Observer-based adaptive sliding mode backstepping output-feedback DSC for spin-stabilized canard-controlled projectiles

Yunchun SHEN,Jinqio YU,*,Gunchen LUO,Xiolin AI,Zhenyue JIA,Fngzheng CHEN

aSchool of Aerospace Engineering,Beijing Institute of Technology,Beijing 100081,China

bSpace Star Technology Co.,Ltd,Beijing 100029,China

Observer-based adaptive sliding mode backstepping output-feedback DSC for spin-stabilized canard-controlled projectiles

Yuanchuan SHENa,Jianqiao YUa,*,Guanchen LUOb,Xiaolin AIa,Zhenyue JIAa,Fangzheng CHENa

aSchool of Aerospace Engineering,Beijing Institute of Technology,Beijing 100081,China

bSpace Star Technology Co.,Ltd,Beijing 100029,China

Available online 16 February 2017

*Corresponding author.

E-mail address:jianqiao@bit.edu.cn(J.YU).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.01.004

1000-9361©2017 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

This article presents a complete nonlinear controller design for a class of spin-stabilized canard-controlled projectiles.Uniformly ultimate boundedness and tracking are achieved,exploiting a heavily coupled,bounded uncertain and highly nonlinear model of longitudinal and lateral dynamics.In order to estimate unmeasurable states,an observer is proposed for an augmented multiple-input-multiple-output(MIMO)nonlinear system with an adaptive sliding mode term against the disturbances.Under the frame of a backstepping design,an adaptive sliding mode output-feedback dynamic surface control(DSC)approach is derived recursively by virtue of the estimated states.The DSC technique is adopted to overcome the problem of‘explosion of complexity”and relieve the stress of the guidance loop.It is proven that all signals of the MIMO closed-loop system,including the observer and controller,are uniformly ultimately bounded,and the tracking errors converge to an arbitrarily small neighborhood of the origin.Simulation results for the observer and controller are provided to illustrate the feasibility and effectiveness of the proposed approach.

©2017 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Backstepping;

Dynamic surface control technique;

Nonlinear systems;

Observers;

Sliding mode control;

Spin-stabilized canardcontrolled projectiles

1.Introduction

Precision guidance for conventional direct f i re munitions is a relatively new and rapidly evolving research field due to such features as its high shell delivery accuracy and low expenditure rate.1,2Many feasibility studies and designs done especially for spin-stabilized projectiles have been proposed and adopted during the last several decades and resulted in advanced applications like the U.S.XM1156 Precision Guidance Kit3and roll-decoupled course correction fuze.4Guided munitions require these devices to overcome the traditionally poor performance of unguided projectiles with respect to the accuracy.

Many researchers have focused on employing similar solutions to those used in missile systems for other guided projectiles.But,precise guidance of spin-stabilized projectiles still faces huge challenges.The coupling between the longitudinal and lateral dynamics in the control channels due to the Magnus effect,gyroscopic effect and servomechanism delay must be taken into account.Accordingly,an autopilot design based on each axis separately is not suitable for spin-stabilized projectiles.Aerodynamic coefficients are relatively nonlinear and uncertain with respect to flight states.Thus,traditional methods based on linearization of dynamic models around fixed operating points are also not applicable to such complex models.Control system design is therefore an especially important field for spin-stabilized guided projectiles since it concerns high nonlinearity,strong coupling and rapidly varying parameters.

Recently,quasi-linear parameter-varying (Quasi-LPV)models have been used to approximate the nonlinearity and coupling of spin-stabilized projectiles,and they have proven their value for inclusion in control modules.1,5Unfortunately,local controllers are still designed for several operating points instead of LPV control.Nevertheless,the advanced results seen from Quasi-LPV use show that the development of nonlinear control theories have pushed new control techniques that deal directly with nonlinearity and coupling.6

The principles of nonlinear control design proposed thus far depend fundamentally on the feedback linearization technique.Their solutions require exact knowledge of a nonlinear system including model structure and parameters to be obtained.However,the nonlinearity and uncertainty of spinstabilized projectiles leads to a complicated mathematical plant model with perturbations or disturbances for control design,which are the main obstructions in the application of nonlinear control.In order to dispose of the nonlinear systems with unstructured uncertainties,numerous control methods have been proposed such as sliding mode control,predictive control,fault tolerant control and fuzzy control.7–10One example,a data-driven supplementary control approach with adaptive learning capability based on action-dependent heuristic dynamic programming(ADHDP),is designed for airbreathing hypersonic vehicle tracking.11In Ref.12,transient properties of non-singular terminal sliding mode control are first investigated from the mathematical and theoretical standpoints.Ref.13 systematically discusses some significant nonlinear control problems.To overcome the limitation that uncertainties should satisfy matching conditions,new nonlinear control methods have been proposed by using backstepping.14–16As demonstrated in Refs.17–19,robust nonlinear methods based on the above techniques have been successfully applied to the design of aircraft controllers and achieved acceptable performance.

It should be pointed out that most of the above applications only deal with single-input-single-output(SISO)cases,ignoring the dangerous coupling and nonlinearity that arises between longitudinal and lateral dynamics.In contrast to finstabilized aircraft,the coupling caused by the Magnus effect,gyroscopic effect and servomechanism delay plays a key role in spin-stabilized projectiles and may lead to unstable coning motion.20Researchers of late have gradually begun to follow this coupling with interest.Yan et al.21established suitable design criteria for spinning missiles with rate loop feedback and employed the dynamic inverse to deal with control system coupling problems.Mattei and Monaco6considered coupling when dealing with the bank-to-turn maneuvers,and in Ref.22,Tol et al.identified coupling using multivariate splines.In these works,the aircraft models were described by multiple-input-multiple-output(MIMO)nonlinear systems that consider coupling.The correlative nonlinear controller was designed for MIMO nonlinear models.

In the above references,the significant states,angle of attack α and angle of sideslip β,are assumed to be measurable.But,it is a well-known fact that α and β are rarely direct measurements of sensor outputs from direct f i re munitions and vary from the angular velocity and acceleration that can be directly measured by an onboard gyroscope and accelerometer.Thus,a robust nonlinear observer should be designed to obtain α and β signals during the flight.Many nonlinear observers have been presented for the sake of state estimations.The work in Ref.23 considers the unknown input in a multivariable nonlinear system and designs a sliding mode observer for the retransformed subsystem.Refs.24,25 analyze the boundedness and asymptotic stability of the sliding mode observer for the SISO nonlinear system with a single disturbance.An extended observer is used in Refs.26,27 to estimate the unmeasurable states for lower triangular nonlinear systems with uncertainties.Ref.28 uses the Lyapunov method to get a new convergence condition for the super-twisting sliding mode observer.Among the existing literature,the adaptive sliding mode observer(ASMO)stands out for its generality and practicability in solving the problem of unmeasurable states for nonlinear uncertain systems.The estimations of these states laid the foundation for a nonlinear controller design.

Observer-based aircraft nonlinear controllers have drawn a lot of recent attention.For example,Mu et al.29proposed a continuous sliding mode controller with disturbance observer to suppress chattering for the tracking control of hypersonic vehicles;Casavola and Mosca30used the observer-based command governor approach to design a bank-to-turn missile autopilot;Chunodkar and Akella31designed a switching angular velocity observer for the stabilization of the rigid-body attitude and tracking control;Zarovy and Costello32introduced an extended state observer for a helicopter mass and centerof-gravity estimation algorithm.However,contrary to the abundance of literatures6,29–35on missile or UAV control systems,the available solutions for spin-stabilized projectiles are,to the authors’knowledge,very few.This paper tries to bridge the gap by proposing an observer-based backstepping outputfeedback nonlinear control law for the dynamics of MIMO spin-stabilized projectiles.

An obvious drawback for using the backstepping technique is the problem of ‘explosion of complexity.” In fact,Refs.36–38 point out that the explosion of complexity is caused by repeated differentiation of certain nonlinear functions,i.e.,the virtual controls in each step.The repeated differentiation compels the complexity of the controller to increase drastically with the order of the nonlinear system.Guided projectiles require the guidance command to be repeatedly differentiated;To solve the above issue with the backstepping technique,a socalled dynamic surface control(DSC)technique has been developed by using a one-order filter at each step of backstepping approach.Thus,the modified controller is much simpler than the existing controller.Ref.39 uses the DSC technique to design an adaptive controller for a class of SISO nonlinear systems,and Ref.40 does the same for MIMO nonlinear systems.In this paper,we combine the nonlinear controller with the DSC technique to increase practicability.

Motivated by the above-mentioned observations,an observer-based adaptive sliding mode backstepping outputfeedback DSC controller is proposed for spin-stabilized projectiles.Compared to existing literature,the main advantages of using this controller type for spin-stabilized projectiles are as follows:(1)A MIMO uncertain nonlinear model of the spin-stabilized projectiles can be constructed progressively without the assumptions from Refs.1,4 that the model is LPV and its parameters are exactly known;(2)By designing a state observer,the unmeasurable states α,β can be estimated directly without additional sensors.Also,the proposed adaptive sliding mode term removes the limitations from disturbances and restrains chattering;(3)An adaptive sliding mode backstepping output-feedback controller can track external guidance commands.Furthermore,the DSC technique is employed to eliminate the higher derivatives of the guidance commands.

The remainder of this paper is structured as follows:Section 2 describes the concept of spin-stabilized projectiles and constructs the uncertain nonlinear model.Section 3 addresses the control issues concerning unmeasurable states and mismatched uncertainties.The main results for adaptive sliding mode backstepping output-feedback DSC design and stability analysis are presented in Section 4.Extensive simulation results demonstrate that the proposed controller is effective in Section 5.In Section 6,conclusions are presented.

2.Guided projectile preliminary

This section presents an uncertain nonlinear model of projectiles with the longitudinal and lateral dynamics desired for control module design.A closed-loop feedback framework of the control module is discussed at the end.

2.1.Nonlinear model with uncertainty

The guided direct f i re munitions studied in this paper belong to a class of spin-stabilized canard-controlled projectiles whose configuration is illustrated in Fig.1.The whole projectile,using the so-called dual-spin structure,is divided into two parts on the basis of function:the aft part rotates at a high spin rate,which provides the gyroscopic effect to ensure the dynamic stability of the whole projectile1;the fore part connects to the aft by bearings and is completely roll decoupled from it.One of the mature control patterns adopts a phase servo equipped in the aft part to control the spin rate and roll angle of the fore to zero,which is the desired state illustrated in Fig.1.At this point,the control surfaces are similar to those in a non-spinning projectile:canards#2–#4 provide effective longitudinal force and pitching moment,and canards#1–#3 provide effective lateral force and yawing moment.Thus,control coupling due to the servomechanism delay is avoided to some extent.

More complete 7-DOF motion equations of the above projectile have been derived,and their details can be found in Ref.41 This paper focuses on the longitudinal and lateral dynamics of the above projectile;hence the equivalent approximated 6-DOF motion equations1fully meet design requirements.The 6-DOF motion equations ignore the relative roll motion between the aft and fore parts and have been proven reasonable under an abundant number of simulations.

The longitudinal and lateral angular motion dynamics embody main characteristics that include inherent motion stability,and their complex response characteristics are also issues that the controller deals with.Eq.(1)represents the longitudinal and lateral dynamics of the above projectile in the body-fixed frame.

where the states α,β are the angle of attack and angle of sideslip,respectively,and the aerodynamic angles are defined by the projection of the vehicle into the body-fixed frame;the statesqandpare the Euler pitch and yaw angular rates,respectively;the moments of inertiaJx,JyandJz,velocityVm,massmand spin rate ωxare known or measurable parameters;δzand δyare canard deflection angles for#2–#4 and#1–#3,respectively.This dynamics model ignores the aerodynamic force directly produced by the canards.The aerodynamic forces and moments have the following expressions6:where the variables ρ,SandLare the air density,reference area and reference length,respectively.The values of the parameters describing the projectile are listed in Table 1.

Table 1 Projectile parameters.

The aerodynamic coefficients depending on the Mach numberMa,aerodynamic angles and spin rate ωxare written as follows:

The aerodynamic coefficients are sums of the static terms(coefficientscys,czs,mzs,mysdamping terms(coefficientsmydandmzd)and Magnus terms(coefficientscym,czmmzm,mym),which may be written as introduced by structural deformation and measurement errors.These unknown deviations and model idealizations are all grouped into the uncertainties or disturbances to simplify the design.Based on this,the longitudinal and lateral dynamics of the projectile can be reconstructed in a more reasonable form as

whered*（*= α，β，p，q）represents the above mentioned disturbances in the corresponding dynamic equation.

Eq.(4)can be rearranged to form the following augmented MIMO nonlinear uncertain dynamical system:

The longitudinal and lateral dynamics contain three main characteristics.Firstly,is the obvious coupling between the longitudinal and lateral dynamics.On one hand,the inertial coupling greatly depends on the high spin rate ωx[term-(Jx/Jy)ωxqfor the longitudinal and term(Jx/Jz)ωxpfor the lateral dynamics]presented in Eq.(1).On the other hand,the aerodynamic coupling caused by Magnus terms relate to the aerodynamic angles(β for the longitudinal and α for the lateral dynamics)and the spin rate ωxpresented in Eq.(3).Secondly,the aerodynamic coefficients appearing in Eq.(3)may possess a heavily nonlinear form or some lookup tables with respect to the tabulating variables.Finally,the normal model developed in Eq.(1)is only applicable to designing a nonlinear controller based on ideal conditions.There exist differences between the actual,true dynamics and the normal ones that are assumed by the designer.For example,aerodynamic coefficients are usually obtained by computational fluid dynamics,aerodynamic parameter identification or the wind tunnel tests,yet these methods will introduce unknown deviations.Other common inconsistencies are the random winds that appear in the actual combat environment and those

where the state vectorx=x2= ［q，p］T;the control input isu= ［δz，δy］T;the disturbance termsd=assumed to be unknown but bounded.The smooth vector fields and matrices of the Eq.(5)take the form of

2.2.Control module structure

The dynamic model provided in the above section is suitable for the design of robust nonlinear stabilizing and tracking control laws based on the signals of states.The sensors equipped on a guided projectile,including accelerometers and gyroscopes,usually measure the longitudinal and lateral accelerationsay,azand the Euler pitch and yaw angular ratesq,p,whereas the significant aerodynamic angles α, β are assumed to be unmeasurable.Thus,the problem of controlling spinstabilized projectiles can be interpreted to design an observer-based nonlinear control law to estimate the unmeasured signals and track the desired guidance commands.This closed-loop system has its feedback structure presented in Fig.2.In the closed-loop system,measurementsh1（x）=［ay，az］Tandh2（x）= ［q，p］Tare chosen to be the output signaly=h（x）=The control input and measurements are imported into the observer to obtain state estimations,and then the controller produces the control input to close the loop by using the estimated states and guidance commands.Detailed design processes are derived in the coming sections.

3.Adaptive sliding mode observer

The guided projectile should be directed by a nonlinear controller capable of coping with high nonlinearity,strong coupling and uncertainties.However,most mature nonlinear techniques require the aerodynamic angles α,β to be available.Under this condition,an ASMO isfirst proposed to estimate system states(Eq.(5)),and correlative proofs are provided.

Before establishing the ASMO observer for estimating the system statex1,we should review a Lipschitz observer for systems without uncertainties.

where x∈Rn,u∈Rm,y∈Rpare the state vector,control input and measurable output,respectively;A∈ Rn×nand C∈ Rp×nare constant matrices;ψ(x,u)is a smooth nonlinear function satisfying the Lipschitz condition with a Lipschitz constantγψ＞ 0.

Then we can construct an estimator

to ensure asymptotic convergence of the estimation residual if there exists a gain matrixLand a symmetric Lyapunov matrixP=PT＞0 satisfying

(1)A-LCis stable;

Remark 1.Lemma 1 is an existence lemma that is the conclusion of Thau’s42well-known results in the area of nonlinear observer design.However,choosing the observer gain matrixLand the Lyapunov matrixPso as to satisfy Lemma 1 is not straightforward.This paper uses a solvable method to obtain the solutions,and interested readers can refer to Ref.43 for details.

Remark 2.Lemma 1 provides a normal observer for nonlinear system Eq.(7)that ensures the estimation errorex=^x-xconverges to zero by importing the input and output of the original system.However,the normal observer greatly relies on exact knowledge of the original system,which is difficult to obtain.When considering modeling uncertainties and disturbances,the normal observer may fail to estimate the state of the original system.

Referring to Lemma 1,the nonlinear uncertain system Eq.(5)is desired to be transformed to a frame similar to Eq.(7)by means of local coordinate transformation.Outputyas well as itsfirst derivative are assumed to be known.Define a nonlinear diffeomorphism transformation Φ :x→z.

where the Jacobian matrix ∂Φ/∂xis assumed to be nonsingular.Now we can define the new coordinatezwith the mapping Φ as follows

Then the nonlinear uncertain system Eq.(5)under the new coordinate can be written as

Remark 3.In the construction of the unknown-input observer in the absence of measurement noise,it is often required that the dimensions of the disturbances do not exceed that of the measured output.44It is a necessary condition in the unknowninput observer design.

Then,we make the following assumptions:

Assumption 1.The known functionsf1（x）,f2（x）andg（x）in Eq.(6)are uniformly bounded with respect to their arguments,and the inputuis also bounded.Furthermore,the nonlinear system Eq.(5)is bounded-input-bounded-states(BIBS)stable.Hence,we can conclude that the nonlinear functionG(z,u)in Eq.(12)is globally Lipschitz with Lipschitz constant γGi.e.for arbitraryz1andz2，z1≠z2,

Assumption 2.The disturbancesd1andd2in system Eq.(5)are bounded,so the unknown input η in Eq.(12)is bounded for some unknown upper bound η,which is ‖η‖ ≤ η.

Theorem 1.Consider the nonlinear uncertain system Eq.(11)underAssumptions 1 and 2;One can construct a robust adaptive sliding mode observer by introducing the compensation terms as

where the output error feedback gain matrix L and the symmetric Lyapunov matrix P=PT＞0satisfying

whereγGis the Lipschitz constant of Lipschitz function G(z,u)defined inAssumption 1;andR（LTP+PL）is the Rayleigh quotient of the matrix LTP+PL.The sliding mode term v in system Eq.(14)is defined by

where^ηis the estimation ofηand the upper bound ofηis unknown.Define the error as~η=η-^ηand^ηupdates according to

Proof.Define the estimation errore=^z-z.It can be derived from Eqs.(11)and(14)satisfying

Differentiating the Lyapunov function with respect to time yields

Furthermore,by using the Assumption 1,Eqs.(15)and(16),the following inequality is obtained

Forthe Lyapunov matrixPsatisfying the condition R（LTP+PL）≥ 2γGλmax（P）,we obtain

Next,we use the definition of the unknown constant η in Assumption 2 and consider the adaptive law given by Eq.(17)to obtain

Therefore,one can conclude that the convergence of the estimation erroreto zero is achieved.□

Remark 4.In the sliding mode term(Eq.(16)),the discontinuous sign function may cause catastrophic chattering.To restrain the chattering and avoid the time derivative of the sign function in the sequel,the sliding mode termvshall be replaced by the nonlinear damping term.

where the updated law is replaced by

However,the cost of the replacement is that the estimation erroreis bounded instead of converging to zero.To see this,note that using the triangle inequalitya2+b2≥2ab,we have

hence,substituting Eqs.(24)–(26)into Eq.(20),one can obtain

From Eq.(29),we can conclude thatV(t)is bounded.Further,the estimation errorewill converge to an arbitrarily small neighborhood of the origin by adjusting the design parameters γ,ε and σ appropriately.

At last,one can obtain the observer for the original system

using the inverse transformationrepresented by

4.Observer-based adaptive sliding mode backstepping DSC controller synthesis

With the help of the observer proposed in the previous section,a nonlinear controller is capable of being synthesized to make the closed-loop system stable and meet performance specifications.From the familiar guidance commands provided by the guidance loop,an adaptive sliding mode output-feedback DSC control law is derived step-by-step through the proposed observer.

The states α,β in the longitudinal and lateral dynamics(Eq.(4))are available by virtue of an ASMO derived in the above section.Based on the ASMO,the longitudinal and lateral dynamics(Eq.(4))will be extended to system Eq.(32)by combining the original system Eq.(5)with the observer Eq.(30).

Thus,we are able to make the longitudinal and lateral accelerations of the projectile centroid

and track the guidance commands provided by the appropriate guidance law.

The following control law aims to make ξ（x1）of the system Eq.(32)keep up with the desired trajectory ξ1dby virtue of the measureable estimate^xfor original statex.The desired trajectory ξ1dsatisfies the following assumption:

Assumption 3.16,18The desired signal ξ1dis a sufficiently smooth function of time,and ξ1d，˙ξ1dand¨ξ1dare bounded,i.e.,there exists a positive constant ξ satisfying

in which ε1＞ 0 is the design gain andis the estimation of bound θ.Define the error as= θ -,and the adaptive law ofis updated by

In the traditional backstepping design,the intermediate control function μ will be repeatedly differentiated,which leads to the so-called explosion of complexity in the sequel.

To avoid this phenomenon,Eq.(39)introduces a new state variable ξ2dobtained through afirst-order filter on intermediate control function μ with a constantc;the dynamics of ξ2dare

Define the output error of the first-order filter Χ = ξ2d- μ,which yields˙ξ2d=-Χ/cand

whereB（s1，s2，Χ）is a continuous function with the following expression:

Choose as the Lyapunov function

The time derivative ofV1along the solutions Eqs.(39)–(41)yields

Step 2.In this step,the actual controluappears.Differentiating the second sliding mode surfaces2with respect to time yields

The time derivative ofV2along the solutions Eqs.(45),(46)and(43)is By using the triangle inequalitya2+b2≥2abagain,one has

From Eq.(52)and invoking the argument in Remark 4,it can be proved that all the signals in the closed-loop system are ultimately bounded.At the same time,the tracking errors1= υ1- ξ1dcan be made to be arbitrarily small by adjusting the design parameters appropriately.

Remark 5.At the beginning of this section,the familiar longitudinal and lateral accelerations of the projectile centroid were chosen as the desired trajectory.As is known to all,it is difficult for the guidance loop to provide the second derivatives of accelerations in practical application.The DSC technique introduces a new state variable to cope with this problem.The part2din actual controluis achieved by afirstorder filter presented in Eq.(42)instead of the derivation of the˙ξ1din the traditional backstepping design.

The aforementioned analyses and design procedures are summarized in the following theorem.

Theorem 2.For the nonlinear uncertain system described by Eq.(5)satisfyingAssumptions 1–3,if the adaptive sliding mode observer is established by Eq.(30)with the feedback gain matrices designed by Eq.(15)and sliding mode terms Eq.(24),and the adaptive sliding mode output-feedback DSC control law is provided by Eq.(47),then the whole control scheme ensures that all signals in the closed-loop system are uniformly ultimately bounded.Further,the tracking error betweenξandξ1dcan be made to be arbitrarily small by choosing the proper design parameters.

5.Simulation results

This section provides the full simulation results of the ASMO and the adaptive sliding mode backstepping output-feedback DSC controller.The simulation results prove the feasibility and effectiveness of the observer-based nonlinear control law.

5.1.Simulation results for ASMO

An ASMO for the longitudinal and lateral dynamics Eq.(4)is simulated,and comparisons of the results are illustrated to validate the feasibility of the proposed observer.The known parameters for the longitudinal and lateral dynamics are chosen asVm=272 m/s, ωx=1206 rad/s and ρ/2=16338-kg/m·s2.The unknown bounded disturbances are chosen asd1=[0.03sin α,-0.03cos β]T,d2=[8.5sin α,-8.5sin β]T.The canard deflection angles are set as δz=1°,δy=1°.

In the simulation,the aerodynamic angles α,β are allowed to vary from-10°to 10°;and，denote the estimation of the aerodynamic angles;the canard deflection angles δz, δyare required to not exceed 25°;the Euler angular rates usually do not reach 0.1 rad/s.Hence,one can obtain the Lipschitz constant γG=2,and the feedback gain matrixLand symmetric Lyapunov matrixPdetermined by the above Theorem 1 are as follows:

The design parameters for sliding mode terms are chosen as γ=10,ε=0.01,σ=0.1.If the initial values are chosen as α=0°，β=0°=1°，=-1°，the estimation results are illustrated in Figs.3 and 4 by comparing the normal observer with the proposed observer.The embedded part is the partially enlarged view of the shadow in the upper part of Figs.3–6.The normal observer,designed by Lemma 1,ignores disturbances.It takes the form of

where the observer gain matrixLhas the same parameters as the proposed observer.

In Case 1,shown in Fig.3,the disturbances are ignored.Through comparison with the simulation results,one can conclude that the normal observer and the ASMO both realize the observation for the aerodynamic angles α,β,and the estimation errors all converge to a small neighborhood in a finite time.The Adaptive sliding mode observer performs faster than the normal observer and improves the convergence time.Case 2,shown in Fig.4,considers disturbances.These uncertainties lead to huge changes in dynamics compared to Case 1.The normal observer can only ensure the estimation be bounded under such conditions.Despite this,the convergence of estimation error to a small neighborhood is still achieved by the sliding mode term when the disturbances are taken into account.

5.2.Simulations results for proposed controller

In this section,the simulation results demonstrate the feasibility and design performance of the proposed adaptive sliding mode backstepping output-feedback DSC controller for two distinct scenarios.In the first scenario,the disturbances are chosen so that the aerodynamic coefficients vary 10%,satisfying normal distribution,whereas the second considers 30%.The longitudinal and lateral accelerations are the pivotal signals in the guidance and control system for the guided projectiles.So the control law synthesis is evaluated by using a demanding command scenario for the accelerations.

By using the above adaptive sliding mode backstepping output-feedback DSC control law with the observed aerodynamic anglesand,the simulation parameters are the same as those in the observer simulation.Design parameters in controller and in adaptive law are chosen ask1=2，k2=5，γ1=50，σ1=0.5，ε1=0.02;fortheDSC technique,the parameter Π is set as Π =0.02,so the filter parameterc=0.04.The simulation results are illustrated in Figs.5–8.

As can be seen from Fig.5,the controller fulfills the performance requirement for tracking of the acceleration commands,and it rejects the coupling effects between the longitudinal axis and the lateral axis.The controller behaves well when dealing with low disturbances,ensuring a rapid convergence to the desired trajectories with small overshoots.However,increasing the level of the perturbation to 30%of the disturbances results in large overshoots from the reference signals.High tracking precision for the two scenarios rely on estimations of the aerodynamic angles α, β shown in Figs.6 and 7,confirming the good performance of the proposed observer.Moreover,a comparison of the canard deflection angles δz,δybetween the two scenarios is also exhibited in Fig.8.δz,δycomplete coordinated movements under the control law with slightly poorer performance when disturbances are at 30%,as shown by the dangerous peaking response at 8 s.

6.Conclusions

(1)A unique control design for a spin-stabilized canardcontrolled projectile is proposed.The control objectives can be fulfilled by a controller achieving high tracking precision for guidance commands.Strongly coupled longitudinal and lateral dynamics used for controller design are given together with a less conservative model that considers high nonlinearity and uncertainty.The nonlinear uncertain system used in this work is more precise and closer to real flight conditions than linearized models or LPV models.

(2)To design the controller for the nonlinear uncertain augmented model,a detailed ASMO is derived to obtain the unmeasurable states of the nonlinear dynamic model.The adaptive law removes the limitations from disturbances to a certain extent.At the same time,the nonlinear damping term used in the algorithm restrains chattering and avoids the time derivative of sign functions.The observer is achieved simply without additional sensors and lays the foundation to obtain a nonlinear controller.

(3)An adaptive sliding mode backstepping output-feedback DSC controller is designed to achieve the tracking and decoupling of the guidance commands.To deal with the problem of ‘explosion of complexity” caused by the backstepping method,the DSC technique isfirst employed to eliminate the higher derivatives of the guidance commands,which may improve practicability in actual application.The closed-loop system is shown to be practically robust and stable,and uniformly ultimately bounded;the control objectives are achieved within a prescribed level of performance.

(4)Future work will involve the study of the controller faced with unknown aerodynamics coefficients and noisy measurements.Actuator dynamics will also be explicitly considered in controller design.

Acknowledgement

This research was supported by the National Natural Science Foundation of China(No.11532002).

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24 May 2016;revised 9 September 2016;accepted 31 October 2016