively,are developed and expressed as follows:

with

where s is the aerodynamic reference area,ρ the function of air density,l the missile characteristic length,g the gravity acceleration in Ziaxis of ICS,Tx(t)the function of thrust component in Xbaxis,m(t)the function of missile mass,Ixx(t)the moments of inertia function in Xbaxis,and Cxthe drag force derivative;Clα,Clβ,Clp,Clδr,Cmα,Cmq,Cmδp,Cnβ,Cnr,and Cnδyare stability derivatives of the rolling,pitching and yawing moments;Cα,Cδp,Cβ,and Cδyare force derivatives of the pitching and yawing forces;δr,δp,and δyare the fin angular deflections in roll,pitch and yaw planes,respectively;t is the missile flight time.

2.2.Aerodynamic coefficients modeling

Obviously,accurate estimation of the aerodynamic coefficients is the corner stone of guidance and control system design.Furthermore,the aerodynamic coefficients evaluation via wind tunnel tests is essential.The coefficients of aerodynamic forces and moments obtained from aerodynamic coefficients' database based on experimental data,are functions of Mach number Ma,α,β,δr,δp,and δy.

2.3.Environmental conditions dynamics modeling

The Lapse rate mathematical model for the troposphere has been used to represent the dynamics of air density and the speed of sound as a function of h as follows:

where ρ0is the air density at mean sea level,L the Lapse rate,T0the absolute temperature at mean sea level,Vs(h)the speed function of sound at altitude h,and R the specific heat ratio.

3.Autopilot design

In this section,the design process of roll-pitch-yaw autopilot based on GFTSMC,the design process of PSNLO and its stability analysis,and the discussion of closed loop system stability are presented as follows.

3.1.Roll-pitch-yaw autopilot design

As a result of autopilot application to missile model with rapid parameters' variation,a GFTSMC based autopilot is supposed as a robust autopilot.The normal GFTSMC makes the convergence velocity to be finite without chattering phenomenon.39,40But it possess a singularity problem,and it needs a relation between the second order derivative of system states and its inputs.To meet the nature of STTNLTV missile model and the structure of the proposed autopilot,a GFTSMC with novel sliding surface is presented.It has the following advantages related to the other sliding surfaces in reference:

(1)Avoiding the usage of sign function which generates the chattering phenomena.

(2)Avoiding the singularity problem of normal GFTSMC.

(3)Avoiding the system output differentiation which is undesirable in the real system due to the noise existence in the system output measurements.

(4)Applicable for systems' model which has a relation between the first order derivative of system states and its input.

(5)In general,the convergent characteristics of GFTSMC are superior to those of the normal SMC.39

Since the missile performs STT maneuver,its outputs to be controlled are(φ,azb,and ayb).A scheme of the proposed autopilot based on GFTSMC with a PSNLO is shown in Fig.2,IMU is the inertial measuring unit.The proposed autopilot consisting of a two-loop structure,controls pitch and yaw accelerations,and stabilizes the roll angle simultaneously.The proposed outer-loop controller generates missile roll,pitch,and yaw angular rate commands(pc,qc,rc)corresponding to roll angle command(φc)and pitch and yaw acceleration commands(azc,ayc).The inner-loop controller generates(δr,δp,δy)corresponding to(pc,qc,rc).

Remark 1.Merit of the two loop autopilot structure is that sum of the relative degree of the states to be controlled and the relative degree of the inner-loop coincides,and therefore,unstable zero-dynamics is not observed as far as the inner-loop is stabilized.2

3.1.1.Outer-loop controller design based on GFTSMC

The derivative of the pitch acceleration can be described as

Substituting Eq.(7)into Eq.(15),we get

where

Similarly,yaw acceleration derivative is obtained as follows:

Substituting Eq.(8)into Eq.(17),we get

where

Re-arranging the equations for the derivative of φ,azb,and aybinto the affine matrix form,we get

wherefa(.)∈ R3X1is smooth function in terms of α,β,x,ur,and t;ga(.) ∈ R3X3is smooth function in terms of α,x,and t;is the measured state vector;

Fig.2 Block diagram of integrated roll-pitch-yaw autopilot based on GFTSMC with PSNLO.

ua∈R3X1is the angular rate commands vector;uris the input vector.Note that the inverse of ga(.)always exists.

The proposed sliding surfaces sithat consider the integral tracking error as a state,and thereaching laws for the outer-loop controller based on GFTSMC,are defined as follows:

where λi,σi,Kpi,and Ksiare positive real designed values;qi0,pi0,qi,and piare positive odd real designed values(qi0lt;pi0,and qilt;pi).For more details,see Ref.39,40.

Differentiating Eq.(22)with respect to time and applying there aching law to Eq.(23),the outer-loop control law is obtained as follows:

where

Remark 2.The control law derived from the normal GFTSMC has the partwhere q and p are positive odd integers,which satisfy qlt;p.This part may cause a singularity problem if x2≠0 when x1=0.41From Eqs.(24)and(25),the partazb,ayb will avoid singularity problem because if ei≠0, the n0,i=φ,azb,ayb.

3.1.2.Inner-loop controller design based on GFTSMC

Re-arranging Eqs.(4)–(6)for derivatives of p,q,and r,respectively into the affine matrix form,we get

where fr(.)∈R3X1,gr(.)∈R3X3are smooth functions in terms of α,β,x,and t;ur∈ R3X1is the input vector.The inverse of gr(.)always exists.

Similarly,the sliding surfaces sjand the reaching laws of the inner-loop controller are defined as

Differentiating Eq.(28)with respect to time and applying the reaching law,the inner-loop control law is obtained as follows:

where

3.2.PSNLO design and its stability analysis

Unfortunately,α and β are unavailable to be measured in the real missile system.Consequently,the partial state observer is necessary.Devaud et al.36presented a design of simple quadratic observer for α and β under some simplifying assumptions.In this paper,an efficient PSNLO for α and β is presented.The proposed PSNLO model equation is expressed as follows:

with

In order to discuss stability of the proposed PSNLO,the candidate Lyapunov function is chosen as follows:

For simplicity in stability analysis,the next assumption can be used because of the smallness of α and β,especially in the case of canard missile control.38

Based on the above,the Lyapunov function derivative can be presented as follows:

Substituting Eqs.(2)and(3)into Eq.(35),we get

Based on the above,we get

Since Vm,m(t),Q,Cα,and Cβare bounded positive functions for canard control missile, Bα,Bβ,Vmgt;0∀tgt;0,axbgt;0∀0lt;tgt;Tswhere Tsis the end time of the sustaining phase.

Then Eq.(37)shows that the proper selection of kαand kβgt;0 guarantee˙Vo(eα,eβ)lt;0,i.e.,the presented PSNLO is asymptotically stable during all the flight envelope.

3.3.Closed loop stability analysis

In order to discuss the closed loop stability of integrated rollpitch-yaw autopilot based on GFTSMC with PSNLO,designed for STTNTV missile model,the candidate Lyapunov function is defined as

Because of the smallness of α,cos α ≈ 1,i.e.,ga(α,x,t)≈ga(x,t).

Then,the outer and inner loops control law of integrated roll-pitch-yaw autopilot based on GFTSMC with PSNLO i.e.,Eqs.(24)and(29)becomes the following:

Differentiating Eq.(22)with respect to time and applying the outer-loop control law in Eq.(40),we get

Similarly,differentiating Eq.(28)with respect to time and applying the inner-loop control law in Eq.(41),we get

with

Substituting Eqs.(42)–(44)into Eq.(39),we get

Since eαand eβare asymptotically stable and the initial values of α and β are zero,eαand eβare very small and bounded.And fa(.)and fr(.)are smooth functions in terms of α,β,x,urand t.Then di,i=p,q,r,φ,azb,ayb are small and bounded because the y are based on eαand eβvalues where di=0 if eαand eβare zero,i.e.,

Based on the above,since(p+q)is even number,should be satisfied,i.e.,

Remark 4.As mentioned above,eα≈ 0,and eβ≈ 0,the n Di≈ 0,i=p,q,r,φ,azb,ayb,i.e.,the sufficient large value of Ksican satisfy the above condition in Eq.(47).

From Laypunov stability theorem,the appropriate choice of kα,kβ,Kpi,Ksi,i=p,q,r,φ,azb,ayb can guarantee asymptotic convergence of PSNLO estimation error and the sliding surfaces siin case of the closed loop system.Since the only way to nullify siis to enforce ei=0,tracking is carried out,and eiis asymptotically stable.Moreover,it is assumed that the missile internal dynamics are stable and bounded,i.e.,the closed loop system is asymptotically stable.

4.Numerical simulation

To verify the proposed autopilot performance and to compare it with the usual sliding mode control (USMC) in Ref.2and the autopilot design in Ref.33,numerical simulation is carried out in a MATLAB®-Simulink©environment with fixed step time 0.02 s(applicable in the real system)for the whole flight time(8 s)considering all the missile parameters' variations in evaluating the proposed GFTSMC based autopilot with PSNLO.USMC based autopilot considered the sliding surface si=ei,andThe presented autopilot in Ref.33is based on classical three loop(CTL)technique with a little modification.The real data of man portable missile type is chosen as a severe case because the missile should be launched in a low-speed,resulting in a dramatic parameters' variation.Never the less,the fin deflection angles are limited due to hardware constraints.12

4.1.Missile dynamical parameters

The parameters value describing the airframe and the environmental conditions are listed in Table 1.In particular,our underlying missile is aerodynamically controlled using canard fins,and contains a two stage rocket motor with boosting and sustaining phases where end time of the boosting phase Tb=1.77 s,and end time of the sustaining phase Ts=7.77 s.During the boosting phase,missile acceleration is extremely high and its velocity increases from 27.4 m/s up to 420 m/s during 1.77 s as shown in Fig.3.Obviously,excessive dynamics in missile velocity produces a great challenge to the autopilot.The aerodynamic coefficients extracted from a database at each time step have a nonlinear dependence on both Ma and the incidence angles.The moments of inertia are approximated as a time function as long as the missile mass is always known at each instant during its flight.For more details,see Ref.42.

In order to reflect the mechanical system physical restrictions,upper and lower limits of the actuator deflection were constrained as

Initial values of the differential equations are chosen as real initial values of the underlying missile as follows:

Table 1 Missile and environmental conditions parameters.

Fig.3 Missile velocity profile.

Moreover,to check robustness of the proposed autopilot,the white noise values are added to all of the measured feedback signals in the numerical simulation.

4.2.Autopilot design parameters

Designed parameters of the proposed roll-pitch-yaw autopilot based on GFTSMC with PSNLO are chosen as follows:

4.3.Simulation results

To investigate the presented autopilot performance against the dynamic acceleration commands with roll stabilization during the entire flight time,the reference roll angle and the reference yaw and pitch acceleration commands are considered as

Fig.4 profile of roll angle,yaw acceleration and pitch acceleration.

Fig.5 profile of angle of attack and side slip angle.

Fig.6 Estimation errors of angle of attack and side slip angle profile.

The comparison of the tracking performance among the presented GFTSMC based autopilot with PSNLO,USMC based autopilot,and CTL based autopilot in roll,pitch and yaw planes is depicted in Fig.4.Fig.4 shows a good tracking performance of the suggested autopilot,and a robustness against the noise in all channels during the entire flight time.Clearly,only a little accepted error in φctracking performance occurs.On the contrary,it shows that USMC based autopilot and CTL based autopilot fail.The angles α and β and their estimations are shown in Fig.5.Fig.5 shows a good performance of PSNLO during the entire flight time,and the smallness of α and β which minimizes the aerodynamic non-linearity and relaxes the coupling between planes of symmetry.The estimation errors of α and β are shown in Fig.6 that shows a very small estimation error during the whole flight time.The related roll,yaw and pitch fin deflections are shown in Fig.7.Fig.7 shows a small and smooth fin deflection which relaxes the actuators.It is evident that overall simulation results of the proposed autopilot have shown a good tracking performance in presence of all of the coupling effect,missile parameters' variation,environmental conditions' dynamics,dynamic acceleration commands,rapid velocity variation,and noises.On the other hand,no demand for the inapplicable measurements of states in the real missile system.

Fig.7 profile of roll,yaw and pitch fin deflection.

5.Conclusions

In this paper,a global fast terminal sliding mode control with partial state nonlinear observer based integrated roll-pitch-yaw autopilot design has been presented for skid-to-turn nonlinear time-varying missile to achieve a good performance in presence of the commonly disregarded factors such as the coupling effect,unavailability of the full state vector,chattering problem,and more missile model dynamics during the entire flight time.The partial state nonlinear observer is designed to estimate the immeasurable states.The stability analysis is performed.The numerical simulation is conducted to verify performance of the proposed autopilot design,and to compare it with an existing approach in literature.The results show robustness and good tracking performance with relative smooth fin deflections in all missile channels in the presence of the aforesaid considerations.Also,it achieves a small angle of attack and side slip angle which minimize the aerodynamics nonlinearity and the coupling effect.Conversely,the compared techniques fail.The future work is suggested to consider a lumped disturbances in the missile model and to design a disturbance observer which can be added to the proposed autopilot to compensate the lumped disturbance effects.Moreover,the proposed sliding surface can be employed in different control applications.

Acknowledgements

This work is co-supported by the National Natural Science Foundation of China(No.61304077),International Scienceamp; Technology Cooperation Program of China (No.2015DFA01710),the Natural Science Foundation of Jiangsu Province of China(No.BK20130765),the Chinese Ministry of Education Project of Humanities and Social Sciences(No.13YJCZH171),the 11th Jiangsu Province Six Talent Peaks of High Level Talents Project of China (No.2014_ZBZZ_005),and the Jiangsu Province Project Blue:Young Academic Leaders Project.

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Awad Ahmedis a Ph.D.candidate at Automation School,Nanjing University of Science and Technology,China.He received his B.S.in Cairo,Egypt in 2002;he was a technical support engineer until 2004,and senior researcher until 2006.In 2010,he received the M.S.degree in electric engineering,missile infrared seeker and flight simulation,from Military Technical College,Cairo,Egypt.His research interests are embedded systems and interfaces,real time systems,hardwarein-loop simulation,visual tracking,and he mostly concerns missile guidance and control.

Wang Haopingis a professor and Ph.D.supervisor at Automation School,Nanjing University of Science and Technology,China.He received his Ph.D.degree in automatic control from Lille University of Science and Technology(LUST),France in 2008.He was research fellows at Modeling Information and System Laboratory of Picardie University and at Automatics,Computer Engineering and Signal Processing Laboratory of LUST,France.His research interests include the theory and applications of hybrid systems,visual servo control,friction modeling and compensation,modeling and control of diesel engines,biotechnological processes and wind turbine systems.

9 November 2015;revised 18 January 2016;accepted 22 February 2016

Available online 26 August 2016

Flight control system;

Global fast terminal sliding mode control;

Integrated autopilot;

Nonlinear state observer;

Skid-to-turn missile

©2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 25 84303971.

E-mail address:hp.wang@njust.edu.cn(H.Wang).

Peer review under responsibility of Editorial Committee of CJA.